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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: $\sum_{i=1}^n L_{X'}[i] - L_X[i] = \chi(\mathcal{S})$.
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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 $|\mathcal{O}_X|_n \leq -N$.

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  • $\begingroup$ 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 $\deg F = \deg F' + n$. $\endgroup$ – windsheaf Jul 23 at 9:46
  • $\begingroup$ (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. $\endgroup$ – windsheaf Jul 23 at 9:46

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