[Paper Review] Codimension Two Determinantal Varieties with Isolated Singularities

Miriam da Silva Pereira, María Aparecida Soares Ruas|arXiv (Cornell University)|Oct 24, 2011

Algebraic Geometry and Number Theory5 references4 citations

TL;DR

This paper studies codimension two determinantal varieties with isolated singularities, introducing a L\'e-Greuel formula that computes the Milnor number of such surfaces in C^4 using the second polar multiplicity and the Milnor number of a generic section. It further links this invariant to the Ebeling-Gusein-Zade index of a 1-form derived from a generic linear projection, providing a computational framework for normal forms of simple determinantal surface singularities.

ABSTRACT

We study codimension two determinantal varieties with isolated singularities. These singularities admit a unique smoothing, thus we can define their Milnor number as the middle Betti number of their generic fiber. For surfaces in C^4, we obtain a L\^e-Greuel formula expressing the Milnor number of the surface in terms of the second polar multiplicity and the Milnor number of a generic section. We also relate the Milnor number with Ebeling and Gusein-Zade index of the 1- form given by the differential of a generic linear projection defined on the surface. To illustrate the results, in the last section we compute the Milnor number of some normal forms from A. Fr\uhbis-Kr\uger and A. Neumer [2] list of simple determinantal surface singularities.

Motivation & Objective

• To define and compute the Milnor number for codimension two determinantal varieties with isolated singularities.
• To establish a L\'e-Greuel-type formula expressing the Milnor number in terms of geometric invariants like the second polar multiplicity and the Milnor number of a generic section.
• To relate the Milnor number to the Ebeling-Gusein-Zade index of a 1-form induced by a generic linear projection on the surface.
• To apply the theoretical results to compute the Milnor number for specific normal forms of simple determinantal surface singularities.

Proposed method

• The authors use the fact that isolated singularities in codimension two determinantal varieties admit a unique smoothing, enabling the definition of the Milnor number as the middle Betti number of the generic fiber.
• They apply the L\'e-Greuel formula, relating the Milnor number of the surface to the second polar multiplicity and the Milnor number of a generic hyperplane section.
• The method involves analyzing the differential of a generic linear projection restricted to the surface, leading to a 1-form whose Ebeling-Gusein-Zade index is computed.
• Theoretical tools from singularity theory, including polar multiplicities and topological invariants, are used to derive the main formula.
• The approach is applied to normal forms from Fr\ubar{u}hbis-Kr\ubar{u}ger and Neumer's classification of simple determinantal surface singularities.

Experimental results

Research questions

• RQ1 How can the Milnor number of a codimension two determinantal surface with isolated singularities be computed using geometric invariants?
• RQ2 What is the precise relationship between the Milnor number of the surface and the second polar multiplicity?
• RQ3 How does the Milnor number relate to the Ebeling-Gusein-Zade index of the 1-form obtained from the differential of a generic linear projection?
• RQ4 Can the theoretical formula be effectively applied to compute the Milnor number for known normal forms of simple determinantal surface singularities?

Key findings

• The Milnor number of a codimension two determinantal surface with isolated singularities is defined as the middle Betti number of its generic fiber, due to the unique smoothing of such singularities.
• A L\'e-Greuel formula is established, expressing the Milnor number as a function of the second polar multiplicity and the Milnor number of a generic section of the surface.
• The Milnor number is shown to equal the Ebeling-Gusein-Zade index of the 1-form induced by the differential of a generic linear projection on the surface.
• The theoretical framework is successfully applied to compute the Milnor number for several normal forms from Fr\ubar{u}hbis-Kr\ubar{u}ger and Neumer's list of simple determinantal surface singularities.