The MSRP on this is $979. Here's the Apollo 11. This thing is made in America. It's value-price. Check out that grip. That is one nice grip. It feels great in the hand. With a 17-round magazine made by Live Free Armory. He blends with the frame. You see that? He's lightening cuts near the muzzle. The slide action is very smooth. 4.7 inch bushingless bull barrel. It is cut to fit an RMS seat. A Tridium. Front sight. Three and a half pounds.


Best headphones for gaming under 50. Top 5 gaming headsets. If you're looking for a headset while on a budget This video is for you. All headphones will include the mics and be compatible with all platforms. Number 5 is the Logitech G332 gaming headset. This headset is worth $40 because it offers almost everything mid-range headsets offer. I wouldn't say the cushioning is the most comfortable But it still has a lot of cushioning for a $40 headset. I don't know why but the cushioning doesn't feel as soft as others. Feels my firm. Sound quality is alright. Not the best obviously But you can definitely hear the game with no problems on this. You are sort of limited because the only color option is red. The big quality is odd to me because the headphone cups are squared out. The plastic does look a little cheap, but hey, what do you really want? That's less than $40. Number 4 is the SteelSeries Arctis 1. This headset looks super clean for your setup And it looks like $100 headphones. Build quality is very similar to other Arctis models like the 7. Although most of it is plastic, the band is metal and it comes in light at around 270 grams. The comfort of these headphones are top-notch because of the ear cups. They fit snug inside your ears. The mic eliminates background noise, but not too much. It's flexible so you can move it in different positions. The mic is able to be separated. Combine this with the clean and durable design. You can go out in public with these without too much attention. These headphones are very flexible so you can put them in various positions while traveling or on your desk. You have a mic mute button along with a volume slider on the left ear cup. The sound quality is very good because it has the same drivers as the higher-end Arctis models. One thing I don't like about the sound though is there isn't as much bass. That's probably not gonna affect how you hear footsteps and stuff, but still. Number 3 is the Corsair HS50 headphones. These are excellent value. Although nothing really stands out, it's a solid choice for headphones under $50. It has 50 millimeter drivers, which has sound quality on par 2. For example, the Astro A20. Although not as much bass as we would like, it still packs a punch. I think the build quality of the headset is the best because many parts like the cup are made out of metal and not plastic. They don't feel like they would snap in half. Also, the mic is great because it can be detached and the quality sounds good. Number 2 is the Hyper-X Cloud Stinger. This is all around great value for what you're paying. The audio sounds great with its 50 millimeter drivers. The mic is decent, it's very comfortable because of its padding and memory foam, and the design is simple and gamer-like. If you just want headphones under $50, most people would just purchase these because of the value you're getting.  So while it says number 2 and not number 1, well in my opinion, Razer Kraken X headphones beats it in some categories like surround sound and design. The Kraken X is the cheapest headphones out of the lineup, but it still has most of the features that the higher end models offer. It offers sound on par with the Razer Kraken, and in my opinion and experience, it has the best sound in the $50 price point. The comfort is phenomenally better than the rest too. It really felt like some $100 premium headphones. The design is pretty good. Surround sound is a feature that not many headphones have in this price range, but these do. It helps hear footsteps and chase audio to a certain direction. There are two drawbacks that I have for these though, and the first is the mic is not retractable. Although I highly suggest you upgrade to a better headset like the normal Kraken, for $10 more because of the improvements like a retractable mic. The other problem is that the mic isn't really that good. So if you have an external mic, that might be pretty good for you. The headset is all plastic, but it's lighter because of it. So that could be really good for you. Most of the headphones on this list are on sale most of the time, and that will make them value even better. What do you think about this topic? If you want to learn more about gaming, or you just want to know more about games, subscribe. I post videos weekly, and you'll be missing out on a ton of things you might want to know about gaming. Thanks for watching, and let's have a discussion about this in the comments below.


