## Tag `0DCD`

## 102.44. Cohomology and Hilbert polynomials

Situation 102.44.1. 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 polynomialof $\mathcal{F}$ 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 26.10.1. In Varieties, Subsection 32.34.12 we have proved the Hilbert polynomial is a polynomial in $t$.Exercise 102.44.2. In Situation 102.44.1.

- 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 102.44.3. In Situation 102.44.1 assume $n \geq 1$. Find a coherent sheaf whose Hilbert polynomial is $t - 101$.

Exercise 102.44.4. In Situation 102.44.1 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 102.44.5. In Situation 102.44.1 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 102.44.6. In Situation 102.44.1 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.)

The code snippet corresponding to this tag is a part of the file `exercises.tex` and is located in lines 4274–4376 (see updates for more information).

```
\section{Cohomology and Hilbert polynomials}
\label{section-cohomology-hilbert-polynomials}
\begin{situation}
\label{situation-hilbert-polynomial}
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 {\it 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 \ref{constructions-definition-twist}.
In Varieties, Subsection \ref{varieties-subsection-hilbert} we have
proved the Hilbert polynomial is a polynomial in $t$.
\end{situation}
\begin{exercise}
\label{exercise-hilbert-pol-easy}
In Situation \ref{situation-hilbert-polynomial}.
\begin{enumerate}
\item If $P(t)$ is the Hilbert polynomial of $\mathcal{F}$,
what is the Hilbert polynomial of $\mathcal{F}(-13)$.
\item If $P_i$ is the Hilbert polynomial of $\mathcal{F}_i$,
what is the Hilbert polynomial of $\mathcal{F}_1 \oplus \mathcal{F}_2$.
\item 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
$\mathcal{F}$?
\end{enumerate}
\end{exercise}
\begin{exercise}
\label{exercise-find-given-hilbert-pol-dim-1}
In Situation \ref{situation-hilbert-polynomial} assume $n \geq 1$.
Find a coherent sheaf whose Hilbert polynomial is $t - 101$.
\end{exercise}
\begin{exercise}
\label{exercise-find-given-hilbert-pol-dim-2}
In Situation \ref{situation-hilbert-polynomial} 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.)
\end{exercise}
\begin{exercise}
\label{exercise-bound-genus-in-degree}
In Situation \ref{situation-hilbert-polynomial} 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 $X$.
\begin{enumerate}
\item Give an upper bound on $e$. (Hints: Use that
$\mathcal{O}_C(t)$ only has cohomology in degrees $0$ and $1$
and study $H^0(C, \mathcal{O}_C)$.)
\item 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.)
\item 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 $t \geq d - 1$.
\item 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.)
\end{enumerate}
\end{exercise}
\begin{exercise}
\label{exercise-three-quadrics-in-plane}
In Situation \ref{situation-hilbert-polynomial} 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
$$
\mathcal{F} = \Coker\left(\mathcal{O}_X(-2)^{\oplus 3}
\xrightarrow{s_1, s_2, s_3} \mathcal{O}_X\right)
$$
List the possible Hilbert polynomials of such $\mathcal{F}$.
(Try to visualize intersections of quadrics in the projective plane.)
\end{exercise}
```

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