The Stacks project

Exercise 111.45.2. In Situation 111.45.1.

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 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 $X$.

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 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

List the possible Hilbert polynomials of such $\mathcal{F}$. (Try to visualize intersections of quadrics in the projective plane.)