Fix integers r, s(1),..., s(l) such that 1 <= l <= r-1 and s(l) >= r-l+1, and let C(r; s(1),..., s(l)) be the set of all integral, projective and nondegenerate curves C of degree s(1) in the projective space P-r, such that, for all i = 2,..., l, C does not lie on any integral, projective and nondegenerate variety of dimension i and degree < s(i). We say that a curve C satisfies the flag condition (r; s(1),..., s(l)) if C belongs to C(r; s(1),..., s(l)). Define G(r; s(1),..., s(l)) = max {pa(C) : C is an element of C( r; s(1),..., s(l))}, where p(a)(C) denotes the arithmetic genus of C. In the present paper, under the hypothesis s(1) >> center dot center dot center dot >> s(l), we prove that a curve C satisfying the flag condition (r; s(1),..., s(l)) and of maximal arithmetic genus pa(C) = G(r; s(1),..., s(l)) must lie on a unique flag such as C = V-s1(1) subset of V-s2(2) subset of center dot center dot center dot subset of V-sl(l) subset of P-r, where, for any i = 1,..., l, V-si(i) denotes an integral projective subvariety of P-r of degree s(i) and dimension i, such that its general linear curve section satisfies the flag condition (r-i+1; s(i),..., s(l)) and has maximal arithmetic genus G(r-i+1; s(i),..., s(l)). This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.

Di Gennaro, V. (2008). Hierarchical structure of the family of curves with maximal genus verifying flag conditions. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 136(3), 791-799 [10.1090/S0002-9939-07-09123-X].

Hierarchical structure of the family of curves with maximal genus verifying flag conditions

DI GENNARO, VINCENZO
2008

Abstract

Fix integers r, s(1),..., s(l) such that 1 <= l <= r-1 and s(l) >= r-l+1, and let C(r; s(1),..., s(l)) be the set of all integral, projective and nondegenerate curves C of degree s(1) in the projective space P-r, such that, for all i = 2,..., l, C does not lie on any integral, projective and nondegenerate variety of dimension i and degree < s(i). We say that a curve C satisfies the flag condition (r; s(1),..., s(l)) if C belongs to C(r; s(1),..., s(l)). Define G(r; s(1),..., s(l)) = max {pa(C) : C is an element of C( r; s(1),..., s(l))}, where p(a)(C) denotes the arithmetic genus of C. In the present paper, under the hypothesis s(1) >> center dot center dot center dot >> s(l), we prove that a curve C satisfying the flag condition (r; s(1),..., s(l)) and of maximal arithmetic genus pa(C) = G(r; s(1),..., s(l)) must lie on a unique flag such as C = V-s1(1) subset of V-s2(2) subset of center dot center dot center dot subset of V-sl(l) subset of P-r, where, for any i = 1,..., l, V-si(i) denotes an integral projective subvariety of P-r of degree s(i) and dimension i, such that its general linear curve section satisfies the flag condition (r-i+1; s(i),..., s(l)) and has maximal arithmetic genus G(r-i+1; s(i),..., s(l)). This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/03 - Geometria
English
Con Impact Factor ISI
projective curves
http://www.ams.org/journals/proc/2008-136-03/S0002-9939-07-09123-X/S0002-9939-07-09123-X.pdf
Di Gennaro, V. (2008). Hierarchical structure of the family of curves with maximal genus verifying flag conditions. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 136(3), 791-799 [10.1090/S0002-9939-07-09123-X].
DI GENNARO, V
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/8074
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