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.)

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