Doctoral Thesis
DOI
10.11606/T.55.2014.tde-13022015-100258
Document
Author
Full name
Nancy Carolina Chachapoyas Siesquén
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2014
Supervisor
Committee
Ruas, Maria Aparecida Soares (President)
Brasselet, Jean Paul
Dutertre, Nicolas Andre Oliver
Hernandes, Marcelo Escudeiro
Title in Portuguese
Keywords in Portuguese
Número de Milnor
Obstrução de Euler
Seções genéricas
Transformação de Nash
Abstract in Portuguese
Title in English
Invariants of determinantal varieties
Keywords in English
Determinantal variety
Euler obstruction
Generic determinantal variety
Generic sections
Milnor number
Nash transformation
Abstract in English
In this work, we study the essentially isolated determinantal singularities (EIDS), which have been defined by W. Èbeling and S. M. Gusen-Zade in the article [23]. This type of singularities is a natural generalization of isolated ones. A generic determinantal variety Mtm;n is a subset of the space of m X n matrices, given by matrices of rank less than t, where t ≤ min{n;m}. A variety X ⊂ CN is determinantal if X is defined as the pre-image of Mtm;n by a holomorphic function F : CN → Mm;n with the condition codim X = codim Mtm;n. Determinantal varieties have isolated singularity if N ≤ (n - t + 2)(m - t + 2) and they admit smoothing if N < (n - t +2)(m - t +2). Several recent works investigate determinantal variety with isolated singularities. The Milnor number of a surface was defined in [35, 31] and the vanishing Euler characteristic was studied in [31]. In this work we study the set of limits of tangent hyperplanes to determinantal varieties X2 ⊂ C4 and X3 ⊂ C5 to give a characterization of this set by the fact that the Milnor number of its section with the surface in the first case or the 3-dimensional determinantal variety in the second case is not minimum. The first case is studied by Jawad Snoussi in [38]. We also prove that if X is a d- dimensional EIDS and H and H' are strongly general hyperplans, if P ⊂ H and P' are linear plans of codimension d - 2 contained in H and H', the Milnor number of the surfaces X ∩ P and X ∩ P' are equal. In the case that the generic section is a curve the result has been proved in [26]. We study the Nash transformation of an EIDS and give sufficient conditions for this transformation to be smooth. Another aim of our study is the Euler obstruction of essentially isolated determinantal singularities. We obtain inductive formulas associating the Euler obstruction with the vanishing Euler characteristic of the essencial smoothing of their generic sections. We study the determinantal variety with singular set of dimension 1 to illustrate the results.