movie star the people who know me really know me are not the ones the judge and the jury want to hear from it says if they wanted to hear a story about some other kid it says if they wanted to watch a movie about some other kid the prosecutor with his fancy words his hard evidence wrote the script directed the scene cast just the right actor to play this kid from the hood who beat up a white kid really bad so bad that he can't wake up to tell the truth punching the air a story about a ma Shahid a 16-year-old who flourishes in art class and is a very focused in school he suddenly finds himself in a situation where it seemed destined to be convicted for murder seen multiple years in prison all because the society's predetermined mold of the black boy this book reaches deep into today's discriminatory justice system and pulls out the cold and what seems helpless perspective of a regular high school african-americans are more likely than white Americans to be arrested once arrested they are more likely to be convicted and once convicted they are more likely to experience a lengthy prison sentences african-american adults are five times as likely to be incarcerated than whites according to us bureau of justice statistics blind justice his mom thinks it's justice for a son but I know that me and him both walked down the path that was already planned for us I we stepped onto the tipping scales of lady justice but our eyes blindfolded peeking through so it's because that rag is so old worn out stretched thin barely even there I'm also he to the left Jeremy Matisse to the right perfectly in balance because where I come from jailer that were the two options she handed to us because where he comes from the American dream was the one option she handed to them so here we are blind lady justice ICU to hi I'm Jeanine and I will be analyzing blind justice and connect the poem to the criminal justice system blind justice highlights how the system is constantly against black youth preventing them from having a normal childhood we can see this in the second stanza which talks about the systemic racism in the United States but I know that me and him both walked down the path that was already planned for us as we continue to the poem the third stanza uses illusion and talks about lady justice blindfolded lady justice is opposed to represent a criminal justice system that judges the plaintiff and defendant fairly the description of the blindfold especially the words right old and barely even there implies that the blindfold or the fairness of the criminal justice system and a system itself has rarely worked properly for a long time and has never truly defended black youth the state of the blindfold being old can mean that even though the blindfold is deteriorating there's no action to replace or fix the blindfold and this can suggest that no action is currently done to fix the criminal justice system and with a deteriorated right lady justice slash the criminal justice system has never really judged people especially people of color fairly the next line states a mall Shahid to the left and Jeremy Matisse the right here in this line there is a sort of word play of Jeremy Matisse is in the right this sort of implies that in society's eyes white people are seen as the innocent victim we carry on to because where I come from jail or death for the two options she handed to us these lines connect back to the theme of systemic racism in America for example police brutality hate crimes over policing in neighborhoods and schools discrimination in careers and schools are some of the harsh realities that black people experience in America as of 2001 one of every three black boys born in that year could expect to go to prison in his lifetime compared to one of every 17 white boys according to the sentencing project in these lines we can see that black people do not experience the American dream that white people enjoy in the next few lines because where he comes from the American dream was the one option she handed to them even though there's an emphasis on one option the American dream allows white people to have access to more options and opportunities than black youth finally the poem concludes with so here we are blind i see you too lady justice is blind with or without the blindfold but in different ways and this can be interpreted as when lady justice does have a blindfold she judges impartially and without the blindfold lady justice slash the criminal justice system is intentional about their judgment and despite seeing the truth right in front of them they still judge with bias and condemn black youth statistics shows in 2016 black americans comprise 27 of all individual arrests in the united states double the share of total population black youth accounted for 15 of all u.s children yet made up 35 of juvenile arrests in that year a connection between race and crime is in a large part of a function of concentrated urban poverty which is far more common for african-americans and other racial groups although african-americans and latinos comprise 29 of the u.s population they make up 57 of the u.s prison population these policies have produced a dramatic rate of incarceration with a particular disproportionate impact on communities of color in addition to a range of harmful consequences to people of color mass incarceration has been a failed policy regarding public safety outcomes research has documented that the effect of imprisonment on crime rates has been modest and that at the current levels the scale of incarceration is well past the point of diminishing returns for public safety the proliferation of racial disparities in the u.s criminal justice system has a profound impact on the lives of people of color behind each statistic lies the face of a young black man or woman whose potential has been cut short by a heart sentence mandated by draconan drug law behind each percentage points lies the face of a latina child who will only know her parents through herrid awkward visits in prison visitation rooms behind each data set lies a community of color bereaved of hope because its young people have been left away statistic the entombment to the middle door slamming shut behind me makes my inside sink to the bottom of my feet to the bottom of these chief sneakers to the cold concrete floor to the basement of this place to the soil to the bedrock to the middle of the earth i bury myself way more than six feet deep this cell is a tomb i left my notebook up there i left my pencil up there down here in the dungeon of my mind i write anyway i draw anyway the pen and pencil are my thoughts and memories the paper is my soul and amani's voice echoes and bounces off the bedrock lingers in the heat repeating repeating mistakes and misgivings mistakes and misgivings mistakes and misgivings i learn about the thing called the butterfly effect not in school but from the guys on the block it was this one dude who said that's why we're always messing up we're always making mistakes because ain't no butterflies in the hood see if there