Equisingular Deformations of Curves and Surfaces in Threefolds

TL;DR

This paper extends classical deformation theory to study equisingular deformations of curves and surfaces in threefolds, providing criteria for the existence and maximality of certain singularities beyond generic cases.

Contribution

It develops deformation-theoretic criteria for equisingular deformations in threefolds, including non-generic nodal behavior, and establishes maximality results under certain unobstructedness conditions.

Findings

Criteria for existence of deformations with isolated singularities

Maximality results governed by global deformation directions

Applications to refined Severi counts and 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.