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
This short paper concerns the existence of curves with low gonality on smooth\ud
hypersurfaces X in P^{n+1}. After reviewing a series of results on this topic, we report on a recent\ud
progress we achieved as a product of the Workshop Birational geometry of surfaces, held at\ud
University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained\ud
that if X is a very general hypersurface of degree d grater than or equal to 2n + 2, the least gonality of\ud
a curve C ⊂ X passing through a general point of X is explicitely given, apart\ud
from some exceptions we list
Consider the Fano scheme F k (Y ) parameterizing k-dimensional linear subspaces contained in a complete intersection Y ⊂ P m of multi-degree d = (d 1 , . . . , ds). It is known that, if t :for Y a general complete intersection as above, then F k (Y ) has dimension −t. In this paper we consider the case t > 0. Then the locus W d,k of all complete intersections as above containing a k-dimensional linear subspace is irreducible and turns out to have codimension t in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general [Y ] ∈ W d,k the scheme F k (Y ) is zero-dimensional of length one. This implies that W d,k is rational.
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