/pdf/the-milnor-number-and-deformations-of-complex-curve-2ufduz5jmb.pdf

The Milnor number and deformations of complex curve singularities

J. Emsalem and the author showed in [18] that a general polynomialf of degree j in the ring SP= k[yl,...,yr] has (jr+-rl 1) linearly independent partial derivates of order i, for i = 0,1, . . ., t = [ j/2]. Here we generalize the proof to show that the various partial derivates of s polynomials of specified degrees are as independent as possible, given the room available. Using this result, we construct and describe the varieties G( E) and Z( E) parametrizing the graded and nongraded compressed algebra quotients A = R/I of the power series ring R = k[[xl sXr]] having given socle type E. These algebras are Artin algebras having maximal length dimk A possible, given the embedding degree r and given the socle-type sequence E = ( els . . . s eS), where ei is the number of generators of the dual module A of A, having degree i. The variety Z(E) is locally closed, irreducible, and is a bundle over G(E), fibred by affine spaces AN whose dimension is known. We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable-have no deformation to (k + + k)-for dimension reasons. For some choices of the sequence E, D. Buchsbaum, D. Eisenbud and the author have shown that the graded compressed algebras of socle-type E have almost linear minimal resolutions over R, with ranks and degrees determined by E. Other examples have given type e = dimk (socle A) and are defined by an ideal I with certain given numbers of generators in R =

/pdf/compressed-algebras-artin-algebras-having-given-socle-4gujrxomua.pdf

Compressed algebras: Artin algebras having given socle degrees and maximal length

In this fourth and final volume the author extends Buchberger's Algorithm in three different directions. First, he extends the theory to group rings and other Ore-like extensions, and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second, he covers similar extensions as tools for discussing parametric polynomial systems, the notion of SAGBI-bases, Grobner bases over invariant rings and Hironaka's theory. Finally, Mora shows how Hilbert's followers - notably Janet, Gunther and Macaulay - anticipated Buchberger's ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugere (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.

Solving Polynomial Equation Systems I

We introduce the notion of the Bernstein–Sato polynomial of an arbitrary variety (which is not necessarily reduced nor irreducible) using the theory of V-filtrations of M. Kashiwara and B. Malgrange. We prove that the decreasing filtration by multiplier ideals coincides essentially with the restriction of the V-filtration. This implies a relation between the roots of the Bernstein–Sato polynomial and the jumping coefficients of the multiplier ideals, and also a criterion for rational singularities in terms of the maximal root of the polynomial in the case of a reduced complete intersection. These are generalizations of the hypersurface case. We can calculate the polynomials explicitly in the case of monomial ideals.

/pdf/bernstein-sato-polynomials-of-arbitrary-varieties-3u9z1xoluw.pdf

Bernstein-Sato polynomials of arbitrary varieties

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p≥0, and 𝔤=Lie G. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let 𝒩=𝒩(𝔤) denote the nilpotent variety of 𝔤, and ℭnil(𝔤):={(x,y)∈𝒩×𝒩 | [x,y]=0}, the nilpotent commuting variety of 𝔤. Our main goal in this paper is to show that the variety ℭnil(𝔤) is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme ℋn⊂Hilbn(ℙ2) is irreducible over any algebraically closed field.

/pdf/nilpotent-commuting-varieties-of-reductive-lie-algebras-rqmfuqzlwt.pdf

Nilpotent commuting varieties of reductive Lie algebras

A complex hypersurface D in C-n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields We classify all LFDs for n at most 4
 
By analogy with Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of C-n \ D We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive For n at most 4, we show that the GLCT holds for all LFDs
 
We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT As a by-product we obtain a topological proof of a theorem of V Kac on the number of irreducible components of such discriminants

/pdf/linear-free-divisors-and-the-global-logarithmic-comparison-3clsnlzmxv.pdf

Linear free divisors and the global logarithmic comparison theorem

Les zéros de b sont rationnels : ceci est démontré par B. Malgrange, pour / à singularité isolée dans [19], en mettant en évidence un lien avec la monodromie de / (les valeurs propres de la monodromie sont les g-2î7i-û^ ^^ ^ Q^ ̂ p^p Kashiwara [11] dans le cas général en utilisant la résolution des singularités. Malgrange généralise ensuite son résultat aux singularités non isolées dans [20], Varchenko [26] interprète les zéros de & à l'aide d'une filtration voisine de la filtration de Hodge asymptotique.

/pdf/algorithme-de-calcul-du-polynome-de-bernstein-cas-non-1o0nwnk0zu.pdf

Algorithme de calcul du polynôme de Bernstein : Cas non dégénéré

The Gröbner fan of an An-module

A complex hypersurface D in complex affine n-space C^n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. 
Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in C^n. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields is reductive. For n at most 4, we show that the GLCT holds for all LFDs. 
We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) fulfill the GLCT. As a by-product we obtain a simplified proof of a theorem of V. Kac on the number of irreducible components of such discriminants.

The analytic standard fan of a D-module

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Papers

Linear free divisors and the global logarithmic comparison theorem

Algorithme de calcul du polynôme de Bernstein : Cas non dégénéré

The Gröbner fan of an An-module

Linear free divisors and the global logarithmic comparison theorem

The analytic standard fan of a D-module