Hey, good morning everybody, Lee Lowell here, smartoptionseller.com today, Saturday, February 6th, 2021. How's everyone doing today? Listen, today we're going to be talking about how to pick stocks to trade. Where to find stocks? How to find stocks? Which ones should you be trading?  But still you have to find the stocks to trade the options on and that's absolutely correct. So today, once again, I'm going to give you and show you how you can go about finding stocks to trade on and what stocks to pick and resources. I'm going to give you resources to help you find your stocks to trade. So let's jump right in, okay? This is the charts that we use and for me, it's all about looking at the charts. When I'm trying to find a stock to trade, it's all about how the charts look. And by doing so, it can give you an idea of where the stock is most likely going to move next.  So for me, looking at the charts is where it's at because I have some indicators, support and resistance levels. Some people will say, what about fundamental analysis? Don't you want to see the sales and the earnings and the price, the PE ratios and all that other stuff. That's called fundamental analysis. To me, yeah, that's fine. Some people like to do that. If you really want to dig deep into the company, sure, you can do that. And I do a little bit of that as well, but to me, all of that fundamental data, all that underlying data of the company is reflected in the charts. I mean, if a company is doing poorly and they're not growing their sales and they don't have any earnings, the stock chart's going to be going down. But if they're doing well, they got earnings, people love their products, the stock chart's going to go up. So all I have to do is look at a stock chart and I like to see where the support and resistance lies and look at some moving averages. And that tells me what the stock is most likely to do next. So that's how I pick my stocks. Now, of course, I've been doing this for 30 years. I pretty much have a system down in my charting platform here. You can see just a smattering of stocks on the side here. And to me, it's all about day after day. I just continually look at these charts. And most of them are the same companies I've been following for a while. So over time, it's not something that you're going to learn overnight. It takes time. It's a process, just like learning a new skill. You have to learn how to read charts if that's how you want to go forth with trying to pick stocks to trade. So when I pick a stock to trade, number one, I look at the chart. Let's look at something like PayPal, for example. This is a chart of PayPal. Now, obviously, if you're going to get long on a stock, you always want to see the stock moving upwards. And that means from the bottom left of a chart to the top right of a chart. Now, I default to looking at a daily bar chart. Now, some people might say, what's a daily chart? Well, daily chart is you see the price action for one day's worth of trading. So each bar that you see on the screen here is one day's worth of trading. Now, some people who are hyperactive will like to look at a one minute chart. Now, this is a one minute chart where each bar represents one minute of trading and they will try to find price patterns based on the one minute chart. So these are hyperactive in and out many times during the day. These are the types of traders that just, looking to scalp a few ticks here and there and make their money and move on. To me, you know, a long time ago, I tried to do that. It just didn't work for me. It's just, it's too time intensive. It's emotionally intensive and it just wasn't working for me. So you have to find psychologically what works for you. Daily bars for me is what works. And when I want to time my entry, when I actually want to get into the trade then I'll narrow it down to I'll look at the one minute chart to see if I can get a better entry. But if you're investing for the long term or, you know, a longer term, you know, six months a year, two years, whatever, daily chart is pretty much all you need to look at. And you want to see the movement of the chart over time. Now this is going back a couple of years. You can see PayPal, you know, kind of went sideways for a while and then it just, after the coronavirus in March, it just went ballistic. So you want to find stocks, quality stocks in their sectors and you want to follow them for a time. You want to see how they move up over time. But once again, it all comes back to, well, how do I find these stocks? How do I know what stocks to look at? No, one of the things you can do, obviously, is if you're interested in a certain sector, whether it's an airline sector, electric vehicle sector, the computer chip sector, you want to find the stocks that are at the top of the game. Now we like to play advanced micro devices in our newsletter, smart option seller because advanced micro devices is a great chip company, computer chip company. For a while it was always Intel, but AMD has really come on strong in the last couple of years. Now we could pull up a chart of AMD right here and we can see over time AMD bottom left to top right. It's just been moving up. It's a great, it's a great company. Now we can dial even deeper into the monthly charts. So this shows the history of AMD over many, many years going back to the early 1980s. And you can see for a while, it just flatlined. It was a sub $10 stock until late 2016 and it started to go up. Look at this beautiful chart for AMD. Now you contrast that to Intel. Here's the monthly chart of Intel. Now Intel since same early 80s had the nice move up before the dot com bust in year 2000, but Intel was always a great, was really the only chip company for computers until AMD really started to get its act together. So Intel really hasn't done all that much over the last three, four years.  This is a daily chart of Intel up, down, up, down, sideways for a while. So Intel to me is not the leader anymore. I mean, it's a great company. It's a great chip maker, but we like AMD as far as the, how the chart looks. So it's all about looking at the charts.  You want to see that long-term trajectory, but once again, how do you find the stocks? How do I go about finding stocks? So let's start by looking at some resources that can certainly help you find stocks to trade. Now I've showcased this website before. This is a website that can help you get started on building a watch list, building stocks that you may want to invest in. It's called the dripinvesting.org website. And these are all about the high-quality dividend companies. Now dividends are part of a company's earnings that they pay out to the shareholders. And companies that continually raise their dividends over time, they pay out more each year are what's called dividend raisers or dividend risers. And if a company is paying out more dividends each year, that means the company is growing, the officers in charge know that the company's doing well so they can give back more of their earnings to the shareholders. And when you open up, when you go to this dividendinvesting.org website, you want to click on the, let me move myself here, you want to click on the Dividend Champions Excel spreadsheet or you can do it in PDF format. I like the Excel spreadsheet because you can manipulate some of the columns and some of the rows. So you click on that and it'll open up to, and let me pull up the Excel spreadsheet here, it'll open up to an Excel spreadsheet that looks like this. Now along the bottom, you have the title, it might open up into, here's the title page, dividend champions list. And along the bottom are more tabs. You want to concentrate on the Champions tab. The Champions tab is based, the list contains companies that have raised their dividends every year for at least 25 years in a row. Okay, so these are solid companies that you can start creating your watch list. And in the list itself, you can scroll down and it has roughly 148 companies that have raised their dividends for 25 years in a row. Now there's another thing that you can do, they're called the dividend kings. The dividend kings are companies that have raised their dividends for at least 50 years in a row. What you can do is you look in column E here, it says the number of years and that tells you how many years they've raised their dividends. And you can filter out by looking at companies that have only raised dividends for at least 50 some odd years or more. And there's not many, there's maybe 15, I keep clicking on my picture here, 15 out of this 148 or so, you can pick out the ones that have raised their dividends for more than 50 years. And you know those are solid companies. So make yourself a list of the dividend kings that have raised their dividends for 50 years. And then you start watching those stocks. You put them in a watch list, you look at the charts. And if it's a company, you could dive deeper into their fundamentals. If it's a company that you think is worthy of maybe a long-term investment, then you could start making some trades. You can buy the stock, you can buy call options, you can sell put options. It's all about trying to find companies that may jive with your outlook. And the dripinvesting.org, the ones that create this Excel spreadsheet, you can see the latest update was January 29th, 2021, right here. And they update it continuously. I don't think it's every day, maybe every