Counting Real Rational Curves onK3 Surfaces

Abstract

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.

Qualitative aspects of counting real rational curves on real K3 surfaces

“…A couple of other instances of real K3 surfaces where the lower bound for the number of real rational curves given by |w g | is optimal were already pointed in our previous paper [15]. One such example was the case of Harnack surfaces of degree 4 in P 3 .…”

“…where e C = 24 is the Euler characteristic of X. As we proved in [15], w g depends only on the Euler characteristic e R of the real part X R of X, and for e R fixed the generating function for w g is as follows:…”

“…In our previous paper [15], we considered the problem of counting real rational curves on primitively polarized real K3 surfaces, introduced an appropriate invariant Z-valued count and expressed the answer in a closed form, which can be viewed as a real version of the Yau-Zaslow formula (see Sect. 1 below).…”

On the lower bounds for real double Hurwitz numbers

“…So in real enumerative geometry, it is important to find the lower bounds for the real enumerative problems and to analyse the properties of these lower bounds. In the study of real algebraic curves in real surfaces passing through certain real points, signed count of real solutions is an effective way to provide such lower bounds [8,12,15,21,22]. This method is also valid in the study of counting covers.…”