Affiliation:
1. Institute of Algebraic Geometry , Gottfried Wilhelm Leibniz Universität Hannover , Welfengarten 1, 30167 Hannover , Germany
Abstract
Abstract
Let
X
⊂
ℙ
4
{X\subset\mathbb{P}^{4}}
be a very general hypersurface of degree
d
≥
6
{d\geq 6}
.
Griffiths and Harris conjectured in 1985 that the degree of every curve
C
⊂
X
{C\subset X}
is divisible by d.
Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of d.
Building on Kollár’s method, we prove this conjecture for infinitely many d, the smallest one being
d
=
5005
{d=5005}
.
The set of these degrees d has positive density.
We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over
ℚ
{\mathbb{Q}}
that satisfy the conjecture.
Subject
Applied Mathematics,General Mathematics
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