week or every two weeks. So the list, these numbers won't change all that much, but if you wanna get an updated list, just check back to see the date that they use. So here's your number one resource, the dripinvesting.org website and look at this spreadsheet here. Let's take a look at another website that I like to use and I've talked about it before is stockfetcher.com. Stockfetcher.com is a fabulous website to kind of create a watch list. And stockfetcher.com is all based on technical analysis. Things like moving averages and bullish crossovers or bearish crossovers or RSI indicators. Now what stockfetcher can do is you can type in, and they have very easy language, you can type in phrases that'll say, find me stocks that are priced $50 and over that have where the 50 day moving average crossed above the 200 day moving average and has done so for the last 30 days. And the volume of the stock is at least a million shares per day. And you can write all that out in the language that they use and it'll show you stocks making that price pattern. And down here at the bottom, it shows you all the technical indicators that you can use and write in a phrase to help you find stocks. Now, just to show you an example and they have a great support system. If you have trouble with trying to come up with a language to find a certain pattern, email them and they email you back pretty quickly within a day or so. And they tell you, here's some suggestions and they have this community, they also have a community, which this general discussion, they have forums here. So you can always go into the forums and see what other people think and you can ask questions. So it's really a great website. And you can use it for free, but if you want to use their paid services, which really gives you more stocks from the filter. If you just use the free version and you type in a filter, they'll give you the top five stocks that meet that requirement. If you want more than that, you have to pay for it. And it's super cheap anyway. I think it's like $9 a month. So it's a great website to start looking for stocks. Now, here they just, they give you some quick examples. You can click on the quick examples here to click the down arrow and it'll give you popular examples. You got to click on it and it'll say a MACD bearish crossover. Let's just click on this MACD bullish crossover. And right here, it'll give you the top five stocks and let's see what happens when we click on view for results. What happens is it'll show you the actual language that they use to find these stocks. It says show stocks were MACD with a fast line and it shows you the parameters of the MACD line crossed above the MACD slow line. And it says and draw the 50 day moving average and draw the 200 day moving average. Now you scroll down here and it'll give you the top five picks out of 752 results. It's showing you the top five picks or five of those picks. I'm not sure if these are the top ones. And you can click on the chart and it'll open up into a bigger chart here and it'll show you, okay. It'll show you exactly what the phrase was typed in where you got the MACD crossover I think is down here and it's got the volume and here's the actual stock chart. So it's a great way to come up with if you have a, if someone told you or you have figured out, okay, I've been looking at charts and every time there's this, you know, MACD crossover the stocks tend to go up. So if you're looking to get, find those specific stocks go into stockfretcher.com you type out the, what you're looking for and it's called a filter. It's a filter and you type in your filters and then you click fetch stocks right here's the button and it'll show you the five stocks. You know, that's a great start. And if you want more than that then you pay for the service. If you wanna, if you wanna take these results down further you can type in and I'll show you right here you type in something like and close meaning the closing stock price is above 50. And that's telling me it's only gonna look for stocks that are over $50 per share and we click fetch stocks and let's see what happens. So here's the top five stocks and see it narrowed it down to 149 results from the 750 results. So you're getting smaller. And if you wanna narrow it down even more you can say, you know, and volume is over a million shares, you know, based on the last 30 days. It's super easy. So it's a great way to find stocks that are, you know, meeting your requirements of what you wanna see a stock do it. So there's your second resource right there. You got the drip investing, you got stockfretcher. Let's see what else we have. We have finviz.com Finviz, a lot of people talk about finviz.com. It's a great place to look at all kinds of news on stocks. You can see insider information, but we're talking about the screener.  And you can look at the screens based on descriptive. Here's you got three tabs right up here. Okay, this'll, this gives you some fundamental information. On this tab right here, fundamental information. And you can click the dropdowns on these and filters as many as you want. Now I look at the technicals, so I would click on this tech. Any of these could change, any of these could change, any of these for the moving hours, for the moving averages, RSI's, all this stuff. This got one for patterns that, what the patterns that say, you know, it's got a wedge. Let's see what we got for wedge up strong and see what that looks like. And I'm up with, it's got a bunch here. And it gives you a lot too. I don't think there's any restrictions on how many it gives you. You got to deal with the advertisements that come up here. So finviz.com is another great place to get started looking for stocks that may meet your requirement. What else we got? Another one I'd like to show you is it's all about looking at patterns. When I look at charts, I look for patterns. I look at rising wedges. I look for congestion patterns, channels. If you go to chartpatterns.com, it's a very super simple site. This one's been here for a long time. I've written about this even in my book that came out 14 years ago. Chartpatterns.com is a great way to recognize patterns on stock charts that can help you decide which way the stock's getting ready to move. Now you can click on, let's say you click on this one right here and it'll give you more of a definition of what it is. It's called an ascending triangle and this is where a stock is moving upwards. It's hitting some resistance and once there's strong enough pressure, it's gonna move above that resistance line here and then keep going. So it's a great way to look at some simple chart patterns. Here's the channel patterns and once it finally breaks out to the upside or to the downside. So you've got little descriptions here. It's a great resource to help you pinpoint if you're starting to see patterns on the charts, which way the stock is likely to break next. And lastly, let's go to stockcharts.com, which is another great free website that you can just look at any kind of chart you want. Let's just say we look at AMD, type in AMD and it'll bring up a chart of AMD. Now this is all free. Let me move myself over here a little bit. This is all free and below, it allows you to change some of the parameters and all that information and all that information and all that information and all that. Or you can take, if you don't want to see any of that, all, all, all, and it'll take away some of those, some of that information so you can kind of see a cleaner chart. But what I wanted to show you was on this website, you click on chart school up here, chart school. And when it comes up, when it comes up with a chart school, it's all about these lessons here. It tells you, you click on any one of these and it'll give you information on how to do chart reading. Let's click on this one that says chart analysis and it'll talk about what are charts, support, resistance, trend lines. You click on one of these and it'll bring you to another page that gives you all this great free information on how to look at charts. So good free information, good free information, good free information, good free information and another resource for you, another resource for you, another resource for you. Let me show you two other books that I've shown in the past. We've got this book, Charting Made Easy by John Murphy. It's a super simple paperback, very small so you could read through it very quickly. And another one that I've had on my shelf for years is this one by seven chart patterns that consistently make money by ed-downs. So two more resources, two more resources for you to get yourself a nice watch list looking and start looking at charts. Okay, so once again, there you go. Once you have your stocks picked out, then you start looking at the charts and you start looking for patterns. And that's it, if I was teaching people when I have people come to my coaching sessions, they ask me these same questions, Lee, how do I get started? How do I even find these stocks? And I tell them exactly what I've just told you. So you can use these free resources to get yourself started. And then once you've got your stocks, you put them in a watch list like I do and you just watch the stocks which now brings me to my next part of the video which is what we call our Saturday Synopsis. We like to take a look at the stock charts, see what the market has done, where it's been and where do we think it's going? We look at the indexes, we look at the some individual stocks that give you some opinions on what I'm seeing and what I'm thinking. So let's jump right into the Saturday Synopsis. And as the market closed yesterday, February 5th, 2021, we can see how the whole week played out. So what do we do? We always look at the SPY exchange-traded fund for the S&P 500. This is the broadest measure of the market as a whole. This is what I like to look at. And if I'm investing, when I do invest long-term, I wanna get some stock index exposure to the whole market. 