111.45 Cohomology and Hilbert polynomials
Situation 111.45.1. Let $k$ be a field. Let $X = \mathbf{P}^ n_ k$ be $n$-dimensional projective space. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Recall that
\[ \chi (X, \mathcal{F}) = \sum \nolimits _{i = 0}^ n (-1)^ i \dim _ k H^ i(X, \mathcal{F}) \]
Recall that the Hilbert polynomial of $\mathcal{F}$ is the function
\[ t \longmapsto \chi (X, \mathcal{F}(t)) \]
We also recall that $\mathcal{F}(t) = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(t)$ where $\mathcal{O}_ X(t)$ is the $t$th twist of the structure sheaf as in Constructions, Definition 27.10.1. In Varieties, Subsection 33.35.13 we have proved the Hilbert polynomial is a polynomial in $t$.
Exercise 111.45.2. In Situation 111.45.1.
If $P(t)$ is the Hilbert polynomial of $\mathcal{F}$, what is the Hilbert polynomial of $\mathcal{F}(-13)$.
If $P_ i$ is the Hilbert polynomial of $\mathcal{F}_ i$, what is the Hilbert polynomial of $\mathcal{F}_1 \oplus \mathcal{F}_2$.
If $P_ i$ is the Hilbert polynomial of $\mathcal{F}_ i$ and $\mathcal{F}$ is the kernel of a surjective map $\mathcal{F}_1 \to \mathcal{F}_2$, what is the Hilbert polynomial of $\mathcal{F}$?
Exercise 111.45.3. In Situation 111.45.1 assume $n \geq 1$. Find a coherent sheaf whose Hilbert polynomial is $t - 101$.
Exercise 111.45.4. In Situation 111.45.1 assume $n \geq 2$. Find a coherent sheaf whose Hilbert polynomial is $t^2/2 + t/2 - 1$. (This is a bit tricky; it suffices if you just show there is such a coherent sheaf.)
Exercise 111.45.5. In Situation 111.45.1 assume $n \geq 2$ and $k$ algebraically closed. Let $C \subset X$ be an integral closed subscheme of dimension $1$. In other words, $C$ is a projective curve. Let $d t + e$ be the Hilbert polynomial of $\mathcal{O}_ C$ viewed as a coherent sheaf on $X$.
Give an upper bound on $e$. (Hints: Use that $\mathcal{O}_ C(t)$ only has cohomology in degrees $0$ and $1$ and study $H^0(C, \mathcal{O}_ C)$.)
Pick a global section $s$ of $\mathcal{O}_ X(1)$ which intersects $C$ transversally, i.e., such that there are pairwise distinct closed points $c_1, \ldots , c_ r \in C$ and a short exact sequence
\[ 0 \to \mathcal{O}_ C \xrightarrow {s} \mathcal{O}_ C(1) \to \bigoplus \nolimits _{i = 1, \ldots , r} k_{c_ i} \to 0 \]
where $k_{c_ i}$ is the skyscraper sheaf with value $k$ in $c_ i$. (Such an $s$ exists; please just use this.) Show that $r = d$. (Hint: twist the sequence and see what you get.)
Twisting the short exact sequence gives a $k$-linear map $\varphi _ t : \Gamma (C, \mathcal{O}_ C(t)) \to \bigoplus _{i = 1, \ldots , d} k$ for any $t$. Show that if this map is surjective for $t \geq d - 1$.
Give a lower bound on $e$ in terms of $d$. (Hint: show that $H^1(C, \mathcal{O}_ C(d - 2)) = 0$ using the result of (3) and use vanishing.)
Exercise 111.45.6. In Situation 111.45.1 assume $n = 2$. Let $s_1, s_2, s_3 \in \Gamma (X, \mathcal{O}_ X(2))$ be three quadric equations. Consider the coherent sheaf
\[ \mathcal{F} = \mathop{\mathrm{Coker}}\left(\mathcal{O}_ X(-2)^{\oplus 3} \xrightarrow {s_1, s_2, s_3} \mathcal{O}_ X\right) \]
List the possible Hilbert polynomials of such $\mathcal{F}$. (Try to visualize intersections of quadrics in the projective plane.)
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