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Curves with canonical models on scrolls. (English) Zbl 1357.14040
Throughout, let \(C\) be a curve (i.e., an integral, complete, one-dimensional scheme) over an algebraically closed field of arithmetic genus \(g\). Let \(C'\subseteq {\mathbb P}^{g-1}\) be its canonical model which is defined by the global sections of the dualizing sheaf of \(C\). It is well-known so far that properties on trigonal Gorenstein curves can be deduced whenever its canonical model is contained in a surface scroll; e.g. [K.-O. Stöhr, J. Pure Appl. Algebra 135, No. 1, 93–105 (1999; Zbl 0940.14018)], [R. Rosa and K.-O. Stöhr, J. Pure Appl. Algebra 174, No. 2, 187–205 (2002; Zbl 1059.14038)].
In this paper the authors study the case where \(C\) is non-Gorenstein and \(C'\) is contained in a scroll surface. Here the concepts “nearly Gorenstein” and “arithmetically normal” become relevant according respectively to Theorems 5.10 and 4 in [S. L. Kleiman and R. V. Martins, Geom. Dedicata 139, 139–166 (2009; Zbl 1172.14019)]. Moreover, as looking at for examples, they consider rational monomial curves and show that for such a curve its canonical model is contained in a scroll surface if and only if the curve is trigonal. This leads to the question when a nonhyperelliptic curve can be characterized by its canonical model; in fact, this is worked out for the case of a nonhyperelliptic curve with at most one unibranched singular point. Finally they generalize some results in [F.-O. Schreyer, Math. Ann. 275, 105–137 (1986; Zbl 0578.14002)].

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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