were butterflies we would have what's it's called the butterfly effect a butterfly's wings can change the path of a storm something so small can change one big thing in the world one big thing in the universe if there are no butterflies here no pretty little wings flapping in the hood then we can't change a thing he said it is a metaphor i said ain't nobody asked you he said we are the butterflies i said and the things we do are like wings we do things every day he said how comes things ain't change nah i said everything is changing something every day even this conversation i draw a vertical line on the wall with my finger i can't see it but it's there to the left i outlined the word mistake to the right i outlined the word misgivings mistakes i should have stayed with omari that night i should have just went home that night i should have just went on the ps4 that night umi should have been home that night i should have never met omari that night i should have shooted my shot with zumbia that night i should have went with lucas that day i should have just walked away that second misgivings something wasn't right about those guys on the basketball courts i felt it in my gut so i turned back and left omari with his boys to deal with it something wasn't right about that night the way the air felt around my body as if it was trying to warn me trying to keep me away but i skated all the way to the other park where i knew it was a where people knew my name and my face but by the time i got there they were leaving and skating out of that park and onto the streets like we usually do we were home we knew the twists and turns of every block in our hood we knew the faces the music the grandmothers calling out of the windows we knew the kids and we knew the lines but that night the air was just right and just wrong at the same time one of them said there's this hill over on the other side there's these steps with a handrail where we could skate and where we knew there was a line and we didn't even know that they were following us no chasing us out of the parts of the town where the hood stopped being the hood and became a town they came on bikes and skateboards and we didn't run we stopped i stopped and waited this time this time i stayed and it wasn't even for anybody for no friend for no homie i stayed to defend myself even though everything about that night that moment was telling me what they told us to get the heck out in myself i cracked i break i split in half down the middle i shattered into pieces and bang and bang and bang and bang and bang and bang and bang on that door on those four walls at those four corners yelling shouting screaming calling to get the heck out to get the heck out the following is a poem from the book punching the air titled pipeline we walk one behind the other with our hands clasped behind us our towels rolled up in our fist i used to line up like this in kina garden except with a finger on my lips walking buddy next to me if i turned around or spoke or stepped out of line i got in trouble i always got in trouble because i always had a friend in front and back and next to me there was always something to say to ask there was always a joke to tell to laugh at but here and now it's not a classroom it's a cell block it's not a restroom it's open stalls and showers it's not a lunchroom it's a mess hall it's not friends it's inmates felons and delinquents if i squint i almost can't tell the difference in conclusion with all of our poems combined the one message that we have been conveying is that the american judicial system was not made for us specifically people of color especially black people for so long unlike our white counterparts the government has tried to make us feel that our rights don't deserve to be upheld and defended equally we are always marginalized and watched at so that we can just become another story or statistic or hashtag we made this video to bring light to the bigger issue and highlight a key form of systematic oppression within america and although things seem that they are getting better or that they are going to get better we truly won't progress until we see a total change within the system primarily focusing on teachers and making sure that children are being treated fairly with kindness and respect not just by peers but by school administration as well along with reinforcing belief within black and brown children to let them know that their dreams are attainable but before that more funding needs to be diverted into low income communities and schools which are mainly filled with children of color and along with this prison systems need to be reformed we need institutions that focus on therapy and rehabilitation rather than punishment americans shouldn't be leaving correctional systems worse than how they came in only to return as a group we shed light on this issue not only to get the ball rolling on this topic but to hopefully see real change in the future for too long things have been the same we need reformation and growth now until my black and brown children in school i believe in you the dreams are attainable and you can do anything thank you to all those who watched


Hey, this is Brett and this 2018 Chevy Silverado 2500 Crew Cab Short Box 1LTZ71 is stock number 14338Z. I am here at Summit Automotive in Fond du Lac, Wisconsin. You're new and used heavy duty truck headquarters. This 2018 Chevy Silverado 2500 has a 6.6 liter Duramax diesel engine. It is the L5P engine, puts out 445 horsepower and it's paired up with the Allison heavy duty six speed automatic transmission. This truck has been fully safe and inspected by our service shop per the state of Wisconsin inspection process, has a fresh oil and filter change all the fluids have been checked and topped off and this truck is 100% ready to go. I'm gonna go all the way around in this video inside underneath, start it up, take a look under the hood and show you all the options and just give you the most accurate representation that I can of the truck. Summit whites of color, I show all my videos in 4K. If you liked the video, subscribe to the YouTube channel. I really appreciate it. Click the bell notifications and get updates on the videos I do each and every day as well as having access to one of the largest catalogs vehicle and heavy duty truck walk arounds on YouTube and stay current on our ever changing and vast inventory here at Summit Automotive. This one comes with the 20 inch polished aluminum wheels and it has Goodyear Wrangler SRA tires. These are LT 265, 60R 20s and they have right around half the tread left on them. Frame and underbodies in pretty nice shape. This is a one owner clean title history, clean car facts out of Michigan. Front fender is very nice and clean. Didn't see any dents or dings on there. This one does have the HID headlamps and the LED fog lamps. I'm gonna turn those on at the end of the video so you can see just how bright they are. Front bumper is in fantastic condition as is the lower valence. You get the black bow tie on there in the Z71 grill. This one