arXiv subjects
Publications and source records attributed to Torsten Ekedahl.
This paper provides explicit closed formulas in terms of tautological classes for the cycle classes of the height and Artin invariant strata in families of K3 surfaces. The proof is uniform for all strata and uses a flag space as the computations in [arXiv:math/0412272] for the Ekedahl-Oort strata for families of abelian varieties, but employs a Pieri formula formula to determine the push down to the base space.
We prove that for every reductive algebraic group $H$ with centre of positive dimension and every integer $K$ there is a smooth and projective variety $X$ and an algebraic $H$-torsor $P \to X$ such that the classifying map $X \to \Bclass H$ induces an isomorphism in cohomology in degrees $\le K$. This is then applied to show that if $G$ is a connected non-special group there is a $G$-torsor $P \to X$ for which we do not have $[P]=[G][X]$ in the (completion of the) Grothendieck ring of varieties.
We introduce a Grothendieck group of algebraic stacks (with affine stabilisers) analogous to the Grothendieck group of algebraic varieties. We then identify it with a certain localisation of the Grothendieck group of algebraic varieties. Several invariants of elements in this group are discussed. The most important is an extension of the Euler characteristic (of cohomology with compact support) but in characteristic zero we introduce invariants which are able to distinguish between classes with the same Euler characteristic. These invariants are actually defined on the completed localised Grothendieck ring of varieties used in motivic integration. In particular we show that there are $\PSL_n$-torsors of varieties whose class in the completed localised Grothendieck ring of varieties is not the product of the class of the base and the class of $\PSL_n$.
We study the class of the classifying stack of a finite group in a Grothendieck group of algebraic stacks introduced previously. We show that this class is trivial in a number of examples most notably for all symmetric groups. We also give some examples where it is not trivial. The latter uses counterexamples of Saltman and Swan to the problem of Noether.
In the Hilbert scheme of points on a scheme X there is an open subset parameterizing distinct points. The closure of that open set is by definition the good component. When X is flat over the base, we show that a certain blow-up of the symmetric product of X is the good component. The center of the blow-up we describe by giving generators for its defining ideal. In the non-flat case we obtain similar result by replacing the symmetric product with the divided power product. For smooth surfaces X the good component equals the Hilbert scheme of points.
We introduce a stratification on the space of symplectic flags on the de Rham bundle of the universal principally polarized abelian variety in positive characteristic and study its geometric properties like irreducibility of the strata and we calculate the cycle classes. When the characteristic $p$ is treated as a formal variable these classes can be seen as a deformation of the classes of the Schubert varieties for the corresponding classical flag variety (the classical case is recovered by putting $p$ equal to 0). We relate our stratification with the E-O stratification on the moduli space of principally polarized abelian varieties of a fixed dimension and derive properties of the latter. Our results are strongly linked with the combinatorics of the Weyl group of the symplectic group.
Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For w in $W^J$, let $f^{w,J}_{i}$ denote the number of elements of length i below w in Bruhat order on $W^J$ (notation simplified to $f^{w}_{i}$ in the case when J=S). We show that $f^{w,J}_{i}$ is less than or equal to $f^{w,J}_{j}$ when i < j and j is less than or equal to the length of w minus i. Furthermore, we express when an initial and final interval of the f's is symmetric around the middle in terms of Kazhdan-Lusztig polynomials. It is also shown that if W is finite then the sequence of f's cannot grow too rapidly. Som result mirroring our first result are obtaind, again in the finite case. The proofs rely for the most part on properties of the cohomology of Kac-Moody Schubert varieties.
This is the written version of a talk at the conference on ``Non-commutative geometry and representation theory in mathematical physics'' held in Karlstad, Sweden, 5--10 July, 2004. In it we show that the duality formula of Rocha-Caridi and Wallach is a simple consequence of the so called cde-triangle of modular representation theory. It tries to reflect the attempt of the talk to cater to the differing backgrounds of its listeners.
We prove a generalisation to any characteristic of a result of Macdonald that describes strict polynomial functors in characteristic zero in terms of representations of the groupoid of finite sets and bijections. Our result will give an analogous description in terms of finite multisets and an extension of the notion of bijection appropriate for multisets. A projected application is to the description of strict polynomial monads that will give a notion generalising (linear algebra) operads.
Only two ways to construct non-liftable Calabi-Yau threefolds are currently known, one example by Hirokado and one method of Schröer. This article computes some cohomological invariants of these examples of non-liftable Calabi-Yau threefolds, in particular it computes their mini-versal deformations. One conclusion is that their mixed characteristic mini-versal deformation spaces are actually smooth over the characteristic $p$ base field. Furthermore, a new family, constructed in the spirit of Schröer, is introduced and the same calculations are performed for it.
We give a generalization to higher genera of the famous formula $12 λ=δ$ for genus 1. We also compute the classes of certain strata in the Satake compactification as elements of the push down of the tautological ring.
This article presents a new approach to the unobstructedness result for deformations of Calabi-Yau varieties by introducing the tangent lifting property for a functor on artinian local algebras. The verification that the deformation functor of a Calabi-Yau variety in characteristic 0 fulfills the tangent lifting property uses, just as the verification of the $T^1$-lifting property, the degeneration of the Hodge to de Rham spectral sequence. Our main use of our methods is however to the mixed characteristic case. In that case to be able to verify the conditions needed one needs an extension of the criterion introducing divided powers.
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.
We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory. Our main result is that the ring of invariants of a splitting algebra under the symmetric group almost always is the ring of coefficients.
In this note we discuss a class of hyperelliptic curves introduced by Abel in a 1826 paper. After some indications of the context in which he introduced them and a description of his main result we give some results on the moduli space of such curves. In particular we compute the dimension of it at each of its points as well as giving a combinatorial formula for the number of components.
This paper takes its starting point in an idea of Grothendieck on the representation of homotopy types. We show that any locally finite nilpotent homotopy can be represented by a simplicial set which is a finitely generated free group in all degrees and whose maps are given by polynomials with rational coefficients. Such a simplicial set is in some sense a universal localisation/completion as all localisations and completions of the homotopy is easily contructed from it. In particular relations with the Quillen and Sullivan approaches are presented. When the theory is applied to the Eilenberg-MacLane space of a torsion free finitely generated nilpotent group a close relation to the the theory of Passi polynomial maps is obtained.
We study each of the 16 types of complete intersection Calabi-Yau threefolds in projective n-space times the projective line, for various n, and prove existence of isolated rational curves of bidegree (d,0) for all positive integers d on a general threefold of each of these types. As a corollary we find isolated rational curves in all degrees on a general CICY of each of the 5 possible types in usual projective n-space.
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula $12 λ_1=δ$ for genus 1.