Miyaoka's bound for conics on K3 surfaces and beyond

TL;DR

The paper establishes that Miyaoka's bound on the number of conics (and more generally smooth degree-d rational curves) on degree-2h K3 surfaces is sharp for infinitely many polarization degrees, by constructing explicit K3 models with 24 such curves when h≥2d(2d+3)−2 and d or h is even. The approach combines elliptic fibrations, Mordell–Weil lattice theory, and Noether–Lefschetz-type lattice enhancements to produce very ample polarizations that realize 24 disjoint rational curves, and it treats both even and odd d, across characteristic zero and p≠2,3. In positive characteristic, the authors supplement the geometric constructions with two explicit K3 surfaces that carry large Mordell–Weil lattices, enabling the same conclusions for p≠2,3. Together, these results provide a comprehensive picture of the asymptotic behavior of low-degree rational curves on high-degree K3 surfaces and confirm the sharpness of Miyaoka’s bound in broad settings.

Abstract

We show that Miyaoka's bound for the number of conics on a degree-2h K3 surface is attained for high h, and analogously for higher even degree (smooth) rational curves.

Theorems & Definitions (23)

• Theorem 1.1
• Corollary 1.2
• Proposition 3.1
• proof
• Remark 3.4
• Proposition 4.1
• proof
• Lemma 4.2
• Lemma 4.3
• proof : Proof of Lemma \ref{['lem:odd_d_va']}