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

2025/07/15 by Choi, Youngook, Iliev, Hristo, Kim, Seonja · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.11304

Abstract

We show that for two classes of m-secant curves X ⊂ S, with m ≥ 2, where f : S = ℙ (O_Y ⊕ O_Y (E)) → Y and E is a non-special divisor on a smooth curve Y, the Tschirnhausen module E^\vee of the covering φ= f_|X : X → Y decomposes completely as a direct sum of line bundles. Specifically, we prove that: for X ∈ |O_S (mH)|, where H denotes the tautological divisor on S, one has E^\vee ≅ O_Y (-E) ⊕ ⋯ ⊕ O_Y (-(m-1)E) ; for X ∈ |O_S (mH + f^∗q))|, where q is a point on Y, E^\vee ≅ O_Y (-E-q) ⊕ ⋯ ⊕ 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 ⊂ ℙ^R induced by the tautological line bundle |O_S (H)|, where R = dim |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.

Citations

• Non-reduced components of the Hilbert scheme of curves using triple covers
• Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components
• On some components of Hilbert schemes of curves
• Components of the Hilbert Scheme of smooth projective curves using ruled surfaces
• Special scrolls whose base curve has general moduli
• Remarks on the Obstructedness of Cones Over Curves of Low Genus
• On the projective normality of complete linear series on an algebraic curve

Cited by

• Decomposition of the Tschirnhausen module for coverings on decomposable ℙ¹-bundles

Related

• Decomposition of the Tschirnhausen module for coverings on decomposable ℙ¹-bundles
• On some components of Hilbert schemes of curves
• Components of the Hilbert Scheme of smooth projective curves using ruled surfaces
• Components of the Hilbert Scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components
• Non-reduced components of the Hilbert scheme of curves using triple covers