Smoothness of equisingular families of curves

Abstract:

Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{|D|}^{irr}\big (\mathcal {S}_1,\ldots ,\mathcal {S}_r\big )$ of irreducible curves in the linear system $|D|_l$ with precisely $r$ singular points of types $\mathcal {S}_1,\ldots ,\mathcal {S}_r$ is T-smooth. Considering different surfaces including the projective plane, general surfaces in $\mathbb {P}_{\mathbb {C}}^3$, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type \[ \sum \limits _{i=1}^r\gamma _\alpha (\mathcal {S}_i) < \gamma \cdot (D- K_\Sigma )^2, \] where $\gamma _\alpha$ is some invariant of the singularity type and $\gamma$ is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the $\gamma _\alpha$-invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.