Generic Linear Form Research Articles

Brasselet Number and Function-Germs with a One-Dimensional Critical Set

The Brasselet number of a function fhnonisolated singularities describes numerically the topological information of its generalized Milnor fibre. In this work, using the Brasselet number, we present several formulas for germs $$f:(X,0)\rightarrow (\mathbb {C},0)$$ and $$g:(X,0)\rightarrow (\mathbb {C},0)$$ in the case where g has a one-dimensional critical locus. We also give applications when f has isolated singularities and when it is a generic linear form.

Block-Krylov techniques in the context of sparse-FGLM algorithms

Consider a zero-dimensional ideal I in K[X1,…,Xn]. Inspired by Faugère and Mou's Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of I in order to compute a description of its zero set by means of univariate polynomials.Steel recently showed how to use Coppersmith's block-Wiedemann algorithm in this context; he describes an algorithm that can be easily parallelized, but only computes parts of the output in this manner. Using generating series expressions going back to work of Bostan, Salvy, and Schost, we show how to compute the entire output for a small overhead, without making any assumption on the ideal I other than it having dimension zero. We then propose a refinement of this idea that partially avoids the introduction of a generic linear form. We comment on experimental results obtained by an implementation based on the C++ libraries Eigen, LinBox and NTL.

On ideals generated by two generic quadratic forms in the exterior algebra

Based on the structure theory of pairs of skew-symmetric matrices, we give a conjecture for the Hilbert series of the exterior algebra modulo the ideal generated by two generic quadratic forms. We show that the conjectured series is an upper bound in the coefficient-wise sense, and we determine a majority of the coefficients. We also conjecture that the series is equal to the series of the squarefree polynomial ring modulo the ideal generated by the squares of two generic linear forms.

Almost Buchsbaumness of some rings arising from complexes with isolated singularities

We study properties of the Stanley–Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley–Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of nonhomologically isolated singularities, providing many examples in which the Stanley–Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.

Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0) and g:(X,0)→(C,0). We give applications when g is a generic linear form and when f and g have isolated singularities.

Inverse systems, generic linear forms, divided powers, and transversality

The main result of this work is a transversality result for linear spaces of forms of polynomials vanishing on generic points. In the process of proving this result, we clarify and refine previous work with Fröberg on the theory of inverse systems and generic forms. The needs for these improvements grew out of our proof of Zanello’s bound for Hilbert functions.

Face rings of simplicial complexes with singularities

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.

The Paley-Wiener theorem for certain nilpotent Lie groupsJean

We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra 𝔤* are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical Properties of Fat Schemes with Special Support

Given a finite subset 𝒮 of ℕ c we construct a standard 𝒮-scheme whose defining ideal is a square-free monomial ideal, and, more generally, we construct 𝒮-schemes with respect to families of generic linear forms. For these schemes we give a characterization on 𝒮 ⊂ ℕ2 in order to be arithmetically Cohen–Macaulay. Moreover, for fat schemes with support on a standard 𝒮-scheme, we produce a minimal free resolution. Using this result, we furnish the graded Betti sequences for such fat schemes and, under some numerical conditions, for fat 𝒮-schemes where 𝒮 has some combinatorial property.

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal I I in a polynomial ring R R and for every p ≤ dim ⁡ R p\leq \dim R we consider the Koszul homology H i ( p , R / I ) H_i(p,R/I) with respect to a sequence of p p of generic linear forms. The Koszul-Betti number β i j p ( R / I ) \beta _{ijp}(R/I) is, by definition, the dimension of the degree j j part of H i ( p , R / I ) H_i(p,R/I) . In characteristic 0 0 , we show that the Koszul-Betti numbers of any ideal I I are bounded above by those of the gin-revlex G i n ( I ) \mathrm {Gin}(I) of I I and also by those of the Lex-segment L e x ( I ) \mathrm {Lex}(I) of I I . We show that β i j p ( R / I ) = β i j p ( R / G i n ( I ) ) \beta _{ijp}(R/I)=\beta _{ijp}(R/\mathrm {Gin}(I)) iff I I is componentwise linear and that and β i j p ( R / I ) = β i j p ( R / L e x ( I ) ) \beta _{ijp}(R/I)=\beta _{ijp}(R/\mathrm {Lex}(I)) iff I I is Gotzmann. We also investigate the set G i n s ( I ) \mathrm {Gins}(I) of all the gin of I I and show that the Koszul-Betti numbers of any ideal in G i n s ( I ) \mathrm {Gins}(I) are bounded below by those of the gin-revlex of I I . On the other hand, we present examples showing that in general there is no J J is G i n s ( I ) \mathrm {Gins}(I) such that the Koszul-Betti numbers of any ideal in G i n s ( I ) \mathrm {Gins}(I) are bounded above by those of J J .

