# Relation between the decomposition invariants of a projective reduced curve and its normalization

Let $$X$$ be a reduced projective scheme over $$k$$ which is of pure dimension 1. Let $$\pi: X \to \mathbb{P}_k^1$$ be a finite (hence affine, surjective and flat) morphism of schemes having degree $$n$$. Since $$X$$ is Cohen-Macaulay, $$\pi_*\mathcal{O}_X$$ is a free $$\mathcal{O}_{\mathbb{P}_k^1}$$-module of finite rank $$n$$ and hence decomposes into a direct sum of Serre's twisted sheaves: $$\pi_*\mathcal{O}_X \cong \bigoplus_{i=1}^n \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_X|_i)$$ The integers $$|\mathcal{O}_X|_1 \geq \ldots \geq |\mathcal{O}_X|_n$$ are uniquely determined by $$\mathcal{O}_X$$. Now we have a similar situation for the structure sheaf of any irreducible component $$X_i$$ of $$X$$: The closed immersion $$j_i: X_i \to X$$ is a finite morphism and hence $$\pi_i = \pi \circ j_i: X_i \to X \to \mathbb{P}_k^1$$ is also finite of degree $$n_i < n$$ if $$X_i \neq X$$. Hence $$(\pi_i)_*\mathcal{O}_{X_i} \cong \pi_* ((j_i)_* \mathcal{O}_{X_i}) \cong \pi_* (\mathcal{O}_X / \mathcal{I}_i) \cong \bigoplus_{j=1}^{n_i} \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_{X_i}|_j)$$ where $$\mathcal{I}_{X_i}$$ denotes the ideal sheaf cutting out $$X_i$$ in $$X$$.

Since $$X$$ is reduced, we have a finite morphism $$X' = \bigoplus_{i=1}^m X'_i \to X$$ where $$X'$$ is the normalization of $$X$$ (and $$X_i'$$ is the normalization of the component $$X_i$$) and a corresponding injective morphism $$\mathcal{O}_X \hookrightarrow \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i$$ with finite index $$\chi(\mathcal{S})$$ where $$\mathcal{S}$$ makes the following sequence exact $$0 \to \mathcal{O}_X \hookrightarrow \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i \to \mathcal{S} \to 0.$$ It is not hard to see that $$\pi_* \mathcal{O}_X \hookrightarrow \pi_*\left( \bigoplus_{i=1}^n \mathcal{O}_{X_i}/\mathcal{I}_i\right) \cong \bigoplus_{i=1}^{m} \bigoplus_{j=1}^{n_i} \mathcal{O}_{\mathbb{P}_k^1}(|\mathcal{O}_{X_i}|_j).$$

My question is: What are the relations between $$L_{X'} := (|\mathcal{O}_{X_i}|_j)_{i,j}$$ and $$L_X := (|\mathcal{O}_{X}|_\ell)_\ell$$ both arranged in descending order? To be more specific: Does $$L_{X'}[i] - L_X[i] \in O\left( \frac{\chi(\mathcal{S})}{n}\right)$$ hold, i.e. are the differences balanced. Does someone know a good read for this kind of situations or any references at all? Any good idea is also welcome.

What I do know so far:

1. $$L_{X} \leq L_{X'}$$, that is for all $$i=1,\ldots,n : L_{X}[i] \leq L_{X'}[i]$$,
2. $$\sum_{\ell=1}^n |\mathcal{O}_{X}|_\ell = \chi(\mathcal{O}_X) -n$$,
3. $$\sum_{i,j} |\mathcal{O}_{X_i}|_j = \sum_{i=1}^m (\chi(\mathcal{O}_{X_i}) -n_i) = \chi(\mathcal{O}_X) + \chi(\mathcal{S}) -n$$
4. Combining 2. and 3. we have: $$\sum_{i=1}^n L_{X'}[i] - L_X[i] = \chi(\mathcal{S})$$.

OK, I think there is no bound of the type you want. Namely, choose $$X$$ to be the nodal curve obtained by glueing $$n$$ copies $$X_1, \ldots, X_n$$ of the base $$\mathbf{P}^1$$ by glueing points as follows: first glue $$X_3, \ldots, X_n$$ each to $$X_1$$ in a single point, next glue $$X_2$$ to $$X_1$$ in $$N$$ distinct points. We assume the points we are glueing at map to $$n - 2 + N$$ pairwise distinct points in $$\mathbf{P}^1$$.
First we observe that $$|\mathcal{O}_{X_i}|_j = 0$$ always.
For a scheme $$Z$$ over $$\mathbf{P}^1$$ denote $$\mathcal{O}_Z(i)$$ the pullback of $$\mathcal{O}_{\mathbf{P}^1}(i)$$ to $$Z$$.
Denote $$X' \subset X$$ the union of $$X_1$$ and $$X_2$$. A computation show that $$H^1(X', \mathcal{O}_{X'}(N - 2))$$ is nonzero. Since $$\mathcal{O}_X(N - 2) \to \mathcal{O}_{X'}(N - 2)$$ is surjective and since $$X$$ is a curve, we see that $$H^1(\mathcal{O}_X(N - 2))$$ is nonzero. This implies that $$|\mathcal{O}_X|_n \leq -N$$.
• If I could follow your instructions correctly, then the curve $X'$ you described is of the form $X' = V_+(F') \subset \mathbb{P}_k^2$ with $F'$ being a homogeneous polynomial of degree $N+1$. And we obtain $X$ as $V_+(F) \subset \mathbb{P}_k^2$ where $F'$ divides $F$. In your notation, we thus have $\deg F = \deg F' + n$. – windsheaf Jul 23 at 9:46
• (continued) The intended bound is $-\chi(\mathcal{S})/\deg \pi = -\chi(\mathcal{O}_X)/\deg F$. Now since $X$ is a local complete intersection, we have $-\chi(\mathcal{O}_X) = p_a(X) -1 = \frac{1}{2}(\deg F -1)(\deg F-2)-1 \in O(\deg F^2)$. Moreover, $\deg \pi = \deg F$ and thus the intended bound is in $O(\deg F)$. But your argument is that we deduce $|X|_n \leq - \deg F' = \deg F - n \leq -\chi(\mathcal{S})/\deg \pi$ which is still in the intended order. – windsheaf Jul 23 at 9:46