On the Tschirnhausen module of coverings of curves on decomposable ruled surfaces and applications

TL;DR

This paper studies the decomposition of the Tschirnhausen module for certain coverings of curves on decomposable ruled surfaces, enabling new insights into the geometry of these curves and their Hilbert schemes.

Contribution

It proves the complete decomposition of the Tschirnhausen module for specific classes of curves on ruled surfaces, leading to new results on the Hilbert scheme components.

Findings

Decomposition of Tschirnhausen module as a direct sum of line bundles.

Calculation of global sections of the normal bundle for embedded curves.

Construction of new families of Hilbert scheme components, including non-linearly normal and nonreduced components.

Abstract

We show that for two classes of $m$-secant curves $X \subset S$, with $m \geq 2$, where $f : S = \mathbb{P} (\mathcal{O}_Y \oplus \mathcal{O}_Y (E)) \to Y$ and $E$ is a non-special divisor on a smooth curve $Y$, the Tschirnhausen module $\mathcal{E}^{\vee}$ of the covering $\varphi = f_{|_X} : X \to Y$ decomposes completely as a direct sum of line bundles. Specifically, we prove that: for $X \in |\mathcal{O}_S (mH)|$, where $H$ denotes the tautological divisor on $S$, one has $ \mathcal{E}^{\vee} \cong \mathcal{O}_Y (-E) \oplus \cdots \oplus \mathcal{O}_Y (-(m-1)E) $; for $X \in |\mathcal{O}_S (mH + f^{\ast}q))|$, where $q$ is a point on $Y$, $ \mathcal{E}^{\vee} \cong \mathcal{O}_Y (-E-q) \oplus \cdots \oplus \mathcal{O}_Y (-(m-1)E-q) $ holds. This decomposition enables us to compute the dimension of the space of global sections of the normal bundle of the embedding $X \subset \mathbb{P}^R$ induced by the tautological line bundle $|\mathcal{O}_S (H)|$, where $R = \dim |\mathcal{O}_S (H)|$. As an application, we construct new families of generically smooth components of the Hilbert scheme of curves, including components whose general points correspond to non-linearly normal curves, as well as nonreduced components.