Singular Subscheme Research Articles

On the Cynk-Hulek criterion for crepant resolutions of double covers

A collection A = { D 1 , … , D n } of divisors on a smooth variety X is an arrangement if the intersection of every subset of A is smooth. We show that, if X is defined over a field of characteristic not equal to 2, a double cover of X ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up are splayed , a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in A and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.

The Valabrega–Valla module of monomial ideals

In this paper, we focus on the initial degree and the vanishing of the Valabrega–Valla module of a pair of monomial ideals [Formula: see text] in a polynomial ring over a field [Formula: see text]. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of J. For edge ideal of graphs, a complete characterization of the vanishing of the Valabrega–Valla module is given. For higher degree ideals, we find classes, where the Valabrega–Valla module vanishes. For the case that J is the facet ideal of a clutter [Formula: see text] and I is the defining ideal of singular subscheme of J, the non-vanishing of this module is investigated in terms of the combinatorics of [Formula: see text]. Finally, we describe the defining ideal of the Rees algebra of I/J provided that the Valabrega–Valla module is zero.

Freeness versus maximal degree of the singular subscheme for surfaces in $$\mathbb {P}^3$$ P 3

We show that a free surface in \(\mathbb {P}^3\) is characterized by the maximality of the degree of its singular subscheme, in the presence of an additional tameness condition. This is similar to the characterization of free plane curves by the maximality of their global Tjurina number given by A. A. du Plessis and C.T.C. Wall. Simple characterizations of the nearly free tame surfaces are also given.

Grothendieck Classes and Chern Classes of Hyperplane Arrangements

We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of Orlik and Solomon relating the characteristic polynomial with the ranks of the cohomology of the complement of the arrangement. We also show that the characteristic polynomial can be computed from the total Chern class of the complement of the arrangement; this has also been observed by Huh. In the case of free arrangements, we prove that this Chern class agrees with the Chern class of the dual of a bundle of differential forms with logarithmic poles along the hyperplanes in the arrangement; this follows from work of Mustata and Schenck. We conjecture that this relation holds for all free divisors. We give an explicit relation between the characteristic polynomial of an arrangement and the Segre class of its singularity (`Jacobian') subscheme. This gives a variant of a recent result of Wakefield and Yoshinaga, and shows that the Segre class of the singularity subscheme of an arrangement together with the degree of the arrangement determine the ranks of the cohomology of its complement. We also discuss the positivity of the Chern classes of hyperplane arrangements: we give a combinatorial interpretation of this phenomenon, and we discuss the cases of generic and free arrangements.

Verdier specialization via weak factorization

Let X⊂V be a closed embedding, with V∖X nonsingular. We define a constructible function ψX, V on X, agreeing with Verdier’s specialization of the constant function 1V when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–Włodarczyk. The main property of ψX, V is a compatibility with the specialization of the Chern class of the complement V∖X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart ΨX, V in a motivic group. The function ψX, V and the corresponding Chern class cSM(ψX, V) and motivic aspect ΨX, V all have natural ‘monodromy’ decompositions, for any X⊂V as above. The definition also yields an expression for Kai Behrend’s constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.

Special subschemes of the scheme of singularities of a plane foliation

From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane P 2 is uniquely determined by its subscheme of singular points (the singular subscheme of the foliation), we pose the problem of existence of proper closed subschemes Z of the singular subscheme which still determine the foliation in a unique way. We prove the existence of such subschemes Z for foliations with reduced singular subscheme. Unlike the degree deg Z of such subschemes is not sharp for the posed problem, we show that it is so in the sense that Z defines the so-called polar net of the foliation. To cite this article: A. Campillo, J. Olivares, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Shadows of blow-up algebras

We study different notions of blow-up of a scheme X along a subscheme Y, depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to an embedding into a nonsingular variety M. We prove that this latter blow-up is intrinsic of Y and X, and is universal with respect to the requirement of being embedded as a subscheme of the ordinary blow-up of some ambient space along Y. We consider these notions in the context of the theory of characteristic classes of singular varieties. We prove that if X is a hypersurface in a nonsingular variety and Y is its `singularity subscheme', these two extremes embody respectively the conormal and characteristic cycles of X. Consequently, the first carries the essential information computing Chern-Mather classes, and the second is likewise a carrier for Chern-Schwartz-MacPherson classes. In our approach, these classes are obtained from Segre class-like invariants, in precisely the same way as other intrinsic characteristic classes such as those proposed by William Fulton and by W. Fulton and Kent Johnson. We also identify a condition on the singularities of a hypersurface under which the quasi-symmetric blow-up is simply the linear fiber space associated with a coherent sheaf.

A plane foliation of degree different from 1 is determined by its singular scheme

We prove that a foliation 877-1 of degree ≠ 1 on P 2 is completely determined by its singular subscheme SingS( 877-2) of P 2. We apply this result to obtain a similar characterization of 877-3 in terms of the configuration of base points associated to its singular scheme, in case every singularity of 877-4 has non-trivial linear part. Our main motivation comes from a well-known fact: in case a foliation 877-5 of degree r ≥ 2 on P n has only isolated singularities of multiplicity 1, then 877-6 is completely determined by its singular set Sing( 877-7).