Counting Real Rational Curves on K3 Surfaces
نویسنده
چکیده
We provide a real analog of the Yau-Zaslow formula counting rational curves on K3 surfaces. ”But man is a fickle and disreputable creature and perhaps, like a chess-player, is interested in the process of attaining his goal rather than the goal itself.” Fyodor Dostoyevsky, Notes from the Underground.
منابع مشابه
Constructing Rational Curves on K3 Surfaces
We develop a mixed-characteristic version of the MoriMukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.
متن کاملDensity of Rational Curves on K3 Surfaces
Using the dynamics of self rational maps of elliptic K3 surfaces together with deformation theory, we prove that the union of rational curves is dense on a very general K3 surface and that the union of elliptic curves is dense in the 1st jet space of a very general K3 surface, both in the strong topology. The techniques developed here also lend themselves to applications to Abel-Jacobi images, ...
متن کاملA Simple Proof That Rational Curves on K3 Are Nodal
The Picard group of S is generated by two effective divisors C and F with C = −2, F 2 = 0 and C · F = 1. It can be realized as an elliptic fiberation over P with a unique section C, fibers F and λ = 2. It is the same special K3 surface used by Bryan and Leung in their counting of curves on K3 surfaces [B-L]. It is actually the attempt to understand their method that leads us to our proof. We wi...
متن کاملMirror Symmetry, Borcherd-Harvey-Moore Products and Determinants of the Calabi-Yau Metrics on K3 Surfaces
In the study of moduli of elliptic curves the Dedekind eta function η = q ∞ ∏ n=1 (1− q) , where q = e plays a very important role. We will point out the three main properties of η. 1. It is well known fact that η is an automorphic form which vanishes at the cusp. In fact η is the discriminant of the elliptic curve. 2. The Kronecker limit formula gives the explicit relations between the regular...
متن کاملK3 Surfaces, Rational Curves, and Rational Points
We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981. Mathematics Subj...
متن کامل