Hypersurface Sections Research Articles

Cohomological and motivic inclusion–exclusion

We categorify the inclusion–exclusion principle for partially ordered topological spaces and schemes to a filtration on the derived category of sheaves. As a consequence, we obtain functorial spectral sequences that generalize the two spectral sequences of a stratified space and certain Vassiliev-type spectral sequences; we also obtain Euler characteristic analogs in the Grothendieck ring of varieties. As an application, we give an algebro-geometric proof of Vakil and Wood's homological stability conjecture for the space of smooth hypersurface sections of a smooth projective variety. In characteristic zero this conjecture was previously established by Aumonier via topological methods.

Modeling the workspace of a three-link planar manipulator

A study of the working space of a three-link planar manipulator was carried out. The basis is taken from analytical dependencies that allow solving the direct problem of kinematics, i.e. determine the coordinates of the gripper center point using three generalized coordinates of the manipulator. The analysis performed made it possible to give a geometric interpretation of the dependencies. It has been established that the workspace of the manipulator is a three-parameter set of points. On a plane, this set of points is represented in the form of two disks consisting of ring cells, for which the corresponding analytical dependencies are obtained. The geometric image of this set is a three-dimensional torus. The resulting models are visualized, which facilitates the solution of this problem. To determine the values of the generalized coordinates of the gripper center points, a mapping was carried out by orthogonal projection of the families of circles obtained in the work into four-dimensional space. As a result, three-dimensional hypersurfaces in four-dimensional space were obtained. It is proposed to study them by constructing hypersurface sections models by hyperplanes. Such models in visualization mode allow solving direct and inverse kinematics problems of the manipulator under study.

Subvarieties of hypersurface sections of generalized Grassmannians

Consider a generalized Grassmannian G / P ⊂ P embedded in a projective space by a complete linear system of a positive generator of the Picard group. For a very general hypersurface H ⊂ P , we study subvarieties of G / P ∩ H that are not of general type (or not of positive geometric genus). When the degree of the hypersurface is not small, we show that, under a certain condition on the parabolic subgroup P, such subvarieties are union of lines. Our result is a generalization of Clemens-Ran’s result concerning G / P = P n .

Bertini theorems for differential algebraic geometry

We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini’s theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties.

Splitting the conormal module for licci ideals

For a licci ideal in a power series ring over a field, it is shown that its conormal module has a free summand precisely when the ideal is a hypersurface section. Using results of B. Ulrich, in the Gorenstein case one can show, up to deformation, that the conormal module is indecomposable.

Motivic random variables and representation stability, I: Configuration spaces

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, and gives explicit universal formulas for the limits in terms of the exponents of a motivic Euler product for the Kapranov zeta function. The result can be thought of as a weak but explicit version of representation stability for the cohomology of ordered configuration spaces. In the sequel we find similar stability results in spaces of smooth hypersurface sections, providing new examples to be investigated through the lens of representation stability for symmetric, symplectic and orthogonal groups.

Effective bounds on the dimensions of Jacobians covering abelian varieties

We show that any abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective and explicit version of Poonen’s Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve E E over a finite field and any n ∈ N n\in \mathbb {N} , there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to E n E^n .

Motivic random variables and representation stability II: Hypersurface sections

We prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of varieties, and deduce from this the stabilization of the Hodge Euler characteristic of natural families of local systems constructed from the vanishing cohomology. We prove explicit formulas for the stable values using a probabilistic interpretation, along with the natural analogs in point counting over finite fields. We explain how these results provide new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. This interpretation of representation stability was studied in the prequel [20] for configuration spaces.

