Tangent Spaces and Singular Points

The semester is starting up again now, so my schedule is a bit irregular. Sadly, I won’t be able to keep up the pace I set over break. However, I should still be putting out a couple of posts a week. So without further ado, tangent spaces. We’ve looked at tangent spaces before, and used them to define what it means for a point to be smooth versus singular. Today, we’re going to start back at the beginning of this stuff and justify the definitions a little.

Back when we talked about Elliptic curves, we defined the formal derivative of a polynomial. We’ll be using this notion, and specifically we’ll need to think about the geometry of a derivative. In calculus, the derivative measures rate of change of one quantity with respect to another. We’ll return to the meaning of derivatives after giving the classical definition of tangent space.

Let p\in V\subset \mathbb{A}^n be a point on an affine variety defined by f_1=\ldots=f_\ell=0. Then we define the tangent space at p=(p_1,\ldots,p_n) to be T_p V, the common zeroes of the \ell polynomials \sum_{i=1}^n \frac{\partial f_j}{\partial x_i}(p) (x_i-p_i)=0. With a bit of linear algebra, it can be checked that \dim T_p V=n-\mathrm{rank}\left(\frac{\partial f_i}{\partial x_j}\right). The matrix whose rank we are taking is called the Jacobian Matrix of f_1,\ldots,f_\ell.

Geometrically, T_p V is a plane which passes through p and is the “closest approximation” of V near p. A property that the tangent plane will have is that it has intersection multiplicity greater than 1 at p. One common trick in the subject of differential geometry is to pass to the tangent space to get a nice linear problem rather than the ones normally encountered directly. We’ll use that trick here too, in the future.

We can call a point smooth or regular if \dim T_p V=\dim V. The algebraic analogue of a smooth point is a regular local ring. That is, let R be a noetherian ring with only a single maximal ideal (we call such rings local) and denote the ideal by \mathfrak{m}. Then R is a regular local ring if \dim R=\dim_{R/\mathfrak{m}} \mathfrak{m}/\mathfrak{m}^2, where the dimension of a ring is just the length of the largest chain of prime ideals (and if there is no largest one, take it to be infinity). So, for instance, in \mathbb{C}[x,y,z], we have 0\subset (x)\subset (x,y)\subset (x,y,z) is a chain of length three. This notion corresponds to defining the dimension of a variety in terms of the longest chain of proper subvarieties.

Oscar Zariski showed that these two notions are the same for us, that is, that for an affine variety and a point p\in V, we have that p is a nonsingular point if and only if \mathscr{O}_{V,p} is a regular local ring. This is precisely the original definition we gave for a point to be nonsingular. It’s also true that the dimension of the tangent space we defined above, as a subvariety of \mathbb{A}^n, is the same as the dimension of the Zariski tangent space discussed earlier. So we’ll take as our definition of nonsingularity that \dim T_p V=\dim V, and think of T_pV as the Zariski tangent space (in general, I’ll make it clear if we’re using the geometric tangent space) and say that a point is singular if \dim T_p V\neq \dim V. It turns out that the dimension of the tangent space can’t be smaller than that of the variety, and though this would only take us on a slight detour into commutative algebra, we won’t bother.

Later on, when we talk about vector bundles we’ll eventually get to an object called the Tangent Bundle. Basically, what this is is that if we’re given a nonsingular variety, there’s a nice way to take the tangent spaces at each point and put them together to create a new variety of twice the dimension, and in fact the spaces \mathfrak{m}/\mathfrak{m}^2 that we earlier referred to as cotangent spaces can also be glued together to make a Cotangent Bundle. In general in the future, smoothness will be a valuable hypothesis to place on varieties to make them easier to work with.


About Charles Siegel


8 Responses to Tangent Spaces and Singular Points

  1. One of the big keys is that really close to p, it doesn’t intersect anywhere else.

    I think I can construct examples of surfaces where the tangent plane intersects the surface in any open neighborhood of p. Do you mean that “generally” it doesn’t?

  2. Charles says:

    Sadly, I now look at this and see that I was very unclear. I meant this statement as a bit of intuition, and now looking back the intuition I wanted to convey is just wrong. I saw your comment and immediately counterexamples started appearing in my head…they’re fairly easy to construct. For instance, every surface in \mathbb{P}^3 intersects any plane in \mathbb{P}^3 in a curve. In fact, if X\subset\mathbb{P}^n is a hypersurface, it will intersect its tangent planes in a variety of dimension n-2.

    That line is being removed. Sadly, though curves are a good place to get intuition sometimes, at other times they can be rather misleading.

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  7. Anonymous says:

    For a curve, can the dimension of the tangent space be arbitrary big ?

    • Yes, at a singular point, you can have an arbitrarily large tangent space. To see this, look at the union of the coordinate axes in \mathbb{A}^n. Then the tangent space is n dimensional.

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