Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G$G$ of rank 2 over a field. We exhibit exceptional collections of the expected length for types _A2$A_2$ and _B2=_C2$B_2=C_2$ and prove that no such collection exists for type _G2$G_2$. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G$G$-varieties for split linear algebraic groups G$G$ of rank at most 2.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

Conics in the Grothendieck ring

The subring of the Grothendieck ring of k-varieties generated by smooth conics is described, giving many zero divisors. The proof uses only elementary projective geometry.     

Birational maps between generalized Severi-Brauer varieties

A conjecture of Amitsur states that two Severi-Brauer varieties V(A) and V(B) are birationally isomorphic if and only if the underlying algebras A and B are the same degree and generate the same cyclic subgroup of the Brauer group. We examine the question of finding birational isomorphisms between generalized Severi-Brauer varieties. As a first step, we exhibit a birational isomorphism between the generalized Severi-Brauer variety of an algebra and its opposite. We also extend a theorem of P. Roquette to generalized Severi-Brauer varieties and use this to show that one may often reduce the problem of finding birational isomorphisms to the case where each of the separable subfields of the corresponding algebras are maximal, and therefore to the case where the algebras have prime power degree. We observe that this fact allows us to verify Amitsur’s conjecture for many particular cases.     

The distance between superspecial abelian varieties with real multiplication

Let L be a totally real field of strict class number one and let OL be its ring of integers. Let p be a rational prime which is unramified in L. We consider the distance between two superspecial abelian varieties with real multiplication in characteristic p, where by “distance” we mean the minimal degree of an OL-isogeny. We give upper and lower bounds on the distance between superspecial abelian varieties with real multiplication by L in characteristic p in terms of p and the degree and discriminant of L.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.     

Toric degenerations of spherical varieties

We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.     

Effective cycles on blow-ups of Grassmannians

In this paper we study the pseudoeffective cones of blow-ups of Grassmannians at sets of points. For small numbers of points, the cones are often spanned by proper transforms of Schubert classes. In some special cases, we provide sharp bounds for when the Schubert classes fail to span and we describe the resulting geometry.     

Varieties with minimal secant degree and linear systems of maximal dimension on surfaces

In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space. Moreover we study varieties for which the bound is attained proving some general properties related to tangential projections, e.g. these varieties are rational. In particular we completely classify surfaces (and curves) for which the bound is attained. It turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus. This is related to classical beautiful results of Castelnuovo and Enriques that we revise here in terms of adjunction theory.     

Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space

The standard Poisson structure on the rectangular matrix variety Mm,n(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T⊂GLm+n(C). These orbits, finite in number, are shown to be smooth irreducible locally closed subvarieties of Mm,n(C), isomorphic to intersections of dual Schubert cells in the full flag variety of GLm+n(C). Three different presentations of the T-orbits of symplectic leaves in Mm,n(C) are obtained: (a) as pullbacks of Bruhat cells in GLm+n(C) under a particular map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products of sets similar to double Bruhat cells in GLm(C) and GLn(C). In presentation (a), the orbits of leaves are parametrized by a subset of the Weyl group Sm+n, such that inclusions of Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is a matrix product of one orbit with a fixed column-echelon form and one with a fixed row-echelon form. Finally, decompositions of generalized double Bruhat cells in Mm,n(C) (with respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves are obtained.     

Towards a theory of classification

The well-known difficulties arising in a classification which is not set-theoretically trivial—involving what is sometimes called a non-smooth quotient—have been overcome in a striking way in the theory of operator algebras by the use of what might be called a classification functor—the very existence of which is already a surprise. Here the notion of such a functor is developed abstractly, and a number of examples are considered (including those which have arisen for various classes of operator algebras).     

Universal deformation rings and generalized quaternion defect groups

We determine the universal deformation rings R(G,V) of certain mod 2 representations V of a finite group G which belong to a 2-modular block of G whose defect groups are isomorphic to a generalized quaternion group D. We show that for these V, a question raised by the author and Chinburg concerning the relation of R(G,V) to D has an affirmative answer. We also show that R(G,V) is a complete intersection even though R(G/N,V) need not be for certain normal subgroups N of G which act trivially on V.     

Discrete Morse theory for totally non-negative flag varieties

In a seminal 1994 paper Lusztig (1994) [26], Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_?0 of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998) [33]. In Williams (2007) [40] the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998) [28], that (G/P)_?0 - the closure of the top-dimensional cell - is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_>0 is homotopic to a sphere, is new for all G/P other than projective space.     

Quadratic degenerations of odd-orthogonal Schubert varieties

This paper is the second in a series leading to a type B_n geometric Littlewood-Richardson rule. The rule will give an interpretation of the B_n Littlewood-Richardson numbers as an intersection of two odd-orthogonal Schubert varieties and will consider a sequence of linear and quadratic deformations of the intersection into a union of odd-orthogonal Schubert varieties. This paper describes the setup for the rule and specifically addresses results for quadratic deformations, including a proof that at each quadratic degeneration, the results occur with multiplicity one. This work is strongly influenced by Vakil’s [14].     

On cohomologies and extensions of cyclic groups

Cohomology groups Hs(Zn,Zm) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Zn by the Zn-module Zm. Further, for each such a group the number of non-equivalent extensions is determined.     

Filament sets and homogeneous continua

New tools are introduced for the study of homogeneous continua. The subcontinua of a given continuum are classified into three types: filament, non-filament, and ample, with ample being a subcategory of non-filament. The richness of the collection of ample subcontinua of a homogeneous continuum reflects where the space lies in the gradation from being locally connected at one extreme to indecomposable at another. Applications are given to the general theory of homogeneous continua and their hyperspaces.     

Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e.  , up to a specified combinatorial factor, the monotone Hurwitz number in genus g$g$ with ramification specified by a given partition is a polynomial indexed by g$g$ in the parts of the partition.     

Universal polynomials for Severi degrees of toric surfaces

The Severi variety parameterizes plane curves of degree d$d$ with δ$\delta$ nodes. Its degree is called the Severi degree. For large enough d$d$, the Severi degrees coincide with the Gromov–Witten invariants of CP²$\mathbb{C}{\mathbb{P}}^2$. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed δ$\delta$, Severi degrees are eventually polynomial in d$d$.