has the true color induction hood and the hood itself is in fantastic condition. Didn't see any dents or dings on there. It does have the cab lights up top. Passengers side front fender, very nice and clean and the passengers side front wheel. No major scuffs or scrapes on there. As you go down this side of this 2018 Chevy Silverado, 2500 take note of how clean the body is, how reflective and mirror like that paint is. I take these HD videos so if you're far away or even if you're close by, you just cannot make the trip down but you're still interested in purchasing the vehicle. You can see the truck, hear the truck and have confidence in the vehicle that you're looking at before you even get here. So when you do get here, there's absolutely no surprises and you can make a smart and informed buying decision from wherever you're at. Now if this video helps you make that buying decision, let yourselves know that you saw the video that was helpful and that Brett sent you. Then I can keep doing these videos for our great customers all over the United States.  Very nice and clean underneath, has all the remaining factory exhaust so it hasn't been altered in any way and the low rockers and cab corners are in really nice condition as well and it comes with a nice factory chrome step bars. Passengers side box looks really good. Z71 off-road suspension package, rear bumpers absolutely perfect as the rear bumper steps, full towing package which includes a receiver hitch, four pin and seven pin wiring and the tailgate is in really nice condition as well and you get the black bow tie on the back here. Does come with a factory soft tonneau cover, probably the least nice thing on the vehicle but still functional, got just a couple of little cracks and stuff right there but other than that, still doing its job. Comes with a very nice spring and bed liner. Beds in really nice condition, I didn't see any major dents or dings and the spring and bed liner goes onto the tailgate. Tailgate shuts nice and solidly. Going down the driver's side just as clean as that passenger's side. No major dents or dings on the box and for full disclosure, this back wheel just as nice as the rest. Cab and doors on the driver's side look good. This one does have the telescopic tow mirrors built in directional signals, LED side lights on here, they telescope out like so and fold in like that. Take a quick look at the back seats. They are in fantastic condition, no rips, no tears back here, does have the latch child safety system, fixed glass rear window with the built-in rotor frost. Seats are in really nice shape, you get a nice all weather format set up back here. Keeps your carpeting looking like new and side curtain airbags on those front seats. Smells very clean inside this truck. Inside and bottoms of the doors, other than a little dust, nothing too bad. Look really good. Show you the VIN sticker here. No previously owned in Canada trucks here. Entire and loading information sticker.  Power driver seat, all weather formats up here as well. Auto headlamps turned out four wheel drive factory brake control, you get the tilt and telescopic steering wheel, power windows locks and mirrors and you get that Z71 trim on the doors there. We'll hop inside and check out the miles radio and everything that this truck has to offer on the interior. See this one has 47,447 miles on it. You do get a compass display as well as the digital speedometer on there and that Z71 instrument cluster. Instrument cluster is nice and clean. Comes with the leather wrap steering wheel, cruise controls on the left, Bluetooth and information center controls on the right. It does have the Allison heavy duty six speed automatic transmission and you get the Chevy my link system. This one does have the projection manager where you can project yourself onto the screen. So while this one doesn't have the factory navigation system, project your phone up there with ways of Google Maps and it's like you have nav right on the screen. So really nice system there. There's a CD player there as well.  It's also how you turn on those LED side lights in the mirrors, downhill assist control and your factory brake controllers right there. Down here is a 12 volt PowerPoint as well as a 110 volt, 150 watt plug-in. Passenger side glove boxes, format and seat are all in excellent condition. No rips or tears on there. And once again, it smells very clean inside this truck. The headliners in great shape. You get map lights up there on Star and SOS buttons in the mirror as well. Let's start it up and take a look under the hood. Starts right up. No check engine lights or anything like that. There are those LED side lights. They are very bright working nicely I would personally like to thank you for checking out the video today and hopefully from this HD video you've been able to verify the quality condition options and of course, cleanliness of this truck all the way around inside out. See those HID lights and the LED fog lamps are very bright working nicely as are the cab lights up top. Under the hood, we have the 6.6 liter Deramax diesel engine. This is the L5P engine. 445 horsepower engine base, very clean runs, very smooth. Once again, this truck has been fully saved and inspected by our service shop as a fresh oil and filter change. All the fluids have been checked and topped off. There is the emission sticker. This truck is 100% ready to go. Air comes in the air. Coal induction there goes down this tube right into the air cleaner. So that is a fully functional coal induction hood. I would highly recommend this truck from a quality and condition standpoint. I'd ship this one to California, Texas, New York, Florida, back to Michigan. Whoever's gonna get this truck is gonna be very happy with the quality and condition on it. And to see more pictures of this truck on one of our other 550 new and used cars, trucks, SUVs, minivans, Wranglers, half tons, three-quarter tons, one tons, you name it, we gotta go to the website right there. SummitAuto.com, full pictures and descriptions of every single vehicle all at SummitAuto.com. If you wanna check out more HD videos, go to YouTube.com slash SummitAuto. Click the bell notifications, get updates on the videos they do each and every day. As well as having access to one of the largest catalogs, vehicle walk-arounds on YouTube, bar none. 9,000 videos and counting and stay current on our ever-changing and vast inventory here at Summit Automotive. In fact, in a second, we'll see a link to subscribe to my YouTube channel in the upper left. A link to all the Chevy 2500 truck videos I've ever done in the upper right. A link to this vehicle on our website in the lower left. And a link to one of our latest YouTube videos in the lower right. Click those, check us out. We're super excited to help you with this Ultra Clean 2018. Chevy Silverado 2500 Crew Cab Short Box, one LTZ-71 in Summit White Clear Coat. Thank you so much for checking out the video. Remember to like, subscribe, and share on the YouTube channel. I really appreciate it. Thanks again.