Особенность отображения гиперповерхности четырехмерного пространства

Investigation of singularity related to a mapping by the orthogonal projection in a four-dimensional space of a hypersurface given by parametric equations is presented in this paper. On this investigation’s basis have been proposed in a united way three approaches to the hypersurface’s discriminant determination. Thus, have been defined conditions which the investigated hypersurface’s discriminant set and criminant are satisfied. Have been obtained dependencies settling the relationship between parameters of the hypersurface at its discriminant points. They are used to determine the singularity of the hypersurface mapping by analytical methods in general form. The complexity of this approach (the first one) is that an equation connecting the hypersurface’s parameters contains its differential characteristics, and often is the transcendental one in applications, that causes certain difficulties when solving it. Have been obtained dependences in carrying out of which the hypersurface’s discriminant has an edge of regression. A study of hypersurface sections by hyperplanes which are parallel to coordinate hyperplanes has been performed. The last ones contain a coordinate axis along which the hypersurface mapping is performed. It has been established that curves obtained in these sections have extreme points belonging to the hypersurface’s discriminant. Such property is used to calculate the points of the hypersurface’s discriminant by numerical methods without using the hypersurface’s differential characteristics, and it is a basis for the second approach to solving the problem posed. It has been also demonstrated the use of 3D modeling for study of hypersurface’s different sections, as well as its discriminant that represents the third approach to the study. All three approaches having a common basis can be used both independently and complement each other in determining the envelope for one-parameter family of surfaces. As an example has been considered a hypersurface formed by a family of spheres. Based on stated results have been obtained equations determining the hypersurface’s discriminant and this family’s corresponding envelope, as well as various sections. These equations have been used for creating of polygonal 3D models of the hypersurface’s discriminant, and some of the hypersurface’s sections.

Random hypersurfaces and embedding curves in surfaces over finite fields

We use Poonen's closed point sieve to prove two independent results. First, we show that the obvious obstruction to embedding a curve in an unspecified smooth surface is the only obstruction over a perfect field, by proving the finite field analogue of a Bertini-type result of Altman and Kleiman. Second, we prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

Divisorial extractions from singular curves in smooth 3-folds

Consider a singular curve [Formula: see text] contained in a smooth [Formula: see text]-fold [Formula: see text]. Assuming the general elephant conjecture, the general hypersurface section [Formula: see text] is Du Val. Under that assumption, this paper describes the construction of a divisorial extraction from [Formula: see text] by Kustin–Miller unprojection. Terminal extractions from [Formula: see text] are proved not to exist if [Formula: see text] is of type [Formula: see text] or [Formula: see text] and are classified if [Formula: see text] is of type [Formula: see text] or [Formula: see text].

Stability of hypersurface sections of quadric threefolds

Let $S$ be a complete intersection of a smooth quadric 3-fold $Q$ and a hypersurface of degree $d$ in ${\mathbb P}^4$. In this paper we analyze GIT stability of $S$ with respect to the natural $G=SO(5, {\mathbb C})$-action. We prove that if $d\ge 4$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-stable. Also, we prove that if $d\ge 3$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-semistable.

Ubiquity of complete intersection liaison classes

In this paper, we provide constructions to enumerate large numbers of CI-liaison classes. To this end, we introduce a liaison invariant and prove several results concerning it, notably that it commutes with hypersurface sections. This theory is applied to the CI-liaison classes of ruled joins of projective schemes, yielding strong obstructions for such joins to lie in the same liaison class. A second construction arises from the actions of automorphisms on liaison classes, allowing the enumeration of many liaison classes of perfect ideals of codimension at least three.

The probability that a complete intersection is smooth

Given a smooth subscheme of a projective space over a finite field, we compute the probability that its intersection with a fixed number of hypersurface sections of large degree is smooth of the expected dimension. This generalizes the case of a single hypersurface, due to Poonen. We use this result to give a probabilistic model for the number of rational points of such a complete intersection. A somewhat surprising corollary is that the number of rational points on a random smooth intersection of two surfaces in projective 3-space is strictly less than the number of points on the projective line.

On some Frobenius restriction theorems for semistable sheaves

We prove a version of an effective Frobenius restriction theorem for semistable bundles in characteristic p. The main novelty is in restricting the bundle to the p-fold thickening of a hypersurface section. The base variety is G/P, an abelian variety or a smooth projective toric variety.

Smooth hypersurface sections containing a given subscheme over a finite field

We use the "closed point sieve" to prove a variant of a Bertini theorem over finite fields. Specifically, given a smooth quasi-projective subscheme X of P^n of dimension m over F_q, and a closed subscheme Z in P^n such that Z intersect X is smooth of dimension l, we compute the fraction of homogeneous polynomials vanishing on Z that cut out a smooth subvariety of X. The fraction is positive if m>2l.

