Evgenii Shustin - Asymptotically proper bounds in the geometry of equisingular families of curves

Given a smooth complex algebraic hypersurface and an ample linear system on it, we consider the family of irreducible curves in this linear system having isolated singular points of given topological or analytic types, and we ask if this family is nonempty, smooth, irreducible and has expected dimension. We present sufficient conditions for the above ``good'' properties of equisingular families, expressed as upper bounds to the sums of certain invariants of singularities, and we show that these bounds are asymptotically proper, i.e. if the sum of the same singularity invariants exceeds the upper bound multiplied by an absolute constant, then the equisingular family is empty, or not smooth, or reducible, or has a nonexpected dimension.

Our approach consists in reducing the problem to h¹-vanishing for the ideal sheaves of zero-dimensional schemes associated to singularities, and in applying various h¹-vanishing criteria which come from Riemann-Roch, Kodaira theorem, Bogomolov's theory of unstable rank two vector bundles on surfaces, Castelnuovo function theory.

(Joint work with G.-M. Greuel and C. Lossen.)