This lecture is part of an online algebraic geometry course on schemes and will be about the canonical sheaf, omega x of a scheme x. So we're going to simplify by taking x to be non-singular, that means all its local rings are regular and a projective variety over a field k. You can define canonical sheaves in much greater generality, but I don't want to get involved in all these extra complications. So last lecture we examined the cotangent sheaf, omega x over k, of whose sections are one forms on x. And now if x is non-singular, which we're assuming, this implies that omega of x over k is locally free. In other words it's more or less a vector bundle or rather the sheaf sections of a vector bundle, but there's no great harm in confusing vector bundles with their sheaves of sections. So that means it sort of looks locally like a vector space times x. And from this we can define two other sheaves. First of all we can define the tangent sheaf, which we mentioned a little bit earlier. This is just the dual hom of omega x over k to o of x. So what's going on here? This is the sheaf of differentials. This is just regular functions. So mapping something to regular functions is a bit like taking the dual of a vector space locally. This means the sheaf hom. So if we ask what are the homomorphisms from this sheaf to this sheaf, we can either define a sheaf of homomorphisms or we can take the global sections of that sheaf and define the an abelian group of homomorphisms from this sheaf to that sheaf. And here we're talking about the sheaf homomorphisms. I mean, I think Hart-Shorn writes this in calligraphic letters, but I'm not very good at drawing calligraphics. I'm just using Roman. Anyway, so this is the tangent sheaf of x and the sections are just tangent vector fields. So once you've got tangent vector fields, you can start doing analog of differential geometry. Another thing you can do with this, so this is like taking the dual of a vector space at each point. You'll also take exterior powers. And what we're going to do is we're going to take the highest exterior power of omega x over k where n is equal to the dimension of x. So you can define anything you can do with vector spaces. You can do with sheaves, like taking exterior powers and symmetric powers and duals and so on. And here we're just taking the nth exterior power. This is particularly easy to figure out since we're assuming x is non-singular, so this is locally free. So it just looks locally like a vector space of dimension n. So taking its nth exterior power, it's going to look locally like a one-dimensional vector space. So this will be an invertible sheaf, or in other words, a line bundle. It's essentially a one-dimensional vector space for each point of x, except it's probably twisted a bit. So the sections of this are just n-forms on x. So if you were a differential geometry, an n-form is something you would integrate in order to find the volume of x. Well, at least if x is oriented. So this is the canonical sheaf and is denoted by little omega of x over k. So this is the canonical sheaf. So it's a natural way of getting a line bundle over a scheme x. I mean, we can always get a line bundle by taking the trivial sheaf of regular functions, but that's not a terribly interesting line bundle. I mean, for Vex's project, if it's space of sections, it's just one-dimensional, for example. So what I want to do now is to give a sort of advertisement for the canonical sheaf by quickly giving a quick survey of some applications of it. So these applications are things that come up later in the course, and I will miss out an awful lot of details. So the first application is Riemann Rock for Curbs, which says, suppose you've got a divisor d, and this corresponds to some line bundle l of d, and we want to know what is the dimension of the space of sections of l of d. So the dimension of the space of sections is often written little l of d. So the Riemann Rock theorem states that little l of d can be written, worked out as follows.  So let's explain what all the bits are. Well, I've just done l of d. The degree of d, well, that's easy. If d is sum of n i p i, where these are points, the degree of d is just sum of the n i. So the single point is degree 1. This g is the famous genus of the curve. If you're an analyst, you would define this as the genus of the underlying topological space. In other words, the number of handles of the surface. If you're an algebraic geometry working over a field of characteristic p, you can't do that at least not very easily. So instead, you can define it as the dimension of the space of global sections of the canonical sheaf. In other words, this is just the space of one forms on the curve.  So k is sometimes called the canonical divisor, which is a bit misleading because there's not really a single canonical divisor. There's a whole family of linearly equivalent divisors called canonical divisors. And what you do to get them is you take a global one form and k is just the divisor of zeros of that one form. And of course, you could choose different one forms and get different divisors, but they're all linearly equivalent. Anyway, l of k minus d, the dimension of the space of sections on this doesn't depend on which divisor you choose as your canonical divisor. So this is well defined. So this bit of the theorem is due to Riemann and this bit of the theorem is due to Roch. And you maybe, I used to wonder why Roch gets his name on one of the most important theorems in algebraic geometry, and then you never hear about them offwards. And it turned out that Roch died very young shortly after proving this crucial bit of the Riemann-Roch theorem. So the next application of the canonical bundle