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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)
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