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

Abstract

We show that for two classes of $m$-secant curves $X\subset S$, with $m\ge 2$, 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 $\phi =f_{{|}_X}:X\to Y$ 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.