Gorenstein liaison in

Abstract One of the main open questions for Gorenstein liaison is whether or not the deficiency module of a curve will determine its even G-liaison class in ℙ n , n ≥ 4. In this paper, we show that this is indeed the situation for many degenerate curves in ℙ4. As a special case we are able to show that any two degenerate Buchsbaum curves will belong to the same even G-liaison class if and only if they have isomorphic deficiency modules (up to shifts). We then explore when this may also be true for curves which are obtained as hypersurface sections of Buchsbaum surfaces in ℙ4.

Generalized Jacobian rings for open complete intersections

AbstractIn this paper, we develop the theory of Jacobian rings of open complete intersections, which are pairs (X, Z ) where X is a smooth complete intersection in the projective space and Z is a simple normal crossing divisor in X whose irreducible components are smooth hypersurface sections on X . Our Jacobian rings give an algebraic description of the cohomology of the open complement X – Z and it is a natural generalization of the Poincaré residue representation of the cohomology of a hypersurface originally invented by Griffiths. The main results generalize Macaulay's duality theorem and Donagi's symmetrizer lemma for usual Jacobian rings for hypersurfaces. A feature that distinguishes our generalized Jacobian rings from usual ones is that there are instances where duality fails to be perfect while the defect can be controlled explicitly by using the defining equations of Z in X . Two applications of the main results are given: One is the infinitesimal Torelli problem for open complete intersections. Another is an explicit bound for Nori's connectivity in case of complete intersections. The results have been applied also to study of algebraic cycles in several other works. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Complex surface singularities with integral homology sphere links

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities of "splice type", generalizing Brieskorn type. We show that a normal singularity with homology sphere link is of splice type if and only if some naturally occurring knots in the singularity link are themselves links of hypersurface sections of the singular point. The Casson Invariant Conjecture (CIC) asserts that for a complete intersection surface singularity whose link is an integral homology sphere, the Casson invariant of that link is one-eighth the signature of the Milnor fiber. In this paper we prove CIC for a large class of splice type singularities. The CIC suggests (and is motivated by the idea) that the Milnor fiber of a complete intersection singularity with homology sphere link Sigma should be a 4-manifold canonically associated to Sigma. We propose, and verify in a non-trivial case, a stronger conjecture than the CIC for splice type complete intersections: a precise topological description of the Milnor fiber. We also point out recent counterexamples to some overly optimistic earlier conjectures in [Trends in Singularities, Birkhauser (2002) 181--190 and Math. Ann. 326(2003) 75--93].

Ideals of general forms and the ubiquity of the Weak Lefschetz property

Let d 1,…, d r be positive integers and let I=( F 1,…, F r ) be an ideal generated by forms of degrees d 1,…, d r , respectively, in a polynomial ring R with n variables. With no further information virtually nothing can be said about I, even if we add the assumption that R/ I is Artinian. Our first object of study is the case where the F i are chosen generally, subject only to the degree condition. When all the degrees are the same we give a result that says, roughly, that they have as few first syzygies as possible. In the general case, the Hilbert function of R/ I has been conjectured by Fröberg. In a previous work the authors showed that in many situations the minimal free resolution of R/ I must have redundant terms which are not forced by Koszul (first or higher) syzygies among the F i (and hence could not be predicted from the Hilbert function), but the only examples came when r= n+1. Our second main set of results in this paper show that when n+1⩽ r⩽2 n−2, there are again situations where there must be redundant terms. Finally, we show that if Fröberg's conjecture on the Hilbert function is true then any such redundant terms in the minimal free resolution must occur in the top two possible degrees of the free module. Closely connected to the Fröberg conjecture is the notion of Strong Lefschetz property, and slightly less closely connected is the Weak Lefschetz property. We also study an intermediate notion, the Maximal Rank property. We continue the description of the ubiquity of these properties, especially the Weak Lefschetz property. We show that any ideal of general forms in k[ x 1, x 2, x 3, x 4] has the Weak Lefschetz property. Then we show that for certain choices of degrees, any complete intersection has the Weak Lefschetz property and any almost complete intersection has the Weak Lefschetz property. Finally, we show that most of the time Artinian “hypersurface sections” of zeroschemes in P 2 have the Weak Lefschetz property.