500 stocks, it's very diversified and it's very passive investing too. So super easy, you know, if you don't have the time or the energy to research individual stocks, S&P 500 or the QQQ, which is the NASDAQ, ETF, it's super easy to just follow along. And as we know, the market goes up over time and that's called passive investing and you can do it in your IRA and 401K and just your regular brokerage account. We know the market goes up over time because companies are making products and people are buying products and their earnings are going up and the S&P 500 over time will kick out the losers and put in better companies. So you know, this index should go up over time and it has, yes, we're gonna have the pullbacks here and there. This is a monthly chart going back to 1993 or so. But if you're in it for the long haul, couple years at minimum, the market will go up. So you wanna stick to the long haul. But anyway, we're looking at the SPY here and we're seeing where the market had, what we did last week and what had happened this week. So I like to keep some of my patterns on the chart so you can see how it's developed over time. We've looked at this W pattern, which we know is a very bullish pattern, especially if it comes from the downside. But even if it's on the upside here, the stock should move above the W pattern once it gets above resistance by a certain amount of time, usually, not guaranteed, nothing's guaranteed, of course. But here we looked at the W pattern, we had the resistance line, it moved above it, was congesting for a little bit and then it started to go up. Now I always have my 20 day, 50 day, 200 day moving average on the chart. And as you get a good momentum moving stock, they will typically bounce off the 20 day or 50 day when there is a pullback. Now we like to see uptrending stocks and we always wanna have pullbacks now and again, because that's healthy for the market. You don't wanna get all this froth and euphoria in the market because then that makes the stocks really drop hard when they do come off and that scares everybody out and we don't like that. We like slow contain movements with nice little pullbacks along the way. It keeps things calm and even and it makes for better investing emotionally and psychologically. So we can see here, let me open this up a little, we can see here the SPY or the SP500 has been moving nicely along the 20 day and then just last Friday, this is where let me show you how we close last Friday. So last Friday we closed right on the 50 day moving averages. This is what we saw last week, last Saturday when I was looking at the charts. So the market had a down week, people were getting a little nervous. Oh my God, what's going on? And had closed just below the 50 day moving average. And last Saturday I said, what we wanna see is we wanna see the market bounce off the 50 day moving average.  We don't wanna see it go all the way down to the 200 day moving average, but that would be the next line in the sand. What I said, we have to see the market bounce. And what happened? Well, right on cue on Monday, the whole week was an up week, all time new highs in the S&P 500. So the market did bounce right off the 50 day moving average here on Monday. We had a great, another up week. So the market is strong. The market is strong. I've been talking about this for a long time now. The market is strong. The coronavirus, although we still have a lot of coronavirus cases around the world, the vaccines are out. The vaccines are being administered. People are getting vaccinated. This is our step, our first step in getting rid of this thing or getting over the hump and getting things back to normal. It's a big deal coronavirus vaccine. So that is the first step for getting things back to normal. The other thing, why I'm bullish is because there's really no other place for you to get any kind of return on your money. CDs, savings accounts, money market funds, government bonds, T-bills, whatever, they're paying nothing. You cannot get a return on your money. The stock market is over the long term. The last 200 years has been the best place to put your money to get any kind of return. Why would you stay out of the stock market? Yeah, it has risks. All investing has risks. But if you know the market goes up over time, you have to have a longer term mindset. You have to, don't play or don't invest with money that you're gonna need tomorrow. If you have to pay the rent tomorrow, don't invest in the stock market because the market could go down tomorrow and you could lose your rent money. We're talking longer term, years of investing over the long term. The market goes up over time. So I'm bullish. It's the best place to get a return on your money. The governments around the world are pulling out all the stops to help economies. Interest rates are cheap. If you have to borrow money, interest rates are super cheap. Companies are doing well. And the stock market's not just going up just because these are prices on a chart and these things don't mean anything. They do mean things. These are underlying companies that are creating products that people are buying and spending money on. That's how the stock market, that's how and why the stock market goes up over time because companies are doing well and people want a piece of the company and they pay for that piece of the company by buying the shares of stock. And if there's more buying pressure, obviously stock prices are going to go up. So the S&P 500, all-time new highs. I'm bullish. If we get pullbacks, look for the pullbacks to the 20-day or 50-day moving average. When you have good momentum stocks, that's where they will usually find support at the 50-day or 20-day. So if you're looking to get in, you can use the pullback to one of these levels to time your entry. And you don't have to go all in. You can put it in increments. If you're worried about it, that it may drop more, put in a little here and wait to see what happens. And if it starts to go up, then you have more confidence. You can put more money in the market. So let's take a look at the NASDAQ. Also, all-time new highs this week. Once again, it closed below the 20-day moving average last Friday and it bounced all this week. So the market is strong. It takes these little pullbacks as a sign to get in. All-time new highs, looking good. You can see on my chart, on my watch list here, these green highlights can't really see the numbers, but that indicates new highs. So we got the NASDAQ, the SMB500. Let's look at the Dow. The Dow very close to hitting all-time new highs. I think we hit an intraday new high on Friday closed just below the highs from two weeks ago. So the Dow on the cusp of making all-time new highs. The market is strong. And it came down to below the 50-day moving average, but then bounced back up. So the market is strong. We look at the VIX. We look at the VIX. I said last week, you get these spikes. They don't last very long. Option prices spike when the volatility spikes. If you're an option seller, you love these spikes because you can sell those options and get a lot more money. You can see spike, spike, spike, spike. And they only last a couple of days at a time because you know why? Because this moves inversely to the general stock market. And when the stock market finds a bottom and starts to rally, the volatility index comes down again. So here we're right at this. Let's draw a little support line here. You got this nice support line right now at near 20%, 20% or so for the VIX and nice support line. For a long time, we was trading around 10% to 15% down here. So we got the support line around 20. So we'll see where it goes from there. It moves inversely to the general stock market. Let's take a look at some individual charts here. We start with GameStop, okay? That obviously was the stock du jour for the last two weeks. A lot of people made a lot of money and then they gave a lot of that money back. GameStop closed at $63 or so yesterday, January, February 5th. It rallied all the way up to almost $500 a share. And then in just a few days came all the way back down. People have been talking about this. It was a huge, huge move. If you read some of the forums or the chat rooms, people up millions and millions of dollars didn't take their profits, left it on the table. And when the market corrected, they gave all that paper money back, paper profits back, sad. But who was making a lot of the money were the insiders and the company executives and family owners of the stocks. Because when's the next time they're gonna be able to sell their shares that they've held forever for $450?  