Abstract
AbstractA 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 $${\mathbb {P}}^r$$
P
r
, under the condition of being not contained in a surface of degree $$<s$$
<
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 $${\mathbb {P}}^r$$
P
r
, not contained in a hypersurface of degree $$<k$$
<
k
. In the present paper we examine the genus of curves C of degree d in $${\mathbb {P}}^r$$
P
r
not contained in quadrics (i.e. $$h^0({\mathbb {P}}^r, {\mathcal {I}}_C(2))=0$$
h
0
(
P
r
,
I
C
(
2
)
)
=
0
). When $$r=4$$
r
=
4
and $$r=5$$
r
=
5
, and $$d\gg 0$$
d
≫
0
, we exhibit a sharp upper bound for the genus. For certain values of $$r\ge 7$$
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.
Funder
Università degli Studi di Roma Tor Vergata
Publisher
Springer Science and Business Media LLC
Reference15 articles.
1. Ballico, E., Ellia, P.: On projections of ruled and veronese surfaces. J. Algebra 121, 477–487 (1989)
2. Chiantini, L., Ciliberto, C.: A few remarks on the lifting problem. Astérisque 218, 95–109 (1993)
3. Chiantini, L., Ciliberto, C., Di Gennaro, V.: The genus of projective curves. Duke Math. J. 70(2), 229–245 (1993)
4. Chiantini, L., Ciliberto, C., Di Gennaro, V.: On the genus of projective curves verifying certain flag conditions. Bollettino U.M.I. 7, 701–732 (1996)
5. Di Gennaro, V.: Hierarchical structure of the family of curves with maximal genus verifying flag conidtions. Proc Am Math Soc 136(3), 791–799 (2008)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献