Logarithmic Equigeneric Smoothing and Maximal Nodal Degenerations

TL;DR

This paper extends classical deformation theory to study equisingular deformations of algebraic curves and surfaces, especially in non-generic nodal situations, providing criteria for the existence of minimal singularities and maximal degenerations.

Contribution

It develops deformation--theoretic criteria for minimal singularities and maximal degenerations, extending Severi theory to non-generic nodal behaviors and degenerations with normal crossings.

Findings

Deformation criteria ensure existence of minimal singularities.

Maximality results relate singularities to global deformation directions.

Applications to refined Severi counts via tropical methods.

Abstract

We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic criteria ensuring the existence of deformations with isolated singularities of minimal type, including cusps on curves and ordinary double points on curves and surfaces in threefolds. Under unobstructedness and surjectivity assumptions for natural global--to--local maps of normal bundles, we prove maximality results showing that the number of such singularities is governed by the global realizability of equisingular deformation directions rather than by numerical invariants alone. Logarithmic semiregularity allows these results to persist in degenerations with normal crossings special fibers. We further explain how these singularities arise as boundary phenomena of equigeneric Severi strata and outline applications to refined Severi counts via logarithmic and tropical methods.