GENERA OF CURVES ON A VERY GENERAL SURFACE IN P3 - Archive ouverte HAL
Article Dans Une Revue International Mathematics Research Notices Année : 2015

GENERA OF CURVES ON A VERY GENERAL SURFACE IN P3

Résumé

In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d\geqslant 5$ in ${\mathbb P}^ 3$ (the cases $d\leqslant 4$ are well known). For all $d\geqslant 4$ we introduce the set ${\rm Gaps}(d)$ of all non--negative integers which are not realized as geometric genera of irreducible curves on a very general surface of degree $d$ in ${\mathbb P}^ 3$. We prove that ${\rm Gaps}(d)$ is finite and, in particular, that ${\rm Gaps}(5)= \{0,1,2\}$. The set ${\rm Gaps}(d)$ is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is ${\rm Gaps}_0(d):=\left[0, \; \frac{d(d-3)}{2} - 3\right]$. We show that the next one is ${\rm Gaps}_1(d):=\left[\frac{d^2-3d+4}{2}, \; d^2 - 2d - 9\right]$ for all $d\geqslant 6$.
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Dates et versions

hal-01062900 , version 1 (10-09-2014)

Identifiants

Citer

Ciro Ciliberto, Flaminio Flamini, Mikhail Zaidenberg. GENERA OF CURVES ON A VERY GENERAL SURFACE IN P3. International Mathematics Research Notices, 2015, International Mathematics Research Notices, 22 (1 January 2015), pp.12177-12205. ⟨10.1093/imrn/rnv055⟩. ⟨hal-01062900⟩

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