So if you're in the game and you see this move up for a $20 stock moves up to $450, why would you not sell? Why would you not lock in your profits? I mean, are you waiting for it to go to 1,000? Some people are waiting for it to go to 1,000. It may have, it could have, but if you've got millions of dollars of profits locked in, you sell. So that's GameStop right there. And then some of the other stocks, some of those big movers, AMC, same thing, up, down, Blackberry, same thing, up, down. So these high flyers that everyone was talking about the last two weeks, what goes up must come down. So let's take a look at some more normal stocks. If I wanna even call it normal stocks, we'd like to take a look at Apple. Apple has been chugging along, moving along the 20-day moving average here. So it's got support here. Last Friday it came down below, but obviously this week moved up above it again. So it's still uptrending, not a lot of movement, just a couple dollars in today. So Apple is moving slowly but surely higher. I just think it's just going to keep going up. It's a slow process, but it does look bullish to me. I'm long some Apple, just letting everybody know. So the stock is going up. You can buy shares, you can buy call options, sell put options, whatever. So you got Apple. Let's take a look at Tesla. Tesla, since its earnings, it hasn't really done much. We had looked at this congestion pattern and I said, it's gonna break higher or lower significantly after earnings, but it really didn't. It went up a little, down a little, and now it's still trading right where it was at the breakout. So Tesla's trying to figure out where it wants to go next. It's finding support on the 20-day moving average. At $850 a share, it's still pretty expensive. I mean, we can draw another pattern here, another triangle just to give some idea of what we're looking at. So it's got another congestion pattern. Once again, it's either gonna move up or move down or it could go sideways for a little longer, but Tesla's trying to figure out its next move. Amazon, we look at Amazon, we look at popular stocks. Now, Amazon had a little bit of a breakout. We had been looking at this congestion pattern within the confines of this larger channel here. Started to, well, it broke out, came back down to the moving average lines, broke out again and started to come back. So Amazon's still trying to figure out what it wants to do. Jeff Bezos is stepping down as CEO, as we heard this week. It doesn't really affect the company too much, but Amazon's still kind of chugging along, still a very strong stock as far as it not dropping. So it has support. It has support. Eventually it's gonna break out most likely higher. So Amazon's not really doing all that much. Let's take a look at Microsoft because that one, yep, actually broke out of our channel pattern that had been in since July or late June. Channel pattern, channel pattern broke out the last two weeks. Got a nice movement out of here. So most likely once a stock starts moving, breaks through resistance, it tends to keep going. So keep an eye on Microsoft. It'll probably just keep going up for the future. So that's cool if you're a long Microsoft. We'd like to look at AMD, one of our favorites here because I just like the stock as a company as a whole. And if you like a company and you're in for the long term, some of these short-term technical movements doesn't bother you so much as long as you believe in the company. Now we've been looking at AMD, it's in this channel pattern, had the W, started to break out, but it keeps falling back. It's down below both 20-day and 50-day moving averages. So AMD is trying to figure out what it wants to do. I'm bullish long-term on AMD. So I'm holding out that it's just, it's gonna find its footing at some point and continue to move higher in the long run. I wanna show you a chart pattern of Cisco. Cisco is another company that looked real good for me on the charts and I'll tell you what that meant. It found a bottom here, started to move up and then it was just hugging along the 20-day moving average, still sloping upwards the moving average, both 20-day, 50-day, 200-day. They're all sloping upwards. So it was just containing all this energy ready to go and just hugging along this 20-day moving average line. And I knew it was just a matter of time before it would go. And I had bought some around $45 or so while it was hugging this line and then yesterday, Thursday and Friday this week, it just, it couldn't contain it anymore. The support was too strong and it just powered higher. Now, it looks like it closed at 48.08 at the end of regular trading, but in the aftermarket that trades until 8 p.m. Eastern time, it had a nice run almost up to $50 a share. Here's where it closed at four o'clock p.m. Eastern time right around the 48, a little over $48. And then for some reason, for the next couple hours, it just rallied up to almost $50 a share, which we go back to the daily chart. So here's 48. So in the after hours, it almost went up to 50. Not sure what happened. We'll see how it opens up on Monday morning. So we like Cisco, I like Cisco and I hope it continues to go higher. I know earnings are coming out soon. This same thing happened, let me show you what happened last time. Cisco was looking good here too. 20-day hugging along and then earnings, boom, knocked it down. Look how far it knocked it down. All the way down to $36, but it's rallied all the way back up. Earnings coming out soon. We'll see if it continues higher or if it has another bad earnings. So that's Cisco. What else do we like to take a look at? Google's been doing well. Google came out with his earnings last week. Just gapped higher, powered higher, all-time new highs for Google. So it's forming this little island top here. We'll see a lot of stocks. They don't like the open spaces. So we may see if it comes down to close the gap as it's called, but it could continue to move higher. Google looking pretty strong, but just beware what's called this island top here or it's island. And some people will say they're looking for an island reversal, meaning they want it, see it come back down to close this empty space and then move back up again. What else? What else we got? Netflix doing pretty good. I like Netflix. Netflix is hugging along this 20-day moving average, popped up on its earnings, came back down. So this could be the support area for a timing entry. If you're looking to get long, Netflix is all about trying to find where to get in, if you want to get in, how do you time it? Well, you want to wait for the pullbacks to the 20-day or 50-day to feel confident that as long as the stock's moving upwards, the uptrending moving average lines, you want to make sure they're uptrending, wait for a pullback, and then that's where you can time your entry. You don't have to go all in, like I said, you can buy a little bit at a time to see where the stock goes. Some of the healthcare stocks, now this is another thing I want to talk about is the indexes themselves can go up. It makes it for easy investing, but individual stocks could go down. We have the healthcare stocks this week. We have Bristol Myers going down, even with the coronavirus vaccine, Pfizer going down, Merck going down. So you have certain individual stocks going down, Verizon, which is the biggest cell phone carrier here, at least in the United States, maybe around the world. Great company, I love Verizon, but it's been in this downtrend. It's just been in this downtrend. So as far as if I was looking to get long Verizon, I wouldn't get long Verizon until I saw it starting to move back up again. Why try to fight the upward battle? The stock's going down. You're gonna try to pick bottoms, but the stock could keep dropping. Wait for the turnaround. Wait to see some kind of up movement before you decide to get in. Something like Kellogg as well. Kellogg just moving down, following along the downtrending moving average line. So these are the things that you wanna look for on the charts. I've talked about this many times in the past. You can watch some of my other videos about how to do technical analysis, how to build your watch list. Just scroll through some of my old videos. I have the titles in there so you can easily pick them out. But it's all about looking at the charts, looking how the stocks are moving on those charts. If you're bullish, you don't wanna buy into a stock that's downtrending.  You don't wanna do that. Use the momentum, use the momentum. Walmart, it's kind of not really, I love Walmart, but it's not really doing anything right now. So stick to the trends of the market. Look at the, here's the SPY, it's going up. If you wanna get long passive investing, you follow along with the SP500. All right, so there's your information on how to start looking for stocks to put in your watch list. Use those free resources I gave you and then start looking at them on the charts. Look for uptrending stocks, look for the uptrending moving averages, and use that to your advantage. So there you go, I hope you liked this video. I hope this was helpful content for you. Please give me a thumbs up, leave me a comment. Don't forget to subscribe in the bottom right hand corner of this video is the red subscribe button. Hit the subscribe button, that way you'll be aware of when I put out a new video. You can usually do these on Saturdays. Send me an email, give me a comment. And lastly, make sure you go to our website. Let me just once again, smart options seller website. Click on the put selling basics link. Get our free guide on how to sell put options. Put your name and email address here. You can go to our services tab, tutters, our newsletters. If you're interested in learning how to do this stuff and want to make trades and have someone do all the heavy lifting for you, that's what I do in these newsletters. I pick out the stocks. I give you all the information. I tell you what options to get into, when to get in, when to get out, what prices, all the work's done for you. So we have our two newsletters and our one-on-one coaching.  So we talk to them on how to get started. So there you go. This is Lee Lowell. I hope everyone has a great weekend and Super Bowl tomorrow. Super Bowl, looking forward to that. All right everyone, have a great weekend.


