Qualitative aspects of counting real rational curves on real K3 surfaces

TL;DR

This paper investigates the growth rate and congruences of real rational curves on real K3 surfaces, showing that their count grows similarly to complex curves and identifying cases of sharp bounds.

Contribution

It establishes the asymptotic growth rate of real rational curves on real K3 surfaces and reveals congruences between real and complex counts, advancing understanding of real enumerative geometry.

Findings

Growth rate of real rational curves matches that of complex curves logarithmically.

Identifies instances where the lower bound for real curves is sharp.

Finds congruences between real and complex curve counts.

Abstract

We study qualitative aspects of the Welschinger-like $\mathbb Z$-valued count of real rational curves on primitively polarized real $K3$ surfaces. In particular, we prove that with respect to the degree of the polarization, at logarithmic scale, the rate of growth of the number of such real rational curves is, up to a constant factor, the rate of growth of the number of complex rational curves. We indicate a few instances when the lower bound for the number of real rational curves provided by our count is sharp. In addition, we exhibit various congruences between real and complex counts.