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# Asian Journal of Mathematics

## Volume 22 (2018)

### Number 5

### Free and nearly free surfaces in $\mathbb{P}^3$

Pages: 787 – 810

DOI: http://dx.doi.org/10.4310/AJM.2018.v22.n5.a1

#### Authors

#### Abstract

We define the nearly free surfaces in $\mathbb{P}^3$ and show that the Hilbert polynomial of the Milnor algebra of a free or nearly free surface in $\mathbb{P}^3$ can be expressed in terms of the exponents. An analog of Saito’s criterion of freeness in the case of nearly free divisors is proven and examples of irreducible free and nearly free surfaces are given.

#### Keywords

Jacobian ideal, Milnor algebra, free divisor, nearly free divisor, Saito’s criterion

#### 2010 Mathematics Subject Classification

Primary 14J70. Secondary 13D02, 13P20, 14C20, 32S22.

The first author was partially supported by the Institut Universitaire de France.

Received 16 September 2015

Accepted 5 September 2017

Published 9 November 2018