\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

103.44 Cohomology and Hilbert polynomials

Situation 103.44.1. 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 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 26.10.1. In Varieties, Subsection 32.34.13 we have proved the Hilbert polynomial is a polynomial in $t$.

Exercise 103.44.2. In Situation 103.44.1.

  1. If $P(t)$ is the Hilbert polynomial of $\mathcal{F}$, what is the Hilbert polynomial of $\mathcal{F}(-13)$.

  2. If $P_ i$ is the Hilbert polynomial of $\mathcal{F}_ i$, what is the Hilbert polynomial of $\mathcal{F}_1 \oplus \mathcal{F}_2$.

  3. 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 103.44.4. In Situation 103.44.1 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 103.44.5. In Situation 103.44.1 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 $X$.

  1. Give an upper bound on $e$. (Hints: Use that $\mathcal{O}_ C(t)$ only has cohomology in degrees $0$ and $1$ and study $H^0(C, \mathcal{O}_ C)$.)

  2. 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.)

  3. 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 $t \geq d - 1$.

  4. 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 103.44.6. In Situation 103.44.1 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.)

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