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.
Best Answer
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):
Find and prove a lemma relating normal bundles to tangent bundles when $Y \subset X$ is a closed immersion of smooth varieties.
Work out what the Hilbert scheme is when $P = 1$.
Is the fact you are interested in true in the case you did in 2?
Work out what the Hilbert scheme is when $P = 2$.
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?
Let $X = \mathbf{P}^3$ and let $P(t) = t + 1$. What is the Hilbert scheme in this case?
Is the fact you are interested in true in 6 and why?
What are the global sections of the tangent bundle of $\mathbf{P}^3$ and why?
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 $X$?
Relate the tangent space of a scheme over $k$ at a $k$-rational point to $\text{Spec}(k[\epsilon])$-valued points.
Apply 10 to the Hilbert scheme. What do you get?
What is a flat deformation of $Y$ over $\text{Spec}(k[\epsilon])$? Write out the definition completely.
Completely understand what it means for a $k[\epsilon]$-algebra to be flat over $k[\epsilon]$.
What are the flat deformations of $\mathbf{A}^n_k$ over $k[\epsilon]$?
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?
In view of your answer to 15 does $\mathbf{A}^2_k$ have a Hilbert scheme in the usual sense?
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.
Combine all of the above and more during your sleep.
Please have the completely written out answers for me in my castle by Monday 8:30 AM.