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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:
  Andreotti, A., Mayer, A.L.: On period relations for abelian integrals on algebraic curves. Annali della Scuolla Normale Superiore di Piza 21(2), 189-238 (1967) · Zbl 0222.14024  Babbage, D.W.: A note on the quadrics through a canonical curve. J. Lond. Math. Soc. 14, 310-315 (1939) · JFM 65.1398.03  Bresinsky, H.: Monomial space curves in $\mathbb{A}^3$ A3 as set-theoretic complete intersections. Proc. Am. Math. Soc. 75, 23-24 (1979) · Zbl 0395.14015  Barucci, V., Fröberg, R.: One-dimensional almost Gorenstein rings. J. Algebra 188, 418-442 (1997) · Zbl 0874.13018  Brundu, M., Sacchiero, G.: Stratification of the moduli space of four-gonal curves. Proc. Edinb. Math. Soc. 57(03), 631-686 (2014) · Zbl 1304.14031  Casnati, G., Ekedahl, T.: Covers of algebraic varieties. I. A general structure theorem, covers of degree 3,4 and Enriques surfaces. J. Algebraic Geom. 5(3), 439-460 (1996) · Zbl 0866.14009  Contiero, A., Stoehr, K.-O.: Upper bounds for the dimension of moduli spaces of curves with symmetric Weierstrass semigroups. J. Lond. Math. Soc. 88, 580-598 (2013) · Zbl 1288.14016  Contiero, A., Feital, L., Martins, R.V.: Max Noether theorem for integral curves. J. Algebra 494, 111-136 (2018) · Zbl 1386.14112  Coppens, M.: Free linear systems on integral Gorenstein curves. J. Algebra 145, 209-218 (1992) · Zbl 0770.14002  Cotterill, E., Feital, L., Martins, R. V.: Dimension counts for singular rational curves via semigroups. arXiv:1511.08515v2 · Zbl 1394.14019  Cotterill, E., Feital, L., Martins, R.V.: Singular rational curves with points of nearly-maximal weight. J. Pure Appl. Algebra 222, 3448-3469 (2018). https://doi.org/10.1016/j.jpaa.2017.12.017 · Zbl 1394.14019  Eisenbud, D., Harris, J., Koh, J., Stillmann, M.: Determinantal equations for curves of high degree. Am. J. Math. 110, 513-539 (1988) · Zbl 0681.14027  Eisenbud, D., Harris, J.: On varieties of minimal degree. Proc. Symp. Pure Math. 46, 3-13 (1987)  Enriques, F.: Sulle curve canoniche di genera $p$ p cello spazio a $p-1$ p-1 dimensioni. Rend. Accad. Sci. Ist. Bologna 23, 80-82 (1919)  Herzog, J.: Generators and relations of abelian semigroups and semigroup rings. Manuscr. Math. 3, 175-193 (1970) · Zbl 0211.33801  Hotchkiss, J., Ullery, B.: The gonality of complete intersection curves. arXiv:1706.08169  Jäger, J.: Längeberechnungen und Kanonische Ideale in Eindimensionalen Ringen. Arch. Math. 29, 504-512 (1977) · Zbl 0374.13006  Kleiman, S.L., Martins, R.V.: The canonical model of a singular curve. Geom. Dedicata 139, 139-166 (2009) · Zbl 1172.14019  Lara, D., Marchesi, S., Martins, R.V.: Curves with canonical models on scrolls. Int. J. Math. 27(5), 1650045-1-30 (2016) · Zbl 1357.14040  Martins, R.V.: On trigonal non-Gorenstein curves with zero Maroni invariant. J. Algebra 275, 453-470 (2004) · Zbl 1060.14036  Matsuoka, T.: On the degree of singularity of one-dimensional analytically irreducible noetherian rings. J. Math. Kyoto Univ. 11, 485-491 (1971) · Zbl 0224.13017  Miró-Roig, R.M.: The representation type of rational normal scrolls. Rend. Circ. Mat. Palermo 62, 153-164 (2012) · Zbl 1268.14014  Reid, M.: Chapters on algebraic surfaces. 6 Feb 1996. Lectures of a summer programm Park City, UT. arXiv:alg-geom/9602006v1 (1993)  Rosa, R., Stöhr, K.-O.: Trigonal Gorenstein curves. J. Pure Appl. Algebra 174, 187-205 (2002) · Zbl 1059.14038  Rosenlicht, M.: Equivalence relations on algebraic curves. Ann. Math. 56, 169-191 (1952) · Zbl 0047.14503  Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Mathematische Annalen 275, 105-137 (1986) · Zbl 0578.14002  Stöhr, K.-O.: On the poles of regular differentials of singular curves. Boletim da Sociedade Brasileira de Matemática 24, 105-135 (1993) · Zbl 0788.14020
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