Varieties In Projective Space Research Articles

Frobenius on the Cohomology of Thickenings

Abstract We investigate the injectivity of the Frobenius map on thickenings of smooth varieties in projective space over a field of positive characteristic. We obtain uniform bounds—that is, independent of the characteristic—on the thickening that ensures an injective Frobenius map when the projective variety is a smooth complete intersection or an arbitrary projective embedding of an elliptic curve. Our bounds are sharp in the case of hypersurfaces, and in the case of elliptic curves.

Mukai flops and Plücker-type formulas for hyper-Kähler manifolds

We study the intersection theory of complex Lagrangian subvarieties inside holomorphic symplectic manifolds. In particular, we study their behaviour under Mukai flops and give a rigorous proof of the Plücker-type formula for Legendre dual subvarieties written down by the second author before. Then we apply the formula to study projective dual varieties in projective spaces.

Existence of embeddings of varieties in projective spaces whose points are spanned by low degree smoothable zero-dimensional subschemes

Let X be an integral projective variety. Set n:= dim X. Let e(X) ≥ 2n+1 be the embedding dimension of X (we may take e(X)=2n+1 if X is smooth). Fix integers δ and r ≥ e(X). We prove the existence of many embeddings j:X↪Pr such that deg(X) ≥ δ and every point of Pr is spanned by a low degree smoothable zero-dimensional subscheme of X.

The Euclidean distance degree of smooth complex projective varieties

We obtain several formulas for the Euclidean distance degree (ED degree) of an arbitrary nonsingular variety in projective space: in terms of Chern and Segre classes, Milnor classes, Chern-Schwartz-MacPherson classes, and an extremely simple formula equating the Euclidean distance degree of X with the Euler characteristic of an open subset of X.

Symmetric differential forms on complete intersection varieties and applications

In this paper we study the cohomology of the symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. As a first application, we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Then, as our main application, we construct varieties with ample cotangent bundle, providing new results towards a conjecture of Debarre.

An infinitesimal condition to smooth ropes

In this article we give a condition, which holds in a very general setting, to smooth a rope, of any dimension, embedded in projective space. As a consequence of this we prove that canonically embedded carpets satisfying mild geometric conditions can be smoothed. Our condition for smoothing a rope \(\widetilde{Y}\) can be stated very transparently in terms of the cohomology class of a suitable first order infinitesimal deformation of a morphism ϕ associated to \(\widetilde{Y}\). In order to prove these results we find a sufficient condition, of independent interest, for a morphism ϕ from a smooth variety X to projective space, finite onto a smooth image, to be deformed to an embedding. Another application of this result on deformation of morphisms is the construction of smooth varieties in projective space with given invariants. We illustrate this by constructing canonically embedded surfaces with \(c_{1}^{2}=3p_{g}-7\) and deriving some interesting properties of their moduli spaces. The results of this article bear further evidence to the complexity of the moduli of surfaces of general type and its sharp contrast with the moduli of other objects such as curves or K3 surfaces.

Regularity, partial elimination ideals and the canonical bundle

We present partial elimination ideals, which set-theoretically cut out the multiple point loci of a generic projection of a projective variety, as a way to bound the regularity of a variety in projective space. To do this, we utilize a combination of initial ideal methods and geometric methods. We first define partial elimination ideals and establish through initial ideal methods the way in which, for a given ideal, the regularity of the partial elimination ideals bounds the regularity of the given ideal. Then we explore the partial elimination ideals as a way to compute the canonical bundle of the generic projection of a variety and the canonical bundles of the multiple point loci of the projection, and we use Kodaira Vanishing to bound the regularity of the partial elimination ideals.

Bounds on the degree of two-codimensional integral varieties in projective space

In this paper we establish new upper bounds on the degree of a two-codimensional non-degenerate integral variety in projective space, depending on the dimension of the variety and on a non-lifting level s for it, i.e. a level such that there is an element of degree s in the defining ideal of a general hyperplane section of the variety which is not restriction of any element of the same degree in the defining ideal of the variety.