Variation of the Milnor Fibration in Pencils of Hypersurface Singularities

Let Φ = (f, g):(Cn+ 1,0) → (C2, 0) be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: Φ defines an isolated complete intersection singularity; g = lN where ℓ is a generic linear form with respect to f and N> 0.) We study how the Milnor fibrations of the germs ϕ(α:β) = α g-β f are related to each other when (α:β) varies in P1. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram. 2000 Mathematical Subject Classification: 32S55, 32S15, 32S30.

High-order vanishing ideals at n+3 points of Pn

The authors conjecture that the ideal IZ[t−1] of functions vanishing to order t−1 at the subscheme Z of Pn, n=2t−1 comprised of 2t+2 generic smooth points, satisfies dimk(IZ[t−1])t=2t, in its initial degree, t. They show that this dimension is at least t+1, by a direct construction of suitable vanishing forms. This result is complementary to those of M.V. Catalisano, P. Ellia, and A. Gimigliano in [5]. The authors also consider related problems, including the Macaulay dual problem, of determining the Hilbert function H(A), A=R/(x12,…,xr2,(x1+⋯+xr)2,L2) — where R=k[x1,…,xr], r=2t and L is a generic linear form — in the socle degree t of A.

Hilbert Functions and Generic Forms

Let A be a homogeneous K-algebra where K is a field of characteristic 0, and h ∈ A a generic form. We bound the Hilbert function H(A/(h),-) in terms of H(A,-) which extends the bound given by M.Green for generic linear forms. We apply this to some conjectures from Higher Castelnuovo Theory and Cayley-Bacharach Theory.

Variation structures: results and open problems

Studying the invariants of isolated hypersurface germs f : (C, 0) → (C, 0), it is very useful to consider composed germs f = p ◦ φ, where φ : (C, 0) → (C, 0) has a manageable discriminant space (for example: φ is an isolated complete intersection singularity, in short ICIS), and p : (C, 0) → (C, 0) is a curve singularity. This gives not only a very large class of examples with powerful testing role (for example, the germs of “generalized Sebastiani– Thom type”, where φ(x, y) = (g(x), h(y)) [9, 10], or the topological series fk = pk ◦ φ → f∞ = p∞ ◦ φ, when φ is an ICIS and pk → p∞ is a topological series [10, 20] of plane singularities), but also clarifies the most general case. To see this, complete the initial, arbitrary germ f to an ICIS (f, g) = φ and take p(c, d) = c. If g is a generic linear form then we recover the classical method of the polar curves, which is an effective inductive procedure. In the composed case, the leading principal is the following: for a given invariant i, find a category C(i) of supplementary structures (“of system of coefficients”) defined either on (C, 0) or on the local complement of an analytic germ ∆ ⊂ (C, 0), (which, in general, is the discriminant space of φ) with the following properties: a) φ defines a structure S(φ) in C(i), and b) the invariant i(f) can be computed in terms of the germ p and the structure S(φ). In this way, one expects that the computation of the invariant i is reduced to lower dimensional topology (link topology of p−1(0)∪∆) with some repre-

The monodromy of a series of hypersurface singularities

Let {f=0} be a hypersurface inCn+1 with a 1-dimensional singular set Σ. We consider the series of hypersurfaces {f+ɛxN=0} wherex is a generic linear form.