Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

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?

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.