Isolated rational curves on K 3-fibered Calabi-Yau threefolds

In this paper we study 16 complete intersection K3-fibered Calabi--Yau variety types in biprojective space ℙn1}×ℙ1. These are all the CICY-types that are K3 fibered by the projection on the second factor. We prove existence of isolated rational curves of bidegree (d,0) for every positive integer d on a general Calabi–Yau variety of these types. The proof depends heavily on existence theorems for curves on K3-surfaces proved by S. Mori and K. Oguiso. Some of these varieties are related to Calabi–Yau varieties in projective space by a determinantal contraction, and we use this to prove existence of rational curves of every degree for a general Calabi–Yau variety in projective space.

Matroids on Partially Ordered Sets

Several generalizations of the notion of matroid have been proposed Ž . Brylawski, Edmonds, Faigle, Korte, Lovasz, Welsh, White . This work proposes yet another one, one that is motivated by two considerations. The first motivation comes from projective geometry. Matroids are the Ž natural setting for the study of arrangements of hyperplanes or, equiva. lently, sets of points in projective space. It is natural to ask whether arrangements of linear varieties of different dimensions in projective space may be ensconced into a similar axiomatic setting, one in which matroid] theoretic arguments, with circuits, rank, bases, etc., may be used. The second motivation is the replacement of a Boolean algebra of sets by the distributive lattice of filters of a finite partially ordered set. This replacement has proved fruitful in other contexts, most successfully in the replacement of algebraic varieties by schemes in algebraic geometry. Our definition of poset matroids allows the extension to this new setting of every notion of matroid theory. In fact, the extension of the notion of matroid to poset matroids sheds light on the mutual relation of the notions of matroid theory. Every family of linear varieties in projective space defines a poset matroid, and a classification of the possible special positions of a set of linear varieties is reflected in the combinatorial structure of the poset matroid thereby obtained. Two languages are available for poset matroids: the language of partially ordered sets and the language of distributive lattices. The translation of poset matroids into the language of distributive lattices leads to the definition of a combinatorial scheme. Again, the translation of matroid

Inverse System of a Symbolic Power, I

The authors determine Macaulay′s inverse system to a symbolic power of the graded ideal of functions vanishing at a union of irreducible varieties in projective space.

Higher-order syzygies for the bracket algebra and for the ring of coordinates of the Grassmanian.

A Poincaré resolution is given for the supersymmetric ring of brackets over a signed alphabet. As a consequence, a resolution is found for the ring of coordinates of the Grassmanian variety in projective space over any infinite field.

Geometry and multiple direction estimation

We consider the problem of the estimation of multiple target directions, given the output of a phased array of identical elements. The natural mathematical setting for this problem is the Grassmannian manifold. In order to understand the problems considered here, a knowledge of exterior forms, lie groups and differential geometry is essential. We review some of the essential ideas relevant to the problem, including the Maurer-Cartan equations, algebraic varieties in projective space and the invariant volume integral for symmetric spaces. Our purpose is to study probability measures on Grassmannians, in order to pose and solve the Bayes problem on Grassmannians.

Free resolutions of certain codimension three perfect radical ideals

Introduction. The original conception of this work included a complete study of certain codimension three perfect ideals that arise naturally in a set-up, with the purpose of enriching the present collection of such ideals. The ones we consider are always radical and their syzygy-theoretic properties are relatively easy to describe. From the viewpoint of the theory of graded resolutions, they share common numerical features with (the projecting cones of) certain codimension two and three varieties in projective space. Their resolutions being seldom pure, they seemed largely convenient for testing the sharpness of some calculations made in the pure case (cf. [19], [25], [26]). Moreover, they yield a large class suited for testing recently developed concepts, such as the notion of strong obstruction, that have its natural place in the theory of linkage ([loc. cit.]) or the more general theory of residual intersection developed in [16]. The authors did not pursue this line of thought, but it seemed worthwhile pointing it out. A brief description of the present contents is as follows. In the first section one obtains the explicit free resolution of a large class of codimension three ideals that appear as presentation ideals of normal cones associated to determinantal ideals fixing a submatrix. As it turns out, the resolutions of these presentation ideals are given by a mapping cylinder of a suitable map to the resolution of the original determinantal ideals. The second section is concerned with deriving some ideal-theoretic results from the knowledge of the presentation given in section one. This procedure, by and large, follows the techniques developed by various authors ([17], [14], [23] and [24]). In the third section one gives the resolution of the determinantal ideal Jr associated to a map m ~ R (m > n), fixing r c m, in the case of the data r = n - 2 and m = n + 2. This case not only extends the previous known ones [1], [2], but mainly gives some genuine clue for the general situation. The construction follows closely the spirit of the scandinavian complex [9] in that one is led to tensor suitable complexes and then to cut down the resulting size by an use of trace maps. This method seems to generalize well in various contexts [21]. Last one establishes exact values for the codimension of the complete intersection locus of the ideals considered in the previous sections. The reason for doing so is that the known estimates to present [11], [27] would give no real grasp, in this case, of the locus.