OK, so I will start. So we had started to talk about products. So I had told you that if we have products, so I first was talking about products of the fine varieties, close varieties of A n, so if x in A n and y in A m are close sub-varieties, then we have that their product, x times y, lies in A n plus m, and is a close sub-variety too. So it's easy to talk about products of close sub-varieties of A n. And in fact, I'm going to need this later, is that this also works if they are locally close. So this is an easy exercise. If x in A n and y in A m are locally close sub-varieties, then it follows that x times y is locally close sub-variety. I didn't prove it, but it's easy to see how you would modify the proof. And then one thing that we had was the universal property. And that said, basically, that if I have such product, p1 from x times y to x and p2 from x times y to y are morphisms. So x subset A n and y subset A m are close sub-varieties, but one can again see that one can easily generalize the same proof. We prove it for locally close sub-varieties. And these are morphisms. And secondly, we have that a map phi from any, so if z is any variety, then a map phi from z to x times y is a morphism, if and only if the compositions with the two projections are morphisms. This is one reformulation of what we stated the other, what we proved the other time. So that's as much as we had done for close or locally close sub-varieties of a fine space to study the products. And now we want to look at the case of quasi-projective varieties. And the problem, as I had said, is that if x is in pn and y in pm are sub-varieties, it is not by itself true that x times y will lie in some projective space. So as the most general things we are studying are quasi-projective varieties, we want the things actually to lie in some projective space and not in the product of projective spaces, although we could also do that. But anyway, that's not how we do things. We somehow have to make this thing lie in some projective space. And we do this, so thus we find an embedding. So this moment just means an injective map from pn times pm to some larger projective space. And we will identify x times y with sigma of x times y. So we study, so instead of actually taking the product, we look at the image here. It's the image under an injective map. And so then this thing has a structure of a quasi-projective variety when x and y are quasi-projective varieties. We will see that. OK, so first I want to define this embedding, which is called the, or this map here, which is called the Sega embedding. Yes? Why is this project this space y? So I mean, so I don't precisely understand the question, but I think you are asking whether, why it is not true that the product of projective space is a projective space? Well, it just isn't. If you look at the definition, it's a different thing. No, it's not the same. No. The product of two projective spaces is not a projective space, no. If you look at the definition, it's really different. You have pairs of n plus 1 tuple and then n plus 1 tuple, where on each of them it's up to multiplying by a constant. And that's really a different thing than taking some other tuple up to multiplying by one constant. It's really a different space. And if you, for instance, if one does algebraic topology, so if you do it over the complex numbers, you can look at the homology of projective space. And you will find that this has z in all even degrees until dimension n for pn. And for pn times pm, it looks quite different. So it has also some homology groups which are larger than z. So it's really not the same. So it's, anyway. So I mean just to connect to that. So you can even see, well, whatever. I will tell you what n is. For some n, which I will now specify. In fact, I think it's the first thing I write down. So now we come to the definition. And as a first statement, I will answer your question.  OK, so it's very precise. n. And so we want to take pn like this when we start with these. So I want to define this Seger embedding. So definition Seger embedding after some Italian mathematician, Benjamino Seger, who defined this. So we take x0 to xn, the homogeneous coordinates on pn. We take y0 to ym, the coordinates on pm. And we also want to somehow take the coordinates on this large pn. So these I call zij, which depends on two indices, where i goes from 0 to n and j goes from 0 to m. Are they coordinates on pn? You can see that the dimensions match. Here we have n plus 1 times n plus 1. And then the projective space has dimension 1 less. So you have a homogeneous coordinates. But somehow I want to not numerate them from 0 to n, but I want to enumerate them by pairs of such numbers, which somehow correspond to the coordinates of pn or pn. OK, now we define our map. So we define a map, sigma. Or if you want also sigma nm, but I will usually forget nm from pn times pm to pn. So this is just a set theoretic map that I'm defining. I'm not defining morphism, no, because we don't know. This is, from our point of view, not a projective variety.  So we just define a map. And how we do it? So if we have two points, a0 to an, a point in pn, and b0 to bm, a point in pm, we send this to all the products. So the ij's coordinate here will be the product of ai times bj. So this is a point here. OK, and sigma is called the Sagan value. So first we want to see that sigma is well defined. I mean, that's kind of trivial. So if you, I maybe just say it, if you multiply all the coordinates here by lambda, you get an equivalent point here. If you multiply all the coordinates here by mu, you get an equivalent thing. And if you do this, for both of them, you multiply all the coordinates by lambda times mu. And so you see that the class depends only on the class. So this is well defined. OK. And I write sigma and m, or sometimes just sigma, if I remember who n and m is, the image. So this would be some subset in pn. So now we want to somehow, so on pn, we have these charts, ui. So these open subsets ui, which cover it, where one of the coordinates is non-zero. We have them also here, and we have them also here. And we want to somehow say something about how they are connected. So and introduce notation. So for i equals 1, 0 to n, we put ui equal to, I mean, the corresponding ui on pn. So the set of all a0 to an in pn such that ai is different from 0. And for j from 0 to m, we put uj, the corresponding thing on pm. So this is nn, p0 to pm in pm, such that pj is non-zero. And you see that, obviously, this is a certain abuse of notation because sometimes i is both lower than n and m. And it defines two different things. But the idea is just that to make life easier for me, whenever I have an i, it means I'm in pn. And whenever I have a j, I'm in pm. So you just, so that one doesn't have to write so much. And obviously, we also have, for the same i and j's, we have uij, which is the set of all points in pn, plargen, whatever aij in plargen. I cannot have the same index, such that aij is not 0. These are the corresponding charts on the plargen, the corresponding open subsets. And we know that these open sets are isomorphic to the corresponding affine spaces. So we have a ui from, which way does it go around? Yeah, an to, or maybe I'll write it like this. We have isomorphisms. So we have an isomorphic via ui to ui. And the inverse map is fi. And the same way we have am isomorphic via uj to large uj. And the inverse map is phj. And finally, a largen is isomorphic via uij to uij. And the inverse map is phij. So this just is to introduce the notation. So what else do we need in order to see whether i? And so we see that, for instance, pn obviously is just the union over the ij of the uijs in the same way as before. These are these open subsets, so one coordinate is non-zero. This is an open cover. And so sigma, which is the image of this map in pn, is equal to the union over all ij sigma intersected uij. And these things, this is also an open cover of sigma. So it's a cover of sigma by intersection with sigma to open sets, such that I will also write this one as sigma ij. So we have a lot of notation. And finally, we have one more piece of notation then we can actually start. So I write sigma ij like this to be the map from an plus m to uij. So how does it go? So I can take a point of an plus m is a point of an. So a pair of points, p in an, q in am. So we take a pair of pq and we send it. So first we go to the corresponding open subset in the projective space. So this is ui of p and uj of q. And then we apply the map sigma. So here we are in, these are some points in pn times, this is a point in pn times pm, we can apply sigma. And this will be a point in uij. In fact, it will be a point in uij