A classical problem in the theory of projective curves is the classification of all their possible genera in terms of the degree and the dimension of the space where they are embedded. Fixed integers r, d, s, Castelnuovo-Halphen's theory states a sharp upper bound for the genus of a non-degenerate, reduced and irreducible curve of degree d in P-r, under the condition of being not contained in a surface of degree < s. This theory can be generalized in several ways. For instance, fixed integers r, d, k, one may ask for the maximal genus of a curve of degree d in P-r, not contained in a hypersurface of degree < k. In the present paper we examine the genus of curves C of degree d in p(r) not contained in quadrics (i.e. h(0) (P-r, I-C(2)) = 0). When r = 4 and r = 5, and d >> 0, we exhibit a sharp upper bound for the genus. For certain values of r >= 7, we are able to determine a sharp bound except for a constant term, and the argument applies also to curves not contained in cubics.

Di Gennaro, V. (2022). The genus of curves in P-4 and P-5 not contained in quadrics. RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO [10.1007/s12215-022-00788-x].

The genus of curves in P-4 and P-5 not contained in quadrics

Di Gennaro, V
2022

Abstract

A classical problem in the theory of projective curves is the classification of all their possible genera in terms of the degree and the dimension of the space where they are embedded. Fixed integers r, d, s, Castelnuovo-Halphen's theory states a sharp upper bound for the genus of a non-degenerate, reduced and irreducible curve of degree d in P-r, under the condition of being not contained in a surface of degree < s. This theory can be generalized in several ways. For instance, fixed integers r, d, k, one may ask for the maximal genus of a curve of degree d in P-r, not contained in a hypersurface of degree < k. In the present paper we examine the genus of curves C of degree d in p(r) not contained in quadrics (i.e. h(0) (P-r, I-C(2)) = 0). When r = 4 and r = 5, and d >> 0, we exhibit a sharp upper bound for the genus. For certain values of r >= 7, we are able to determine a sharp bound except for a constant term, and the argument applies also to curves not contained in cubics.
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/03
English
Projective curve
Castelnuovo-Halphen theory
Quadric and cubic hypersurfaces
Veronese surface
Projection of a rational normal scroll surface
Maximal rank
Di Gennaro, V. (2022). The genus of curves in P-4 and P-5 not contained in quadrics. RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO [10.1007/s12215-022-00788-x].
Di Gennaro, V
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/306915
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