On gonality, scrolls, and canonical models of non-Gorenstein curves

Abstract

Let C be an integral and projective curve; and let C be its canonical model. We study the relation between the gonality of C and the dimension of a rational normal scroll S where C can lie on. We are mainly interested in the case where C is singular, or even non-Gorenstein, in which case C C. We first analyze some properties of an inclusion C ⊂ S when it is induced by a pencil on C. Afterwards, in an opposite direction, we assume C lies on a certain scroll, and check some properties C may satisfy, such as gonality and the kind of its singularities. At the end, we prove that a rational monomial curve C has gonality d if and only if C lies on a (d − 1)-fold scroll.

On the normal sheaf of Gorenstein curves

“…The birational morphism 𝜋 is such that each fiber of 𝜗 : P(ℰ) → P 1 is mapped to a ruling. Moreover, one can show that Pic(P(ℰ)) = Z H ⊕ Z R, where In [LMS19] the authors show that the canonical model of any 𝑘-gonal singular curve 𝐶 can be embedded in a (𝑘 − 1)-fold scroll 𝑆. In addition, they also show that the possible base points of a g 1 𝑘 are exactly those lying on the vertex of 𝑆.…”

“…We also note that singular curves may admit linear systems of degrees bigger than ⌊(𝑔 + 3)/2⌋, c.f. [LMS19].…”

“…has genus 𝑔 and degree 2𝑔 − 2. More recently, Kleimann & Martins in [KM09], followed by Lara, Martins & Souza in [LMS19], shown that if a singular curve has gonality 𝑘, then its canonical model lies on a (𝑘 −1)-fold rational normal scroll of degree 𝑔 − 𝑘 + 1. In particular, assuming that 𝐶 is Gorenstein, its canonical model coincides with the one given by the dualizing sheaf.…”

“…In the last decades many techniques and notions for singular curves have been carried out, such as gonality, canonical models, Petri's analysis, and Max Noether Theorem, see [LMS19], [CF18], [Sto93-2] and [CFV18]. In this thesis, we will study the stability of the normal sheaves on singular curves, starting from Gorenstein ones, and trying to avoid the normalization of these curves.…”

Characterizing gonality for two-component stable curves

“…There are a few possible ways to extend the notion of gonality to singular curves. One way is to simply take the same definition as in the smooth case, as is done in [16] and [14]. The drawback here is that this definition does not work very well in families, since not all curves that are limits of smooth k-gonal ones have maps of degree k to P 1 .…”