intersected sigma. And so in fact, by definition, we have that the image of this map is precisely sigma ij. We see that if you, so one direction is clear, we know that this maps it into this. But it's clear that for the ij coordinate here to be nonzero, we need that the i's coordinate here and the j's coordinate here is nonzero, which means precisely that it comes from here. OK, and now we are in a position to state the result about this. So now I have to wipe it out, but you have to know all, you have all the notations in your mind. So theorem, sigma is injective, and the image is a closed subset. A closed subset, yeah. And sigma is closed in pn. Later see, it's also closed sub-variety, but that's not immediate. And in fact, we can see how it's given. So sigma can be written as a 0 set of the following set of equations of degree, of polynomials of degree 2. So I take zij times zkl minus zilzkj. And this is for all possible ij's and so on. So i comma k run from 0 to n, and j and l run from 0 to m. Because first is always the coordinate, which, OK. So we have a certain, so by itself, this is n plus 1 times m plus 1 equations, but some of them are maybe trivial. But anyway, so this is what we have. So I claim that this set, the image, is the common 0 set of these equations, of these polynomials. Well, yeah, it would, let me see. I expect it's, you can write in matrix of which it is a determinant. But this is just, these are just four terms. So it's not, by itself, a determinant. So I expect you can write down some matrix so that it is all the 2 by 2 minus. But somehow that's not particularly useful, I think. So I mean, by itself, you just have this equation. The point is here is that you see here, you have ijkl. And then you exchange. So here i goes with change, that goes with l. So you exchange how they are matched. But I expect you can somehow most like find some matrix so that this will be the determinant of some minus. But anyway, OK. So I think I will, if I now state the whole theorem and then prove it, the obvious disadvantage is that I state the theorem, then I have to wipe it out and prove it. So maybe I will now prove the first part, then I state the second part and prove it and so on. I think that's easier to follow. OK, so let me do that. So we start with proof. So first, the injectivity is kind of trivial. Let's just, in order to get started slowly, we can still do it with sigma from a0 to an b0 to em. If that is equal to the corresponding thing for some other tuple, say, a0 prime to an prime b0 prime to em prime, well, then what? What does it mean? It means there exists some lambda in k, which is not 0, such that I can get the corresponding point here. I mean the sigma of it, which we know what it is, by multiplying by lambda. So such that lambda times ai bj, maybe with a prime, I do it like this, is equal to ai bj for all i and j. So there's one lambda which does it for all of them. After all, this here is the vector which has these products as entries, and this is the vector which has those as entries. OK, so exists one such. Now these are non-zero vectors, so we certainly find that one coordinate, there certainly is one coordinate which is not 0. So choose, say, i0 and j0 such that ai0 times bj0 is non-zero. So then we can just, you know, this holds for all i and j. So then we can do the following. So we have, so then we have for all i, we have that if we put, I mean we can write out, ai times bj0 is equal to lambda times ai prime bj0 prime. And so we can divide here, ai is equal to, what is it, lambda bj0 prime divided by bj0 times ai prime, times ai prime. So what is it, and this holds for all i. So that means, so this is some non-zero constant, OK. So we have some non-zero constant so that if I multiply the vectors of the i primes with this non-zero constant, I get the other vector. So that means that a0 to an is equal to a0 prime an prime, OK. And obviously the same proof with the role of i and j exchanged will show in the same way we have that b0 to bm is equal to b0 prime until bm prime. OK, so this shows the injectivity. Now we have this more interesting fact that the image is closed, actually given by some explicit polynomials. So if I, so let w be the zero set here on the right-hand side. So on the right-hand side of the equation, on the left-hand side of the blackboard. So clearly we have that sigma is contained in w. Why is that clearly? So if you're in the image of sigma, then the ij's coordinate is ai times bj, where the a is a vector in pn, and b is a vector in pm. So if I have here ai times bj times ak times bl, obviously it's equal to ai times bl times ak times bj. Multiplication happens to be commutative. So that is clear. But now we have to see the other direction. So now let aij, again index indices go from 0 to n and 0 to m, ij, be an element of the point in w. We have to show it in sigma. So we choose a coordinate where this thing is not zero. So we choose k and l such that akl is non-zero. Obviously this is a point in projective space, so it certainly has a non-zero coordinate. And then we just write it down. So we have our vector. This is a reasonable vector. We can certainly multiply it by a number, by a non-zero number. We take this one. This is the same vector. We just have multiplied that. But now we can use this equation. So we can move the indices from one side to the other. So this is, so I can take akj, but using this equation, times a, what is it, i, l. What's this in l? It's still i and j are running. So here something has really changed, but this is OK by the equation. But what is that? I'm saying here we just have the ij coordinate of this thing. Is the product something where we have j here? And the product of something where we have an i here? So that means that this is actually equal to sigma of the vector. So here we have the one that runs here, the i. So a 0l until a nl, ak0 until akf. This is how the map sigma was defined. And so we see it's in the image of sigma. OK, so this proves the first part. Now I should state the second. So I had this map sigma ij, which I'm sure you will remember. So sigma ij from a n plus m. So this is now the second statement of the theorem. From a n plus m to this ij, I say is an isomorphism. So I find that this map sigma ij, so this sigma ij, after all remember this was sigma intersected u ij, we find that this is actually isomorphic to a n plus m. As it should maybe be, this somehow is supposed to be what we get for p n times p m. And so locally, p n is equal to the open subset u i of a n is isomorphic to a n. The open subset uj of p m is isomorphic to a m. So we might hope that. And remember that the map was we take a map pq and it's mapped to sigma of ui of p to j of k. OK. So let's see. So it is correct here. I only canceled it. OK. So obviously we can assume, so we do the proof. So we can assume that i is equal to j is equal to 0. Obviously this makes no difference. So if I look at the map p0,0 composed with sigma 0,0, so what does it do? This will do what it takes an n tuple a1 to a n. And another one b1 to bm and sends it to what? So maybe I can first, maybe I can do it slowly. So if I apply this to this, what will we get? So first we apply to sigma 0,0, we map this into protective space. So this is p0,0 of sigma of the thing starting with 1 a1 to n 1 b1 to bm. And then the sigma will consist in taking all possible products of these elements. So this is equal to the vector c ij. ij goes from i from 0 to 1, j from 0 to m such that c0,0 is equal to 1. c i0 is equal to ai, c0,j is equal to bj. This is for i bigger equal to 1, j equal to 1. And c ij is equal to ai bj for i and j bigger equal to 1. That's just the definition. And then we apply p0,0 to it. So this is the same thing. So this will be c ij, ij where i,j is different from 0,0. So we just leave out this one. So maybe I write it like this. So if you write it in coordinates, it means, you know, so I have here just the coordinates are just either ai or bj or ai times bj. So it's always a polynomial in the coordinates on a n. So it follows that this is a morphism because it's given by regular functions on a n times a n. So phi,0,0 composed with sigma,0,0 is a morphism. And thus sigma,0,0 is a morphism because we just composed with the inverse of an isomorphism. And we know now that by definition, this map sigma,0,0 is surjective onto sigma,0,0, large sigma,0,0. Because that's the large sigma,0,0 was defined as the image. So sigma,0,0 and a n plus m is irreducible. So we know that if we have a surjective morphism from something irreducible, I mean a surjective map. So this is by itself is a morphism. If you have a surjective morphism, then the image is irreducible. Because this is a continuous map. And if this was a union of two closed subsets, then the inverse image would be the union of two closed subsets and so on. So it follows that sigma,0,0 is irreducible. So thus it is a quasi-protective right. So we can define the inverse map. So sigma,0,0 is an isomorphism because the inverse map. For instance, if I just define it in coordinates, I claim it's just z n,0 divided by z,0,0 until z 1,0 until z n,0 divided by z,0,0 and then z,0,1. Now if you start with a1 to an and b1 to bm, and we do this, so the first coordinate will then become 1. And we see that if we apply this to it, we get back precisely what we had here and also the other way around. So this is the inverse. And this is a