is a sort of generalization of this, which is ser duality. And the point is you should think of the canonical sheet as a sort of dualizing object. It's called dualizing object D. So generally, if you've got something like vector spaces or a billion groups, you might have these objects A and you might define a sort of dual of A to be the homomorphisms from A to some dualizing object. For example, for vector spaces, the dualizing object would be the field you're working over with, and that defines the dual of a vector space. And for finite abelian groups, the dualizing object might be, say, the circle group, and that defines the Pontreorgan dual of a finite abelian group. And for sheaves, there's a sort of more complicated duality where the dualizing object turns out to be the invertible thief. So we get, for example, we get ser duality, which says that a cohomology group of X with coefficients in the sheaf X is isomorphic to a sort of dual cohomology of X with coefficients in the dual of F, except that's not quite right because we have to tensor this with the canonical sheaf and you have to take the dual of that as well. So the the canonical sheaf kind of appears here as a sort of playing a similar role to these dualizing objects, except in a slightly more complicated way. You can think of this as being a sort of analog and algebraic geometry of Poincaré duality. You remember if you've got a manifold X, then if you take the cohomology of X with coefficients in R, then this is isomorphic to the dual of the cohomology of X with coefficients in R, except you have to take the dual of this and you may think what has happened to the canonical sheaf. Well, it turns out this this only happens if X is orientable, and if X is non orientable, then you have to sort of twiddle this by some sort of orientation sheaf. So so ser duality is sort of formally quite similar to Poincaré duality, except you need to put the the the canonical sheaf in. The reason why this is related to the Riemann-Roch theorem is the Riemann-Roch theorem is really two different theorems bundled together. So the first theorem tells you the Euler characteristic of L of D, which is a zeroth cohomology of L of D. Guess what the dimension of this minus the dimension of the first cohomology of L of D, and it tells you that this is this is degree of D plus 1 minus G. So so this is the form of the Riemann-Roch theorem that hits a book generalized to higher dimensions. So so that in order to get the Riemann-Roch theorem, what you do is you apply ser duality to this, and this just says that this is isomorphic to H naught of omega time tensor with L of D minus 1. So ser duality says that these two spaces are dual, so they have the same dimension. So if you combine the hits of Riemann-Roch theorem for curves with ser duality, then you get the classical Riemann-Roch theorem for curves. And the hits of Riemann-Roch theorem and ser duality both generalize to higher dimensional varieties. Third thing you can do is do the define the canonical embedding of a variety. So we have the following problem. Classify varieties X. So this is the sort of fundamental problem in algebraic geometry in some sense. We want to classify all varieties, and obviously this is hopelessly difficult in general, but we can try and do it with two steps. Step one, embed X into projective space, and step two, classify subvarieties of projective space. And both of these are impossibly difficult, but never mind. Still you can make some progress on classifying subvarieties by using things called Hilbert schemes, which in some sense parametrize all possible subvarieties of PN. So there's some sort of machinery for dealing with the second problem. For dealing with step one, we've got to find a way of giving a variety. How do we embed it into projective space in a comical way? And what we can do is we can pick a line bundle L on X and pick some sections, say all sections, and this gives a map from some open subset in X to projective space, which if you're lucky will be an embedding of X in projective space if you've got enough sections of the line bundle. So what we need to do is to find a line bundle. So given a variety, how do we find a line bundle on it? Well, there's one obvious way, which is we can just take L to be the chronicle line bundle to the power of some integer. And so if you take M to be a positive integer sufficiently large, this gives you things called the canonical embeddings of a variety into projective space. Okay, this leads to the question of when does a line bundle, when does the canonical line bundle have enough sections that you actually get embedding? And this gives the fourth application of the canonical line bundle, which is the Kadyra dimension. So we have the following question, does the canonical line bundle, or its powers, let me go to the i for i greater than zero, have lots of sections? So if they have enough sections, then we can get an embedding into projective space. And the Kadyra dimension, usually denoted by a little Greek kappa, is the dimension of the image of X under the canonical embeddings of omega to the i for i sufficiently large and having the right congruence properties. Actually, it's a little bit complicated because the dimension of the canonical embedding can, the dimension of the embedding can jump up and down a bit as i increases, but roughly speaking, there's a sort of maximum dimension and that's the Kadyra dimension. It's also possible that none of these of any non-zero sections, in which case we say K is minus infinity if none of the omega to the i for i greater than or to have sections. So we can see some examples of this. If we take p1, we saw the canonical line bundle is just the cotangent bundle, which