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Let $X$ be a smooth projective curve of genus $g>0$ and $L \to X$ a line bundle of degree $d>2g-1.$ Let $\mathcal{I}_X$ be the ideal sheaf of $X \hookrightarrow L$ (embedded by the zero section.) Let $X_n$ be the subscheme defined by the ideal sheaf $\mathcal{I}_{X_n}:=\mathcal{I}^n_X.$

  1. How to compute the arithmetic genus of $X_2$ in terms of $g,d?$
  2. What about the arithmetic genus of $X_n?$

I would appreciate any help.

Added after Matthew Emerton's hint:

Here is my computation for $p_a(X_2).$

Consider the following SES of $\mathcal{O}_L$-modules;

$$0\to \mathcal{I}_X/\mathcal{I}^2_X \to \mathcal{O}_L/\mathcal{I}^2_X \to \mathcal{O}_L/\mathcal{I}_X \to 0$$

Indeed, $\mathcal{O}_L/\mathcal{I}_X$ and $\mathcal{O}_L/\mathcal{I}^2_X$ can be identified with $i_{\star} \mathcal{O}_X$ and $j_{\star} \mathcal{O}_{X_2},$ respectively, where $i_{\star} \mathcal{O}_X$ and $j_{\star} \mathcal{O}_{X_2}$ are extensions by zero outside of $X$ and $X_2$ with the inclusions $i:X \hookrightarrow L$ and $j: X_2 \hookrightarrow L.$ These identifications are useful, because $H^i(X,\mathcal{O}_X) \cong H^i(L, i_{\star}\mathcal{O}_X)$ by Hartshorne's lemma III, 2.10.

The associated long exact sequence is as follows

$$0\to H^0(\mathcal{I}_X/\mathcal{I}^2_X) \to H^0(\mathcal{O}_{X_2}) \to H^0(\mathcal{O}_{X}) \to$$

$$\to H^1(\mathcal{I}_X/\mathcal{I}^2_X) \to H^1(\mathcal{O}_{X_2}) \to H^1(\mathcal{O}_X) \to 0$$

Clearly the support of $\mathcal{I}_X/\mathcal{I}^2_X$ is $X.$ Intuitively, $\mathcal{I}_X/\mathcal{I}^2_X \cong \mathcal{L}^{\vee}$ where $\mathcal{L}$ is the locally free sheaf of sections of $L \to X,$ which I need to make it precise!

By Serre's duality, (or the special case of Kodaira's vanishing theorem for curves) for the assumption $\text{deg}L >2g-1,$ we get $H^0(\mathcal{L}^{\vee})=0,$ hence by Riemann-Roch $h^0(\mathcal{L}^{\vee})-h^1(\mathcal{L}^{\vee})=\text{deg} \mathcal{L}^{\vee} +1-g,$ we obtain $h^1(\mathcal{L}^{\vee})=g+d-1.$ Also, $h^0(\mathcal{O}_X)=1$ since $X$ is smooth projective, $h^1(\mathcal{O}_X)=g.$

Since the alternating sum of $h^i$s is zero in the above long exact sequence, we will have, $p_a(X_2)=h^1(\mathcal{O}_{X_2})=2g+d-2+h^0(\mathcal{O}_{X_2}).$ Now,

What is $h^0(\mathcal{O}_{X_2})?$ where $X_2$ is a projective non-reduced curve. Is there a method to compute the global sections of non-reduce curves in general?

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1 Answer

There is a filtration $$\mathcal O_L/I_X^{n+1} \supset I_X/I_X^{n+1} \supset I_X^2/I_X^{n+1} \supset \cdots \supset I_X^n/I_X^{n+1} \supset 0.$$ This induces a collection of short exact sequences $$0 \to I_X^{i+1}/I_X^{n+1} \to I_X^i/I_X^{n+1} \to I_X^i/I_X^{i+1} \to 0,$$ and so proceeding inductively, your question comes down to computing the cohomology of $I_X^i/I_X^{i+1}$ on $X$. (For this, it helps to remember that $X$ and $X_n$ are the same underlying topological space.)

This quotient is actually an invertible sheaf, and is the $i$th tensor power of $I_X/I_X^2$ (the conormal bundle of $X$ in $L$). So you should begin by computing this conormal bundle (which will admit a description in terms of the invertible sheaf attached to $L$).

Can you fill in the details?

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Dear @Matt E, intuitively, I think, the normal bundle on $X$ is isomorphic to $\mathcal{L}$ where $\mathcal{L}$ is the loc. free sheaf of sections of $L \to X.$ How can I make this precise? –  Ehsan M. Kermani Mar 11 '13 at 19:16
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