morphism, no, because it's given by regular functions on the open subset u,0,0. So this was the second part. Now we come to the third part of the theorem, which I should find a way to answer here. So the third one is something very simple, but anyway. So for if I take a point q is a point in pm, then the map iq, actually I shouldn't call it, maybe I call it iq bar, from pn to p large n, which sends p to sigma of pq is a morphism. In fact, we'll see it's given by polynomials of degree 1, and the image will be a projective subspace of pn, so just a linear subspace. Anyway, so this is quite simple. And you just write down the definitions. So this is kind of obvious. So let q be given in so as equal as p0 to pm. And then what is iq? Well, it's just a point is just mapped to all the point in p large n, which is given by all the products of these coordinates with these coordinates. So in other words, iq as a map is just I take bj xi ij. So the ij coordinate is bj times xi. And so you see this is a map which is given by polynomials of degree 1, so it certainly is a morphism. And we can see that the image is just, in some sense, also projective space. So the image is the set of all ej ai, where a0 ij such that a0 to an is n. This is the projectivization of some linear subspace of an plus 1, of tan plus 1 with a large n. So it's a projective linear subspace. OK, so this was the third one. Now, we proved this just because we want to use it. I mean, this is not really a big deal. But we want to use it to prove that the product is irreducible. Because if you remember, we had this lemma for what we used for affine varieties, that if you have the topology on the product such that maps like this are continuous, then and both factors are irreducible, then the product is irreducible. And so therefore, this will follow. So let's see what the statement is for x in pn, y in pm, quasi-projective varieties. We have that sigma of x times y is a quasi-projective variety. And if x and y are both projective, then sigma of x times y is projective. I think that is all. OK, so this is a statement. So I maybe will, for simplicity, only prove the statement with a projective. So we assume x and y are projective. And then we prove that sigma of sigma of x times y is a projective variety. It's easy. You can easily adapt the proof for the quasi-projective case. If it's quasi-projective, it is something closed minus something other closed. And then you have to write a little bit more. But it's the same thing. So first, I want to show that x and y be projective algebraic sets. So we first want to show that sigma of x times y is a projective algebraic set. And it's similar in the quasi-projective case. So sigma of x times y, I can write, if I want, this union over all ij of sigma of x times y intersected uij. That's not very exciting. But I know that this thing lands in uij, if and only if we are in the image of the map phi ij. So I can also write it like this. Maybe you can see whether you believe me. I take sigma ij of phi i of x phi j of y times phi j of y. So this would mean I take x intersected ui, then I map it to an, and then I apply sigma ij. And the same like this. So this is the same thing. So now, in order to show that something is a closed subset, it's enough to prove that the intersection with any open subset of an open cover is closed in that open subset. So let's see. So phi i of x times phi j of y is a closed subset of a n times a n plus m. Because x is closed in, say, in a n, then x is closed in pn. If I intersect it with ui, it's closed in ui, and then phi is an isomorphism. And so this thing will be closed in phi of x is closed in a n, phi j of y is closed in a n, and the product of two closed subset like this is closed in a n plus m. And sigma ij is an isomorphism. So sigma ij from pn times pm to pn to sigma ij subset pn is an isomorphism. Thus, it follows that sigma ij of phi i of x times phi j of y is closed in sigma ij. And so we have an open cover of sigma by this open subset sigma ij. And the intersection of the image is closed in the open subset for each open subset in the cover. So it follows that sigma of x times y is closed in sigma. And sigma is closed in n, such that sigma of x times y is closed in pn. It's closed in closed. So this proves that. So it is closed if we have two projective algebraic sets, then the image of the product under the sigma embedding is also a projective algebraic set in p large n. And now we have to show it's irreducible. And we want to use a lemma that we had last time. This said that if we have a topology on the product, x times y, such that if x and y are both irreducible and we have a topology on the product, such that this map iq, p is sent to the pair pq and the map jp, q is sent to pq, is continuous for all pq, then the product will be irreducible. So we apply this here. So we have sigma from x times y to sigma of x times y is a bijection. So we could just say that we, so thus we only need that iq, which is the same as sigma composed with iq, iq is a map p maps to pq, that this map is continuous and also jp continuous for all pq. So we only need that. Then it will follow that as x and y are irreducible, it will follow that this is irreducible. Well, but we know that it's continuous because we have seen it's a morphism. So therefore, I mean, if we wanted to directly apply the previous lemma, you would have to argue on x times y. So then you would have to say we take this bijection and we use it to put topology here. That comes to make this a homeomorphism. And then it would follow directly here. But then it's the same statement here if we do it composed with sigma. OK, so this proves the, so this was the last part of the theorem. So we have, what have we managed to do? So we have this map, sigma, the Seguin betting, from pn times pm to pn, p large n. And we see that it's injective. The image is a closed sub-variety. And in fact, if we apply it to a product of two sub-varieties of pn and pm, the image will be a sub-variety of p large n. And so therefore, we have somehow, if instead of looking at x times y, we look at the image, which is bijective to the thing we had before, then we get a variety, a projective variety in the sense that we had before. And so we just want to do that. We want to identify the product x times y with its image in p large n under this map. So the alternative, you could, so that, otherwise you would say this is not really an amplification, but we define as what we call the product of x and y, the image of the product. So that it's a variety. And then, by abuse of notation, this image, we just call x times y. So let me write this down or something. No swear m. So we have again. So if x and y, x subset pn, y subset pm, projective varieties, we identify x times y with the image pn. And in particular, we identify sigma of pn times pm with the image sigma. And so these are then, this is a quasi-projective variety. This would be a projective variety. So if we do it like this, we find that the product of varieties is, again, a variety. And let's see. So now we want to see that we can work with this in a nice way. So first, so in principle, obviously one could always, I could do without this identification, it's just, but if you always have to remember the sigma and so on, it makes it a little bit more complicated. So it's easier to think that we're actually working on pn times pm. It's just that what are the close subsets are just the inverse images of the close subsets in pn, in p large n under this map, and so on. And a function on this thing would be regular if and only if it's the pullback of a regular function on the corresponding set in the image. So a remark, if you still remember part 2 of the theorem, this said that, what was it? It said that this map sigma ij from an plus m to sigma ij is an isomorphism. So if we now do this, translate this with this identification that we instead of saying we are in sigma ij, we are in the corresponding subset ui times uj of pn times pm, then this just says, so this part just says that if I take ui times uj in pn times pm is open, namely this would be the intersection of sigma with uij. And if I take the map phi i times phi j from, and I'll take the inverse of this map anyway, it doesn't matter, ui times uj to an plus m is also an isomorphism. Because this is just the sigma ij is just the image of ui times uj. And this map, the inverse of this map is just phi i times phi j, if you look at how it is defined. So phi i times phi j is just a map which sends map pq to phi i of p, phi i pj of q. And this is, this is identification, the inverse map of sigma ij. OK. So we also have our charts, which are the obvious charts. So the things that we have proven in this theorem translate into some very simple statements. I mean, if we do it in these coordinates, so that somehow just tells us that we can work with the products in the kind of obvious way. You know, everybody would think that the product of two open sets is a product. And if you take the product of two such charts, then this will again be a chart on the product. And this is true if we make this identification. This is what we have proven. OK. We can maybe look at the simplest example, which is p1 times p1, because in particular we, so if I take, now I take sigma of p1 times p1, which I have also, if I want just 4p1 times p1, this lies in p large n, where n is 1 plus 1 times 1 plus 1 minus 1. So this is 