Gonality of curves on fundamental loci of first order congruences (Appendix to article of Ein, Lazarsfeld and Ullery) (original) (raw)

A note on gonality of curves on general hypersurfaces

Bollettino dell'Unione Matematica Italiana, 2017

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces X ⊂ P n+1. After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome "Tor Vergata" on January 11th-15th, 2016. In particular, we obtained that if X ⊂ P n+1 is a very general hypersurface of degree d 2n + 2, the least gonality of a curve C ⊂ X passing through a general point of X is gon(C) = d − √ 16n+1−1 2 , apart from some exceptions we list. The research leading to these results has received funding from the European Research Council under the

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.

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.

On the canonical degrees of curves in varieties of general type

Geometric and Functional Analysis, 2012

In this paper, we work in the framework of complex analytic varieties; without contrary mention, varieties are assumed to be irreducible (and reduced). If C is a projective curve, we let g C be its geometric genus (namely, the genus of its desingularization) and χ(C) = 2 − 2g C its geometric Euler characteristic; we also write deg C L for the degree of a line bundle L on C.

On postulation of curves in ?4

Mathematische Zeitschrift, 1985

We feel it is interesting to know how many hypersurfaces of a certain degree contain a given curve C in lP". In this paper we continue our study ([1, 2]) about the postulation of "general" curves embedded in a projective space.

On Property NpN_pNp for algebraic curves

Kodai Mathematical Journal, 2000

Let C = C x U U C s be a reduced, but reducible, curve and let Se in Pic(C) be very ample. Here we give conditions on deg(j£?|c,) insuring that the embedding of C induced by j£f satisfies Property N p .