Varieties Of Arbitrary Dimension Research Articles

The Fulton-MacPherson compactification is not a Mori dream space

We show that the Fulton-MacPherson compactification of the configuration space of n distinct labeled points in certain varieties of arbitrary dimension d, including projective space, is not a Mori dream space for n larger than \(d + 8\).

Wild ramification and restrictions to curves

We prove that wild ramification of a constructible sheaf on a surface is determined by that of the restrictions to all curves. We deduce from this result that the Euler–Poincaré characteristic of a constructible sheaf on a variety of arbitrary dimension over an algebraically closed field is determined by wild ramification of the restrictions to all curves. We similarly deduce from it that so is the alternating sum of the Swan conductors of the cohomology groups, for a constructible sheaf on a variety over a local field.

Rational homotopy and intersection-formality of complex algebraic varieties

A homotopical treatment of intersection cohomology recently developed by Chataur–Saralegui–Tanre associates a perverse algebraic model to every topological pseudomanifold, extending Sullivan’s presentation of rational homotopy theory to intersection cohomology. In this context, there is a notion of intersection-formality, measuring the vanishing of higher Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We then use mixed Hodge theory to prove some intersection-formality results for large families of complex projective varieties, such as isolated surface singularities and varieties of arbitrary dimension with ordinary isolated singularities.

Contribution of jumping numbers by exceptional divisors

We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson [17] and Tucker [18]. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.

Irrationality issues for projective surfaces

This survey retraces the author’s talk at the Workshop Birational geometry of surfaces, Rome, January 11–15, 2016. We consider various birational invariants extending the notion of gonality to projective varieties of arbitrary dimension, and measuring the failure of a given projective variety to satisfy certain rationality properties, such as being uniruled, rationally connected, unirational, stably rational or rational. Then we review a series of results describing these invariants for various classes of projective surfaces.

Ramification theory for varieties over a local field

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an ℓ-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification locus. We prove a formula of Riemann-Roch type for the Swan conductor of cohomology together with its relative version, assuming that the local field is of mixed characteristic.We also prove the integrality of the Swan class for curves over a local field as a generalization of the Hasse-Arf theorem. We derive a proof of a conjecture of Serre on the Artin character for a group action with an isolated fixed point on a regular local ring, assuming the dimension is 2.

Ramification theory for varieties over a perfect field

For an l-adic sheaf on a variety of arbitrary dimension over a perfect field, we define the Swan class measuring the wild ramification as a 0-cycle class supported on the ramification locus. We prove a Lefschetz trace formula for open varieties and a generalization of the Grothendieck-Ogg-Shararevich formula using the Swan class.

On the probability of having rational $$\ell$$ -isogenies

We calculate the chance that an elliptic curve over a finite field has a specified number of \(\ell\)-isogenies which emanate from it. We give a partial answer for abelian varieties of arbitrary dimension.

Asymptotic Growth of the Number of Classes of Real Plane Algebraic Curves as the Degree Grows

The work presents some results on the asymptotics of the number of real plane algebraic curves as the degree grows. In particular, we obtain the asymptotics of the number of curves considered up to the isotopy and rigid isotopy, as well as the number of isotopic classes of maximal curves realizable by T-curves. Some results are generalized to hypersurfaces in nonsingular algebraic varieties of arbitrary dimension. Bibliography: 18 titles.

A GEOMETRIC VERSION OF DYSON'S LEMMA FOR FACTORS OF ARBITRARY DIMENSION

This paper generalizes the geometric part of the Esnault–Viehweg paper on Dyson's Lemma for a product of projective lines. Using the method of weak positivity from algebraic geometry, we are able to study products of smooth projective varieties of arbitrary dimension and to prove a geometric analogue of Dyson's Lemma for this case. Our main result is in fact a quantitative version of Faltings' product theorem.

Error-Correcting Codes from Higher-Dimensional Varieties

In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric error-correcting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. And still, the bounds obtained are usually as good as the ones previously known (at least of the same order of magnitude with respect to the size of the ground field). Several examples coming from Deligne–Lusztig varieties, complete intersections of Hermitian hyper-surfaces, and from ruled surfaces (or more generally, projective bundles over a curve) are given.

The quadric threefold cone of S 4

In a recent paper [1], we have given some account of theories of equivalence and intersection on a singular algebraic surface and have shown that such theories share many of the simple properties enjoyed by corresponding theories on non-singular surfaces. Another paper [2], now in preparation, will extend this work to singular varieties of arbitrary dimension. In the meantime, Zobel [3] has drawn attention to some suspect arguments of Samuel [4] concerning the specialization of intersections on a singular variety.

On the Topology of Certain Birational Transformations

In the study of the topology of algebraic varieties, the problem arises: When a variety undergoes a birational transformation, what changes take place in its homology groups? This problem has been completely solved, in the case of a surface, by Lefschetz and Zariski ([4] p. 115): but about the case of a variety of arbitrary dimension, little seems to be known. The present paper therefore considers an m-dimensional variety, acted on by a transformation of the simplest possible kind: namely, one of those which B. Segre [2] has named dilatations. In this case, the relationship of the new variety to the old resembles the relationship of a fibre-bundle to its base-space. In fact, a theorem (due to Spanier and Chern) about fibre-bundles can be extended to cover the present situation. It enables us to determine the homology groups of the new variety, if we are given those of the old one, and those of the variety which serves as base for the dilatation.