[Paper Review] Free and nearly free surfaces in $P^3$

Alexandru Dimca, Gabriel Sticlaru|arXiv (Cornell University)|Jul 13, 2015

Geometric and Algebraic Topology21 references10 citations

TL;DR

This paper introduces and studies free and nearly free surfaces in $\mathbb{P}^3$, generalizing concepts from free curves to surfaces. It establishes that the Hilbert polynomial of the Milnor algebra of such surfaces is determined by their exponents, proves an analog of Saito’s freeness criterion for nearly free surfaces, and provides explicit examples, including irreducible and rational surfaces, with applications to vector bundles and logarithmic sheaves.

ABSTRACT

We define the nearly free surfaces in $P^3$ and show that the Hilbert polynomial of the Milnor algebra of a free or nearly free surface in $P^3$ 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.

Motivation & Objective

• To define and characterize free and nearly free surfaces in $\mathbb{P}^3$ as a higher-dimensional analog of free curves in $\mathbb{P}^2$.
• To express the Hilbert polynomial of the Milnor algebra of such surfaces in terms of their exponents.
• To establish an analog of Saito’s criterion for freeness in the case of nearly free surfaces.
• To construct explicit examples of irreducible free and nearly free surfaces, including rational surfaces and those related to discriminants of binary forms.
• To investigate the structure of the first local cohomology group $H^1_Q(M(f))$ and its implications for the logarithmic sheaf $Der(-log D)$.

Proposed method

• Define free and nearly free surfaces in $\mathbb{P}^3$ via the syzygy structure of the Jacobian ideal and the Milnor algebra $M(f) = S/J_f$.
• Use the Hilbert polynomial $P(M(f))$ to characterize the algebraic and geometric properties of the singular locus and the surface.
• Prove that for free surfaces, the exponents $d_1 \leq d_2 \leq d_3$ uniquely determine the Hilbert polynomial, and vice versa.
• Establish a Saito-type criterion for nearly free surfaces by relating the second-order syzygy to determinants of first-order syzygies.
• Construct examples via cones over free/nearly free curves and via discriminants of binary forms.
• Use computer algebra systems (CoCoA, Singular) to compute minimal resolutions and verify freeness and nearly freeness.

Experimental results

Research questions

• RQ1 How can the concept of freeness for curves in $\mathbb{P}^2$ be generalized to surfaces in $\mathbb{P}^3$?
• RQ2 What is the precise relationship between the exponents of a free or nearly free surface and the Hilbert polynomial of its Milnor algebra?
• RQ3 Can an analog of Saito’s criterion for freeness be formulated and proven for nearly free surfaces in $\mathbb{P}^3$?
• RQ4 Under what geometric conditions is an irreducible surface in $\mathbb{P}^3$ free or nearly free?
• RQ5 When is the first local cohomology group $H^1_Q(M(f))$ finite-dimensional, and what does this imply for the logarithmic sheaf $Der(-log D)$?

Key findings

• For a free surface in $\mathbb{P}^3$, the Hilbert polynomial $P(M(f))$ is completely determined by its exponents $d_1 \leq d_2 \leq d_3$, and vice versa.
• An analog of Saito’s freeness criterion holds for nearly free surfaces: the unique second-order syzygy is expressed via determinants of the first-order syzygies.
• The first local cohomology group $H^1_Q(M(f))$ is finite-dimensional if and only if the four polynomials $a_1, a_2, a_3, a_4$ (from the resolution) form a regular sequence.
• Examples of nearly free surfaces exist where $H^1_Q(M(f))$ is infinite-dimensional, implying that the logarithmic sheaf $Der(-log D)$ is not locally free.
• For certain nearly free surfaces (e.g., $D_6, D_7, D_8$, and $D_{a,b}$), the sequence $a_1, a_2, a_3, a_4$ is regular and $H^1_Q(M(f))$ is finite-dimensional, leading to locally free but non-split logarithmic sheaves.
• The examples include irreducible rational surfaces, such as $f = x^4 - xyw^2 + zw^3 = 0$, and families like $D_{a,b}: f = x^{2a+2b-1} + x^{a+b-1}y^a z^b + y^{2a-1}z^{2b-1}w = 0$ with $a>1, b>1, a+b>4$.