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A note on the Severi problem for toric surfaces

Lionel Lang, Ilya Tyomkin

TL;DR

This work advances the Severi problem on toric surfaces by (i) constructing two reducible families—quotients and kite polygons—and (ii) introducing two invariants (a sublattice pair and a kite-specific signature) to distinguish and bound irreducible components of $V_{g,\mathcal{L}}^{\mathrm{irr}}$. The authors develop both tropical and topological methods, including liftings of tropicalizations via $M$-integral triangulations and refined tropicalizations, to produce explicit components realizing admissible invariants. They prove general lower bounds on the number of components in terms of lattice data and verify sharpness in genus one for kite cases; the appendix links the kite Severi problem to the classification of Laurent polynomials by passport, suggesting broader connections to polynomial topology. Overall, the paper advances a structural, invariant-based framework for understanding when Severi varieties on toric surfaces decompose, and lays groundwork for a wider classification governed by lattice-theoretic and tropical data with potential polynomial-topology intersections.

Abstract

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of [Lan19, Tyo13, Tyo14], and provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

A note on the Severi problem for toric surfaces

TL;DR

This work advances the Severi problem on toric surfaces by (i) constructing two reducible families—quotients and kite polygons—and (ii) introducing two invariants (a sublattice pair and a kite-specific signature) to distinguish and bound irreducible components of . The authors develop both tropical and topological methods, including liftings of tropicalizations via -integral triangulations and refined tropicalizations, to produce explicit components realizing admissible invariants. They prove general lower bounds on the number of components in terms of lattice data and verify sharpness in genus one for kite cases; the appendix links the kite Severi problem to the classification of Laurent polynomials by passport, suggesting broader connections to polynomial topology. Overall, the paper advances a structural, invariant-based framework for understanding when Severi varieties on toric surfaces decompose, and lays groundwork for a wider classification governed by lattice-theoretic and tropical data with potential polynomial-topology intersections.

Abstract

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of [Lan19, Tyo13, Tyo14], and provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

Paper Structure

This paper contains 20 sections, 19 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Discussion
  4. The idea of the proofs
  5. Preliminaries, notation, and conventions
  6. Toric geometry
  7. Severi varieties
  8. Conventions
  9. The case of genus one
  10. The invariants
  11. The sublattices associated to the irreducible components
  12. The topological point of view
  13. The tropical point of view
  14. The nodal partition and the signature
  15. Tropical proofs of the main results
  16. ...and 5 more sections

Key Result

Theorem A

If $g\geqslant 1$, then the number of irreducible components of the Severi variety $V_{g,\mathcal{L}}^\mathrm{irr}$ is bounded from below by the number of affine sublattices $M\subseteq\mathbb{Z}^2$ for which the following two conditions hold: (a) $\partial\Delta\cap M=\partial\Delta\cap\mathbb{Z}^2

Theorems & Definitions (41)

  • Theorem A
  • Theorem B
  • Theorem C
  • Proposition 3.1
  • Remark 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 31 more