Nodal surfaces with obstructed deformations

TL;DR

This paper investigates the deformation space of nodal surfaces of degree d, showing smoothness and expected dimension under certain conditions, and providing explicit examples where deformations are obstructed.

Contribution

It establishes conditions for smoothness of the deformation space of nodal surfaces and constructs explicit examples with obstructed deformations for degrees d ≥ 8.

Findings

Deformation space is smooth for d ≤ 7.

Deformation space is smooth for d ≥ 8 with ≤ 4d-5 nodes.

Explicit examples with 4d-4 nodes show 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 $d\leq 7$ or $d\geq 8$ and $X$ has at most $4d-5$ nodes. (The case $d\leq 7$ was previously covered by Alexandru Dimca by using different techniques.) For $d\geq 8$ 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 $f_1f_2+f_3^2f_4$, where $f_1$ is a linear form.