16
$\begingroup$

We have the following theorem:

Let $X$ be a projective scheme over an algebraically closed field $k$, and $Y \subset X$ a closed subscheme with Hilbert polynomial $P$. Then$$T_{[Y]}\text{Hilb}_P (X) = \text{Hom}_Y (I_Y/I_Y^2, \mathcal{O}_Y).$$In particular, if both $X, Y$ are both smooth, then$$T_{[Y]}\text{Hilb}_P (X) = H^0(Y, N_{Y/X}).$$My question is, what is the intuition behind this theorem? What are some examples to keep in mind when thinking of this theorem? Thanks.

$\endgroup$
  • 12
    $\begingroup$ A normal vector field on $Y$ lets you wiggle $Y$ in $X$ (infinitesimally) without changing the Hilbert polynomial. $\endgroup$ – S. Carnahan Aug 28 '15 at 17:25
  • 2
    $\begingroup$ To build on Scott's comment, any vector field allows you to flow $Y$ along it (wiggle $Y$). If the vector field is tangent to $Y$, then $Y$ flows to itself. So sections of $T_X/T_Y$, which is the normal vector field, describe moving $Y$. $\endgroup$ – David E Speyer Sep 2 '15 at 13:57
28
+100
$\begingroup$

My answer to this question and any question of this nature is the following: proving lemmas about the material you are thinking about and working through examples (which is in some sense the same as proving lemmas) is the preferred way of developing intuition for a subject. Instead of asking others what your intuition should be, roll your own. Potential bonus: if your intuition is different from others, your intuition may help you prove different (new?) theorems.

How would this work in this case? Well, I suggest proving some lemmas (not an exhaustive list):

  1. Find and prove a lemma relating normal bundles to tangent bundles when $Y \subset X$ is a closed immersion of smooth varieties.

  2. Work out what the Hilbert scheme is when $P = 1$.

  3. Is the fact you are interested in true in the case you did in 2?

  4. Work out what the Hilbert scheme is when $P = 2$.

  5. For which points in the Hilbert scheme in 4 is there a normal bundle? Is the fact you are interested in true in those cases?

  6. Let $X = \mathbf{P}^3$ and let $P(t) = t + 1$. What is the Hilbert scheme in this case?

  7. Is the fact you are interested in true in 6 and why?

  8. What are the global sections of the tangent bundle of $\mathbf{P}^3$ and why?

  9. Say $X$ is smooth over the ground field is $k$ and let $k[\epsilon]$ be the dual numbers. Can you relate automorphisms of $X \times_{\text{Spec}(k)} \text{Spec}(k[\epsilon])$ to sections of the tangent bundle of $X$?

  10. Relate the tangent space of a scheme over $k$ at a $k$-rational point to $\text{Spec}(k[\epsilon])$-valued points.

  11. Apply 10 to the Hilbert scheme. What do you get?

  12. What is a flat deformation of $Y$ over $\text{Spec}(k[\epsilon])$? Write out the definition completely.

  13. Completely understand what it means for a $k[\epsilon]$-algebra to be flat over $k[\epsilon]$.

  14. What are the flat deformations of $\mathbf{A}^n_k$ over $k[\epsilon]$?

  15. What are the flat deformations of the closed embedding $Y = \mathbf{A}^1_k \subset \mathbf{A}^2_k = X$ where we are holding $X$ fixed?

  16. In view of your answer to 15 does $\mathbf{A}^2_k$ have a Hilbert scheme in the usual sense?

  17. When $Y$ is smooth over $k$ is there a relationship between flat deformations and the tangent bundle? Open your copy of Hartshorne at a random place (maybe the index?) and start reading till you found it.

  18. Combine all of the above and more during your sleep.

  19. Please have the completely written out answers for me in my castle by Monday 8:30 AM.

$\endgroup$
  • 7
    $\begingroup$ Count, will you be awake at 8:30 AM? I would expect you to be back in your coffin by then. $\endgroup$ – David E Speyer Sep 3 '15 at 18:53
6
$\begingroup$

I'll write $H$ for the Hilbert scheme and $[Y]$ for the point of $H$ corresponding to a subscheme $Y \subseteq X$. When $Y$ and $X$ are both smooth, the claim is that the tangent space to $H$ at $[Y]$ is the global sections of $(T_X)|_Y/T_Y$.

Intuition 1 (This is basically Scott's comment.) Given a holomorphic vector field $\theta$ on $X$ near $Y$, we can flow $Y$ along $\theta$ to get a family $Y(t)$ of complex submanifolds of $X$, and thus a path $[Y(t)]$ in $H$. It is plausible (and true!) that the first order variation $\frac{d}{dt} [Y(t)]$, which is a tangent vector to $H$, only depends on $\theta|_Y$. Moreover, if $\theta$ is tangent to $Y$, then $Y$ flows to itself, so it makes sense that only the image of $\theta$ in the quotient $T_X/T_Y$ matters.

Intuition 2 Let's go the other way: Let $[Y(t)]$ be a path through $H$ with $Y(0) = Y$. For $t$ in some open disc $D$ around $0$, we can smoothly (not holomorphically!) trivialize the family and thus get a map $\phi: Y \times D \to X$. For every point $y \in Y$, we have a path $\phi(y,t)$ in $X$, whose derivative at $0$ lies in $T_X(y)$. But changing the trivialization can change this derivative by vectors in $T_Y(y)$, so the intrinsic thing is a vector in $T_X(y)/T_Y(y)$ for each $y \in Y$ -- in other words, a section of the normal bundle.

Bonus We have a long exact sequence $0 \to T_Y \to (T_X)|Y \to N_{Y/X} \to 0$ of vector bundles on $Y$. So we have a long exact sequence $H^0(Y, T_X|Y) \to H^0(Y, N_{Y/X}) \to H^1(Y, T_Y)$. The final vector space, $H^1(Y, T_Y)$, describes deformations of the complex structure of $Y$ and, indeed, the map $H^0(Y, N_{Y/X}) \to H^1(Y, T_Y)$ says, when you wiggle $Y$ inside $X$, how the complex structure on $Y$ changes. The tangent vectors coming from $H^0(Y, T_X|Y)$ are the ones that don't change complex structure. Intuitively, if we have an actual section of $T_X$, then we can flow the embedding $Y \to X$ along it to get a holomorphic map $Y \times D \to X$.

Of course, all of this is intuition only in the case that $X$ and $Y$ are smooth. In order to see that $\mathrm{Hom}(I_Y/I_Y^2, \mathcal{O}_Y)$ is the version which is still right when you put singularities in, I recommend Count Dracula's method.

$\endgroup$
0
$\begingroup$

A good source to start with (or if you get stuck) is Shafarevich's book 'Basic Algebraic Geometry II'. See Theorem 4 in Chapter VI. The information encoded in the tangent space can be looked up in the proof. After the proof there is a small demonstration for what the tangent space of the Hilbert scheme can be used. More information on that can be found in Section 2 resp. 13 of the book 'Hartshorne, Deformation Theory' (I also recommend to take a look at Exercise 2.4).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.