Nonsingular deformations of a determinantal scheme

Journal of Algebra, 2001

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a m × n-matrix. In this note, we consider G-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade [7]. In this note, we prove that matrices parametrized by generic homogeneous forms of degree d define EIDS. It follows that G-finite determinacy of matrices hold in general. As a consequence, EIDS of a given type (m, n, t) holds in general.

Pacific Journal of Mathematics, 1983

Bulletin of the American Mathematical Society, 1970

COROLLARY 1. If !#,» has grade gH, n then it is grade unmixed, i.e. the associated primes of I H , n all have grade gH,n* COROLLARY 2. If R is Cohen-Macaulay (locally), and lH, n has grade gH t n> then In, n is rank unmixed, i.e. the associated primes all have rank (== altitude) gH, n ; moreover, R/I is Cohen-Macaulay. COROLLARY 3. The rank of any minimal prime of In,n is at most gst, n (with no conditions on the grade of I).

Communications in Algebra, 1999

Journal of Pure and Applied Algebra, 1995

Any complete intersection ladder determinantal ring (LDR) is shown to possess the property that all ideals are tightly closed. This implies that the associated ladder determinantal variety over C has rational singularities. Ladder determinantal varieties (LDVs) were first introduced by Abhyankar in his study of the singularities of Schubert varieties [ 11. In fact, a large subclass of the ladder determinantal varieties (those arising from "one-sided" ladders) were shown by S.B. Mulay to be isomorphic to open affine neigborhoods of Schubert subvarieties of flag varieties. The rich algebraic and combinatorial structure of the LDVs has since inspired further study from various points of view. Because the one-sided ladder determinantal varieties are afline neighborhoods of ' The second author is supported by the NSF and was supported by the Alfred R Sloan Foundation during much of the preparation of this paper.

MATHEMATICA SCANDINAVICA, 2014

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 $\mathsf{C}^4$, we obtain a Lê-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 Frühbis-Krüger and Neumer [7] list of simple determinantal surface singularities.

Advances in Mathematics, 1981

We study the blow-ups X of P3 along a proj. normal curve C. We look for very ample divisor classes on X of low degree, and we study the ideal of the embedding of X. Some result is generalized to higher dimensions.

Journal für die reine und angewandte Mathematik (Crelles Journal)

The determinantal variety Σ p ⁢ q {\Sigma_{pq}} is defined to be the set of all p × q {p\times q} real matrices with p ≥ q {p\geq q} whose ranks are strictly smaller than q. It is proved that Σ p ⁢ q {\Sigma_{pq}} is a minimal cone in ℝ p ⁢ q {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.