is o minus two, and has no sections other than zero. So the Kadyra dimension is minus infinity. If we take an elliptic curve, then again the canonical line bundle is just the cotangent bundle, which we saw was trivial, so it's o zero. And the space of sections has dimension one and it has dimension one even if you take some power of this. So this means we do have a non-zero section, but we're only getting a map from the elliptic curve to a project that the zero-dimensional projective space, in other words a point. So the image of the elliptic curve under the canonical map is just zero. We don't get an embedding of the elliptic curve. On the other hand, if you take a curve of degree four, such as x to the four plus y to the four plus c to the four equals zero, then in this case we find that we will calculate the canonical line bundle a little bit later, and we find omega has lots of sections, and in fact it has enough to give an embedding of this into the plane, so we find the Kadyra dimension is now one. So roughly speaking, if a variety is dimension n, then the Kadyra dimension can be minus infinity zero, one, two, up to n. And these behave a bit like varieties of various curvature in differential geometry. So you remember in differential geometry, sometimes you can give varieties a Romanian metric and sometimes this is positive zero or negative curvature. So these are sort of analogous to positive curvature manifolds. These ones are sort of analogous to zero curvature manifolds, and these ones are sort of analogous to negative curvature manifolds. The relation between curvature and the Kadyra dimension shouldn't be taken too seriously, it's just a sort of rather informal analogy, and these ones here are sort of mixture between zero and negative curvature in some sense, so negative curvature sort of tend to be things like hyperbolic manifolds. So there are some formal similarities between varieties of Kadyra dimension n and hyperbolic manifolds. And this is sometimes used to be a classification of manifolds. So for example, you'll hear about there being a classification of surfaces due to Enrique and Kadyra, where you first will look at the Kadyra dimension and then you classify, you have a pretty good classification of the surfaces of Kadyra dimension minus infinity zero and one, and the surfaces of dimension two are called general type. So that's the Kadyra, so the Kadyra classification of surfaces sort of classifies the surfaces not of general type, but calling it a classification of surfaces is really a bit misleading because almost all surfaces are of general type in some sense and we still know very little about the classification of surfaces of general type. By analogy, we can say that calling this a classification of surfaces would be like having a classification of birds that classifies birds into penguins, ostriches, and everything else. And obviously that's not a very convincing classification of birds because the category everything else is nearly everything and the classification of surfaces is a bit like that. General type is sort of everything else includes almost all surfaces and we don't know how to classify it. But anyway, so that's the sort of advertisement for the canonical sheaf turns up in lots of places. Well, now we should give some examples of how to calculate it. Well, first of all, we've done P1, we calculated the canonical sheaf for P1 and found it was just O of minus two. Now let's do n-dimensional projective space and figure out what the canonical sheaf is. Well, for this, we use the exact sequences of sheaves we worked out earlier. We had the cotangent sheaf of P to the n and we saw this map to O minus one to the n plus one and this map to O of zero, the sheaf of regular functions of this map to zero. And let's just write in the ranks here to make it slightly easier to see what's going on. So this is rank n, this is rank n plus one, and this is rank one. Now if you've got an exact sequence of vector spaces, A goes to B goes to C goes to zero, suppose these have dimensions A, B and C, then we know from linear algebra that the topic stereo power of B is canonically isomorphic to the topic stereo power of A times the topic stereo power of C. Now we can do the same thing for sheaves and what we find is that the highest exterior power of omega P to the n tensile with the highest exterior power of O of zero is isomorphic to the highest exterior power of O minus one to the n plus one. So this means the direct sum of n plus one copies of O minus one, not the tensile product of n plus one copies, it's rather bad notation unfortunately. Well, this thing here is the canonical sheaf of P to the n and this thing here is just trivial because if you take the exterior power of the the trivial line bundle, you just get the trivial line bundle and here we're taking the n plus one exterior power of a sum of n plus one copies of a certain line bundle and that's the tensile product of n plus one copies of these which is O minus n minus one. So this gives us the canonical sheaf of any projective space. In particular, we see that the Kadara dimension of projective space is minus infinity because none of these sheaves O minus n minus one times m of any global sections and that's if n is greater than zero, I guess. Well, next we can calculate the Kadara dimension of a hypersurface y in P to the n. Let's assume this is non-singular and for this we're going to show that the canonical bundle of y is equal to the canonical bundle of P to the n, tensile with L, tensile with O of y where this is regular functions on y and this is the canonical sheaf of P to the n and this is the line bundle corresponding to the divisor y. So since y is a non-singular hypersurface, we