A note on gonality of curves on general hypersurfaces

Abstract

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

Multiplicative bounds for measures of irrationality on complete intersections

“…. , (e + 1)!q e−1 , d e ), the inegers q i are distinct primes, and 1 Next, degenerate X ′ to a union of varieties, with one component consisting of a very general complete intersection X ⊂ P e+2 of type (a 1 , . .…”

“…We will first use a dimension count to approximate C as a complete intersection curve. The key point is that if we can find an effective divisor V ∈ |O Y (ℓ)| of degree ℓ ≤ 1 3 a e passing through all of the points p i , then…”

“…gon(X) ≥ d − n. The central theme of [2] is that the positivity properties of the canonical bundle yield lower bounds for measures of irrationality. Using different methods, Bastianelli, Ciliberto, Flamini, and Supino [1] later computed cov. gon(X) ≈ d − 2…”

Measures of irrationality for hypersurfaces of large degree

“…By starting with some of the ideas used in the proof of Theorem C in § 3, the first author, Ciliberto, Flamini and Supino [BCFS17] have computed the covering gonality of a very general hypersurface of degree in almost all cases. Specifically, they show that The numerics here are essentially what one finds by looking for plane curves with singular points covering , as in Examples 1.7(iii) and (iv).…”

On irrationality of surfaces in P3

“…[24]). Besides, in the light of [4, Theorem A] and [6,Theorem 3.3], it would be interesting to understand the behavior of those invariants when X ⊂ P n+1 is a very general hypersurface of large degree and arbitrary dimension.…”