Equisingular Deformations of Curves and Surfaces in Threefolds
Mounir Nisse
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.