Angular momentum, convex Polyhedra and Algebraic Geometry

The three families of classical groups of linear transformations (complex, orthogonal, symplectic) give rise to the three great branches of differential geometry (complex analytic, Riemannian and symplectic). Complex analytic geometry derives most of its interest from complex algebraic geometry, while symplectic geometry provides the general framework for Hamiltonian mechanics.These three classical groups “intersect” in the unitary group and the three branches of differential geometry correspondingly “intersect” in Kähler geometry, which includes the study of algebraic varieties in projective space. This is the basic reason why Hodge was successful in applying Riemannian methods to algebraic geometry in his theory of harmonic forms.

On the embedding of abelian varieties in projective spaces

In this note it is proved that forα ⩾3 an abelian variety of dimension d cannot be embedded in a projective space of dimenrion 2d.

Volumes of Images of Varieties in Projective Space and In Grassmannians

If V is a complex analytic subvariety of pure dimension k in the unit ball in ${{\mathbf {C}}^n}$ which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of $(n - k)$-complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.

Volumes of images of varieties in projective space and in Grassmannians

If V is a complex analytic subvariety of pure dimension k in the unit ball in C n {{\mathbf {C}}^n} which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of ( n − k ) (n - k) -complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.

Neighbourhood manifolds and their parametrization

The main purpose of this paper is the representation by algebraic varieties in projective space, of certain aggregates, in which each element is a sequence P 0 ... P n of n +1 points consecutive on an algebraic (or algebroid) branch. Two types of aggregate are to be considered: W r , n , of all sequences of n + 1 points with a given origin P 0 in S r ; and W r ,* n , of all such sequences with origin anywhere in S r . The problem has been studied by various writers from 1901 onwards, and a complete solution for n = 1,2 only was given in 1955. In this period no progress at all had been made for higher values of n , and its intractability led to conjectures that the aggregates in question were not irreducible, or had some other defects inhibiting the type of representation sought. In the present paper a complete method is given of parametrizating both W r , n and W r * n for all values of r , n , the co-ordinates of a point of the model being expressed systematically in terms of certain invariants of an arbitrary branch through the sequence represented. These models are irreducible and non-singular, and their points are in one-one correspondence without exception with the sequences in the appropriate aggregates. In addition, the geometrical properties of the models are studied in some detail for r = 2 (the clarity of the picture obtained naturally fading off as n increases) and (JV2,3 having been dealt with in the author’s short contribution to a symposium in honour of Bompiani) a complete geometrical description is given of W 2 , 4 , W 3 , 3 , and W 2 * 3 , including the base and intersection theory on these. The main importance claimed for the work is twofold: (i) it offers in theory (i.e. by a well defined method, which however involves sharply increasing labour with increase of n and of r ) a complete solution of a problem which has proved intractable for 60 years; (ii) it gives detailed descriptions of some, and less full description of some others, of a family of varieties which have the same type of importance for local algebraic geometry that for instance Grassmannians have for line geometry, etc. It is possible also that the very ample system of invariants defined in the course of the parametrization of the models may prove to be of use in differential geometry (the formal power series from which they are obtained being interpreted as Taylor series) but this is a pure surmise, being rather outside the author’s sphere of interest.