3. So OK. And this can be written. I claim this can be written as the 0 set of just one equation, say, 0, 0, 0, 0, 1, 1, minus 0, 1, 0, 1, 0. So a power i, if you look at it, there will be more equations. I mean, the equations that I wrote down, it's for all, you know, there should be, what is it, for equations or something. But if you write them down, most of them are just 0 is equal to 0. And this is the only one that remains. There's one equation. OK. So you see that this thing is really the 0 set of polynomial of degree 2 in p3 gives us p1 times p1. And as I told you, for instance, we have here kind of some p1's in here. We have the point p times p1, and for a point p here, and the points q times p1, the point q here. And so sigma of p1 times p1 contains two families of lines, namely, so sigma of p times p1. This will actually be a line in p3, and for all p in p1, and sigma of p1 times q for all q in p1. So through every point in p1, there will be two lines of this kind. And I have some kind of picture in the notes. I mean, it's difficult to imagine how this should go. But anyway, this thing is covered by these two families of lines. OK, so finally, for the moment finally, I want to come to the universal property. So in the case of a fine varieties, we had the product. We had the universal property of the product. We want to show that the same universal property also holds here is as if, as our definition of the product, we take sigma of the product. And as I kind of tried to mention the other time, the thing which makes something into the product and behave like the product is the universal property. So therefore, if sigma of x times y fulfills the universal property of the product, it is for all means and purposes, the product. OK, so we have the universal property. It's the same as before. And it's also not much more difficult. We will reduce it to the fine case. So let x and y be quasi-projective varieties. You can assume that x is in PM, y is in PM. So then, first, the two projections, p1 from x times, yeah. So I now write it in this kind of abuse of notation. So I write p1 from x times y to x and p2 from x times y to y are morphisms. So this rather means you take sigma to the minus 1 composed with p1. Anyway, these are morphisms. And the second statement is if z is any variety, the morphisms from z to x times y, by which I really mean sigma of x times y, are precisely given by pairs of morphisms 1 to x and 1 to y. They're fg from z for x times y given by sending a point p to the pair f of p, g of p. And if one wants to not use the abuse of notation that the morphisms would be z goes to sigma of x times y, and this would be fg composed with sigma. You just take sigma of this. So it's the same thing. So to be a morphism, it's enough that its restriction to any open subset on a chart on an open cover is a morphism. So it's enough to see that p1 is restricted to x times y intersected ui times uj, so that this is a morphism. But we could use what I just said. We have this map phi i times phi j from ui times uj to an plus m. And then we have here the map p1 to an. And this certainly is a morphism. No phi i times phi j was a morphism, so it's a morphism. And the p1 given here is just a restriction of this map. So this is not here, so this phi i times phi j. And the same, obviously, for p2. So this shows that p1 and p2 are morphisms, because in the charts, they are morphisms. And now for the second part. So we have two statements. So the morphisms are precisely those. So if we are given a morphism, we have to show it's of this form. And if we are given something of this form, we have to show it's a morphism. So first, let h from that to x times y be a morphism. Now we know that p1 and p2, the two projections, are morphisms. So then f, which is p1 composed with h and g, which is equal to p2 composed with h, are morphisms. And by definition, h is fg. It's the map which sends the first component. We have f of this and the second of this. We have n, h is equal to fg. OK, so this is the trivial direction. So now we want to go the other route. So let f from z to x and g from z to y be morphisms. We have to show that this fg is a morphism. So we put zij to be, what is it, the inverse image by f of the open subset. So ui, so x lies in pn, y lies in pm, and we have the ui and the uj there. So the inverse image of ui intersects it. As f and g are morphisms, this is an open subset of z and this is an open subset. So this is open in z. And clearly, z is covered by these zijs, if I go over all i and j. And so thus, fg is a morphism, if and only if. It's restriction to each of the zijs is a morphism. Because we know that something is a morphism if it's restriction to all open subsets of an open cover as a morphism. So if and only if fg restricted to zij is a morphism for all ij. OK, now we just look what it is. So I take this map zij and I apply to it the map f comma g. So point p is sent to f of p comma p of p. Well, this will send us to x times y intersected ui times uj. OK, we want to show that this map is a morphism. We compose with the phi i times phi j. This phi i and phi j, we know, are isomorphisms from ui times uj to an times n. So this is phi i of x times phi j of y. So this map is an isomorphism. So in order to see that the original map fg was a morphism, it's enough to show that the whole composition is a morphism. But notice now we are in the affine case. So we are just looking. So this map is just phi i composed with f phi j composed with g, which is the map zij to an plus m. And so now we are looking at a morphism from zij to an, and the morphisms are from zij to am. And this is the product morphism according to the universal property for affine varieties. This is a morphism. And actually, OK, this is not necessarily closed. So for quasi-affine varieties, like I said in the beginning of the lecture, so we can reduce the statement to the affine case by using this cover. So that means we have this universal property. So again, this says just that it's very easy to see what the morphisms from anything to the product are. They are just given by pairs of morphisms to the factors. So I can give a couple of trivial examples I mean to just what morphisms are. So for instance, if f from x to y and g from z to w are morphisms, then I can take the product of these morphisms, f times g from x times c to y times w is a morphism. And here, obviously, what I mean by this is that the map pq is mapped to the pair f of pg of q. And this is essentially trivial because what is f times g? So as a map from x times z to this thing, I'm saying that f times g is just equal. So we start here. We take f composed with p1. So we first project to x, and then we apply f. It just means we take f of p, comma g composed with p2. And so therefore, it is a morphism. And I mean, this is not very exciting. So in particular, if x is isomorphic to y and z is isomorphic to w, then x times c is isomorphic to y times w. That's kind of clear. So if f, so assume you have f from x to y is an isomorphism, g from z to w isomorphism, then f times g from x times z to y times w is a morphism. But it is even an isomorphism because the inverses, obviously the product of the inverses. So this is all very simple.  So in the next lecture, we will talk about two important properties that are already kind of hinted at before. So one is separateness. So if you have, so actually the statement is that if you have the diagonal in x times x, so if x is a variety, if you look at its diagonal, so this is a close sub-variety. That is actually a non-trivial statement. So and this is the, this is called separateness. And it has some nice properties. And you can somehow imagine that it is related to things behaving a little bit like, it's a little bit something like a Hausdorff property, that the diagonal is closed. It is, for instance, if the topology on the product of x with itself was the product topology, then it would be equivalent to then Hausdorff would be equivalent to the diagonal being closed. But the risk topology on the product is not the product of topology for the risk topology. So that's not quite true. But somehow still the fact that the diagonal is closed will give us some properties which are similar to Hausdorff if we study maps. For instance, we find that the graph of morphisms is closed like you used to, you know, and similar things. And the second statement will be completeness. And that is what should replace the compactness properties. So it is, so completeness is a bit more difficult to describe. But one consequence is that, you know, if you have a variety which is complete, then the image of any morphism starting from that variety will be closed. And this is, in some sense, you can view as a new way to make projective varieties. You just take the image of any morphism. And it is not true that all varieties are complete. In fact, what is true is that a variety is complete if and only if it is projective. So it's really a special property of being projective. And that's also compatible with what I have tried to say, that somehow projective varieties are what we want to replace compactness. So OK, and we will prove that. And this actually is slightly tricky proof. And then with that, we should be more or less done. And afterwards, we come to rational maps. Anyway, so the next time will already be on Friday. Because I think I have to skip some lectures, so I have to catch up.