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On gonality, scrolls, and canonical models of non-Gorenstein curves. (English) Zbl 1446.14016
Let $$C$$ be a curve (i.e. an integral and complete one-dimensional scheme over an algebraically closed field) of (arithmetic) genus $$g$$ and let $$C'\subseteq {\mathbb P}^{g-1}$$ be its canonical model. In this paper the authors study the relation between the gonality of $$C$$ and the dimension of a rational normal scroll $$S$$ where $$C'$$ can lie on, in particular when $$C$$ is singular, or even non-Gorenstein, in which case $$C'\ncong C$$. First, they analyze how to get an inclusion $$C'\subset S$$ from any pencil on $$C$$, in particular they get that $$S$$ is $$(d-1)$$-dimensional if $$C$$ is $$d$$-gonal, thus extending to any gonality results by R. Rosa and K.-O. Stöhr [J. Pure Appl. Algebra 174, No. 2, 187–205 (2002; Zbl 1059.14038)]. They also give an upper bound for the dimension of the singular set of $$S$$ in terms of some invariants of the pencil, and look for sufficient conditions for $$S$$ to be in fact singular. Then, in an opposite direction, they assume that $$C'$$ lies on a given scroll $$S$$ with prescribed dimension $$d$$ and intersection number $$l$$ with a generic fiber of $$S$$; varying $$l$$, they are able to relate properties of $$C$$, such as gonality and the kind of its singularities, with $$d$$ and other invariants of $$S$$. This leads to a generalization to arbitrary d of some results by D. Lara et al. [Int. J. Math. 27, No. 5, Article ID 1650045, 30 p. (2016; Zbl 1357.14040)]. At the end, they consider rational monomial curves and prove that such curves have gonality $$d$$ if and only if their canonical model lies on a $$(d -1)$$-fold scroll, and does not lie on any scroll of smaller dimension.
##### MSC:
 14H20 Singularities of curves, local rings 14H45 Special algebraic curves and curves of low genus 14H51 Special divisors on curves (gonality, Brill-Noether theory)
##### Keywords:
non-Gorenstein curve; canonical model; gonality; scrolls
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##### References:
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