Freeness versus maximal degree of the singular subscheme for surfaces in $P^3$

TL;DR

This paper characterizes free surfaces in projective 3-space by the maximal degree of their singular subscheme under a tameness condition, extending known results from free plane curves.

Contribution

It establishes a new characterization of free surfaces in P^3 based on singular subscheme degree and introduces characterizations for nearly free tame surfaces.

Findings

Free surfaces are characterized by maximal singular subscheme degree.

Nearly free tame surfaces are also characterized.

The results extend plane curve characterizations to surfaces.

Abstract

We show that a free surface in $P^3$ is characterized by the maximality of the degree of its singular subscheme, in the presence of an additional tameness condition. This is similar to the characterization of free plane curves by the maximality of their global Tjurina number given by A. A. du Plessis and C.T.C. Wall. Simple characterizations of the nearly free tame surfaces are also given.