A note on gonality of curves on general hypersurfaces (original) (raw)
Related papers
Gonality of curves on general hypersurfaces
Journal de Mathématiques Pures et Appliquées, 2019
This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if X ⊂ P n+1 is a hypersurface of degree d n + 2, and if C ⊂ X is an irreducible curve passing through a general point of X, then its gonality verifies gon(C) d − n, and equality is attained on some special hypersurfaces. We prove that if X ⊂ P n+1 is a very general hypersurface of degree d 2n + 2, the least gonality of an irreducible curve C ⊂ X passing through a general point of X is gon(C) = d − √ 16n+1−1 2 , apart from a series of possible exceptions, where gon(C) may drop by one.
The gonality theorem of Noether for hypersurfaces
Journal of Algebraic Geometry, 2013
It is well known since Noether that the gonality of a smooth curve C ⊂ P 2 of degree d ≥ 4 is d − 1. Given a k-dimensional complex projective variety X, the most natural extension of gonality is probably the degree of irrationality, that is the minimum degree of a dominant rational map X P k . In this paper we are aimed at extending the assertion on plane curves to smooth hypersurfaces in P n in terms of degree of irrationality. We prove that both surfaces in P 3 and threefolds in P 4 of sufficiently large degree d have degree of irrationality d − 1, except for finitely many cases we classify, whose degree of irrationality is d − 2. To this aim we use Mumford's technique of induced differentials and we shift the problem to study first order congruences of lines in P n . In particular, we also slightly improve the description of such congruences in P 4 and we provide a bound on degree of irrationality of hypersurfaces of arbitrary dimension.
arXiv (Cornell University), 2015
This short paper is an appendix to [6]. We prove an existence result for families of curves having low gonality, and lying on fundamental loci of first order congruences of lines in P n+1. As an application, we follow the ideas of the main paper, and we present a slight refinement of a theorem included in it. In particular, we show that given a very general hypersurface X ⊂ P n+1 of degree d ≥ 3n − 2 ≥ 7, and a dominant rational map f : X P n , then deg(f) ≥ d − 1, and equality holds if and only if f is the projection from a point of X.
Curves on Rational Surfaces with Hyperelliptic Hyperplane Sections
In this article we study, given a pair of integers (d, g), the problem of existence of a smooth, irreducible, non-degenerate curve in P n of degree d and genus g (the Halphen-Castelnuovo problem). We define two domains from the (d, g)-plane, D n 1 and D n 2 , and we prove that there is no gap in D n 1 . This follows constructing curves on some rational surfaces with hyperelliptic hyperplane sections and from some previous theorems of Ciliberto, Sernesi and of the author. Moreover, in the last section, based on some results of Horrowitz, Ciliberto, Harris, Eisenbud, we conjecture that D n 2 is the right lacunary domain.
0 Fermat Hypersurfaces and Subcanonical Curves
2016
We show that any Fermat hypersurface of degree s + 2 is apolar to a s-subcanonical (s + 2)-gonal projectively normal curve, and vice versa. Moreover, we extend the classical Enriques-Petri Theorem to s-subcanonical projectively normal curves, proving that such a curve is (s + 2)-gonal if and only if it is contained in a rational normal surface.
Fermat hypersurfaces and Subcanonical curves
Arxiv preprint arXiv:0908.0522, 2009
Abstract: We extend the classical Enriques-Petri Theorem to $ s −subcanonicalprojectivelynormalcurves,provingthatsuchacurveis-subcanonical projectively normal curves, proving that such a curve is −subcanonicalprojectivelynormalcurves,provingthatsuchacurveis(s+ 2) −gonalifandonlyifitiscontainedinasurfaceofminimaldegree.Moreover,weshowthatanyFermathypersurfaceofdegree-gonal if and only if it is contained in a surface of minimal degree. Moreover, we show that any Fermat hypersurface of degree −gonalifandonlyifitiscontainedinasurfaceofminimaldegree.Moreover,weshowthatanyFermathypersurfaceofdegree ...
Brill–Noether theory of curves on Enriques surfaces I: the positive cone and gonality
Mathematische Zeitschrift, 2009
We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov-Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality. 1
An Extension of the Cartan–Nochka Second Main Theorem for Hypersurfaces
International Journal of Mathematics, 2011
In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in ℂPn relative to a possibly degenerate set of hyperplanes. In this paper, we generalize Nochka's theorem to the case of curves in a complex projective variety intersecting hypersurfaces in subgeneral position. Further work will be needed to determine the optimal notion of subgeneral position under which this result can hold, and to lower the effective truncation level which we achieved.