Wronski algebra systems on families of singular curves

The Geometry of Families of Singular Curves

Remarks on Families of Singular Curves with Hyperelliptic Normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus g ≥ 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result whose proof is “vector bundle-free” and relies on deformation theory and bending-and-breaking of rational curves in Sym(S). We also give examples of fa...

Characterizing singular curves in parametrized families of biquadratics

We consider families of biquadratic curvesB = 0 onC, defined with respect to arbitrarily many complex parameters. Due to the fact that these families include curve intersections across different parameter combinations, they represent a generalization of the non-intersecting foliations of one-parameter invariant curves associated with the QRT mapping. We use algebraic methods involving discrimin...

Families of Regular Solutions of Singular Systems

The seminal 1969 paper of W. A. Harris, Jr., Y. Sibuya, and L. Weinberg provided new proofs for the Perron-Lettenmeyer theorem, as well as several other classical results, and has stimulated renewed consideration of families of regular solutions of certain singular problems. In this paper we give some further applications of the method developed there and, in addition, examine some connections ...

Enumerating singular curves on surfaces

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and up to three nodes. The curves must also pass through appropriately many general points. The number of curves is given by a universal polynomial in four basi...