Nodal surfaces with obstructed deformations

Abstract

In this text we show that the deformation space of a nodal surface $X$ of degree $d$ is smooth and of the expected dimension if $d\le 7$ or $d\ge 8$ and $X$ has at most $4d-5$ nodes. (The case $d\le 7$ was previously covered by Alexandru Dimca by using different techniques.) For $d\ge 8$ we give explicit examples of nodal surfaces with $4d-4$ nodes, for which the tangent space to the deformation space has larger dimension than expected. We give a short discussion on the shape of the deformation space of surfaces of the form $f_1{f}_2+f_3^2{f}_4$, where $f_1$ is a linear form.

v2: Added a reference to a similar result by Alexandru Dimca and a discussion on the difference between Dimca's result and ours v3: Expanded several arguments

References in corpus (2)

• Maximal families of nodal varieties with defect
• Geometry of obstructed equisingular families of projective hypersurfaces

Cited by in corpus (1)

• Limits of nodal surfaces and applications