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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 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 32.34.12 we have proved the Hilbert polynomial is a polynomial in $t$.

Exercise 102.44.2. In Situation 102.44.1.

  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 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 $X$.

  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 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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