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 at least 5 in P^3 (the cases d at most 4 are well known). For all d at least 5 we introduce the set 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 P^3. We prove that Gaps(d) is finite and, in particular, that Gaps(5) = {0, 1, 2}. The set Gaps(d) is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is Gaps_0(d) := [0, (1/2) d(d−3)− 3]. We show that the next one is Gaps_1(d) := [(1/2) (d^2−3d+4), d^2 − 2d − 9] for all d at least 6.

Ciliberto, C., Flamini, F., Zaidenberg, M. (2015). Genera of curves on a very general surface in IP^3. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 22, 12177-12205 [10.1093/imrn/rnv055].

Genera of curves on a very general surface in IP^3

CILIBERTO, CIRO;FLAMINI, FLAMINIO
;
2015-11-01

Abstract

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 at least 5 in P^3 (the cases d at most 4 are well known). For all d at least 5 we introduce the set 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 P^3. We prove that Gaps(d) is finite and, in particular, that Gaps(5) = {0, 1, 2}. The set Gaps(d) is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is Gaps_0(d) := [0, (1/2) d(d−3)− 3]. We show that the next one is Gaps_1(d) := [(1/2) (d^2−3d+4), d^2 − 2d − 9] for all d at least 6.
nov-2015
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03 - GEOMETRIA
English
Con Impact Factor ISI
projective hypersurfaces; geometric genus; algebraic hyperbolicity
to be included in a volume
https://academic.oup.com/imrn/article/2015/22/12177/2357480
188642
Ciliberto, C., Flamini, F., Zaidenberg, M. (2015). Genera of curves on a very general surface in IP^3. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 22, 12177-12205 [10.1093/imrn/rnv055].
Ciliberto, C; Flamini, F; Zaidenberg, M
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/90297
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