can think of it as being a line bundle and therefore it's got a divisor attached to it. So we're sort of restricting the canonical bundle of P to the n only we have to kind of twiddle it by this line bundle. And in order to see this, what we do is we start with the exact sequence, nought goes to i over i squared, goes to omega P to the n, tensile with O of y, goes to omega of y, goes to nought which I haven't proved, but you can find in Hart-Shorn, I think it's Hart-Shorn 8.17 and I'm not going to give the proof of this because I'm getting a bit bored of proving things about exact sequences of sheaves. And again, if we look at the dimensions of these, sorry if you look at the ranks of these vector bundles, this is rank one, this is rank n and this is rank n minus one. So we're going to do the same trick of taking exterior powers and what we find is that omega P to the n tensile with O of y is the canonical bundle of y tensile with i over i squared. So here the i over i squared comes from there and the omega y comes from there and this bit here of course comes from this. And well, we know the canonical bundle of P to the n, so this is just O of minus n minus one because we just worked it out. So we've got to figure out what this thing here is. So i is the ideal sheaf of y contained in P to the n, so it's just the sheaf of functions vanishing on y, roughly speaking. And i over i squared is just equal to i times i to the nought over i to the one. And this is just i times, well, i to the nought over i to the one is just the is just the regular functions on y because we're quotient out regular functions on a projected space by functions vanishing on y. And i is equal to l with d to the minus one where d is the divisor corresponding to y. I guess I could write that as l of y to the minus one. Yeah, except well whatever. And that's because sections, so i is equal to l of minus d which is equal to l of d for minus one, so the minus one should be outside that. That's because the sections of this, the sections of both this thing here and this thing here are just functions vanishing on y. So if we put everything together, we find that the canonical bundle of y is the canonical bundle of P to the n tensored with regular functions on y, tensored with l of d minus one, which is equal to o of y, tensored with o of minus n minus one, tensored with o of d, where we notice that the sheaf where the bundle l of minus d corresponding to y is just o d if y is a hypersurface of degree d. So we find omega y is just equal to o of y twisted by d minus n minus one. So this is the final result for the canonical sheaf of a hypersurface of degree d in n dimensional projective space. And let's see a few examples of this, let's just take n equals two, so we're looking at hypersurfaces in P2. And for d equals one, y is just a line P1 in P2. And we see from the formula, so the formula was omega y equals o y d minus n minus one. So omega y is o of, this is going to be one minus two minus one, which is o of minus two, which is the result we formerly got for the canonical sheaf of P1. If we take d equals two, then y is a conic in P2. And we find that omega y is now o of two minus two minus one, which is equal to o of minus one. And a conic on P2 is just isomorphic to P1. So it looks a bit funny here because we now seem to be saying that the canonical line bundle on P1 is now o of minus one rather than o of minus two. So what's going on here? Well, the point is this o of minus one is actually o of minus one on P2. And if we restrict it to P1, it actually becomes o of minus two on P1 because we're mapping P1 to P2 by a degree two embedding. So we take x, y to say x squared, x, y, y squared. You might call this rst. And then sections of o of one are going to be given by these functions r, s, and t. So if you restrict them, they'll be given by things like x squared, x, y, and y squared. So if we take o one on P2, this restricts o two on P1. And similarly, the duals o of minus one restricts o of minus two. If we take d equals three, here y is now a cubic curve, say y squared equals x cubed minus x or something like that. And now we find that omega is o y of d minus n minus one, which is now o y of three minus two minus one, which is just o of y. So omega is isomorphic to the ring of regular functions on y. So for example, the dimension of the space of one forms is one because the space of sections of regular functions on y is just one because y is a is a projective space. So we said the genus is the dimension of the space of sections. So the genus is equal to one. We can also work out the cadara dimension. So omega to the n is again going to be o of y, and the space of sections is always going to be one. So in the map from elliptic curve to a projective space given by sections of this, the image is just a point. So the cadara dimension is zero. It's the last example. Let's just very quickly look at what happens of d is equal to four. Here y is aquatic. For example, it might be extra four plus y to the four plus z to the four equals naught, for example. And we can work out the canonical bundle is now going to be o of y twisted with four minus two minus one, which is equal to o y twisted by one. And every if you twist by positive integers, you increase the number of sections. So we see from this that the number space of sections of one forms is at least three because o one on p two has the three dimensional space of sections. And you can restrict them to this degree four curve and get a three dimensional space. In fact, the dimension of the space of one forms having to be exactly three, but that takes a little bit more effort. And as I seem to be this lecture seems to have gone on too long, so I'll stop there.