Severi curves of rational elliptic surfaces

TL;DR

This work analyzes Severi curves $V(R,L)$ of rational bisections on a general rational elliptic surface $R$, proving they are connected and reduced and providing an upper bound on their geometric genus via quasi-modular data tied to Noether-Lefschetz theory. The authors construct a family of K3 surfaces from a RES, compute the Noether-Lefschetz generating series, and relate NL coefficients to the degree of the branch map $f_L:\widetilde{V}(L)\to \mathrm{Sym}^2(\mathbb{P}^1)$. They show $f_L$ is birational onto its image in many cases, enabling a plane-curve genus bound of $O(g^{12+\varepsilon})$, and propose a multiplicity conjecture for branch-point collisions that could yield sharper bounds, expressed via divisor-sum formulas of collision multiplicities. The results connect enumerative geometry of Severi curves to modular forms and lattice-theoretic Noether-Lefschetz data, with potential for precise degree formulas in $\mathbb{P}^2$ through a refined modular framework.

Abstract

We study Severi curves parametrizing rational bisections of elliptic fibrations associated to general pencils of plane cubics. Our main results show that these Severi curves are connected and reduced, and we give an upper bound on their geometric genus using quasi-modular forms. We conjecture that these Severi curves are eventually reducible, and we formulate a precise conjecture for their degrees in $\mathbb{P}^2$, featuring a divisor sum formula for collision multiplicities of branch points.

Theorems & Definitions (116)

• Definition 1.1
• Conjecture 1.2
• Theorem \ref{thm:severi-connected-with-proof}
• Theorem \ref{thm:reducedseveri-with-proof}
• Theorem \ref{thm:genusbound-with-proof}
• Definition \ref{thm:genusbound-with-proof}
• Proposition \ref{thm:genusbound-with-proof}
• proof
• Lemma \ref{thm:genusbound-with-proof}
• proof