Rational Singularities Research Articles

Littlewood Complexes and Analogues of Determinantal Varieties

One interesting combinatorial feature of classical determinantal varieties is that the character of their coordinate rings give a natural truncation of the Cauchy identity in the theory of symmetric functions. Natural generalizations of these varieties exist and have been studied for the other classical groups. In this paper we develop the relevant properties from scratch. By studying the isotypic decomposition of their minimal free resolutions one can recover classical identities due to Littlewood for expressing an irreducible character of a classical group in terms of Schur functions. We propose generalizations for the exceptional groups. In type G_2, we completely analyze the variety and its minimal free resolution and get an analogue of Littlewood's identities. We have partial results for the other cases. In particular, these varieties are always normal with rational singularities.

Effective non-vanishing of global sections of multiple adjoint bundles for quasi-polarized n-folds

We find natural numbers m such that the dimensions of global sections of multiple adjoint bundles h0(m(KX + L)) are strictly greater than zero for any quasi-polarized n-folds (X, L) for which X is a complex normal Gorenstein projective variety of dimension n with only rational singularities and KX + L is nef.

Secant Cumulants and Toric Geometry

We study the secant line variety of the Segre product of projective spaces using special cumulant coordinates adapted for secant varieties. We show that the secant variety is covered by open normal toric varieties. We prove that in cumulant coordinates its ideal is generated by binomial quadrics. We present new results on the local structure of the secant variety. In particular, we show that it has rational singularities and we give a description of the singular locus. We also classify all secant varieties that are Gorenstein. Moreover, generalizing (Sturmfels and Zwiernik 2012), we obtain analogous results for the tangential variety.

Wonderful resolutions and categorical crepant resolutions of singularities

Abstract Let X be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of X by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove that if X has rational singularities and has a wonderful resolution of singularities, then X admits a categorical crepant resolution of singularities. As an immediate corollary, we get that all determinantal varieties defined by the minors of a generic square/symmetric/skew-symmetric matrix admit categorical crepant resolution of singularities.

Jacobian discrepancies and rational singularities

Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call {Jacobian \: discrepancy} , is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert–Riemenschneider canonical sheaf of a normal variety. As an application, we give characterizations for rational and Du Bois singularities on normal Cohen–Macaulay varieties in terms of Jacobian discrepancies. In the case when the canonical class of the variety is \mathbb Q -Cartier, our result provides the necessary corrections for the converses to hold in theorems of Elkik, of Kovács, Schwede and Smith, and of Kollár and Kovács on rational and Du Bois singularities.

Generically Trivial Azumaya Algebras on a Rational Surface with a Non Rational Singularity

Elementary examples are presented of normal algebraic surfaces X with singular points x such that at the local ring 𝒪 X, x there exist Azumaya algebras of all orders in the Brauer group that are split by the field of rational functions on X. These algebra classes correspond to elements of torsion in the class group of the henselian local ring . The surfaces X are affine normal rational and the singularities x are nonrational.

ALMOST ADJACENT IDEALS IN TWO-DIMENSIONAL MUHLY RATIONAL SINGULARITIES

Let (R, m) be a two-dimensional Muhly rational singularity, i.e. the residue field R/m is algebraically closed and the associated graded ring is an integrally closed domain. The goal of this paper is to use immediate quadratic transforms and degree coefficients to investigate complete ideals that are almost adjacent to m, i.e. [Formula: see text].

Brauer groups of singular del Pezzo surfaces

We describe the effect of rational singularities on the Brauer group of a surface, and compute the Brauer groups of all singular del Pezzo surfaces over an algebraically closed field.

Positroid varieties: juggling and geometry

AbstractWhile the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

On the $F$-rationality and cohomological properties of matrix Schubert varieties

We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the $F$-rationality of matrix Schubert varieties. Although it is known that such varieties are $F$-regular (hence $F$-rational) by the global $F$-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott–Samelson resolution of Schubert varieties. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.

Finiteness of cominuscule quantum $K$-theory

The product of two Schubert classes in the quantum K-theory ring of a homogeneous space X = G/P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when X is cominuscule, all boundary Gromov-Witten varieties defined by three single points in X are rationally connected.

On the classification of rational surface singularities

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely certain types of graphs. This allows to extend the classification of rational singularities to multiplicity 8. We also discuss the complexity of rational resolution graphs.

Degenerate flag varieties of type A: Frobenius splitting and BW theorem

Let \(\mathcal F ^a_\lambda \) be the PBW degeneration of the flag varieties of type \(A_{n-1}\). These varieties are singular and are acted upon with the degenerate Lie group \(SL_n^a\). We prove that \(\mathcal F ^a_\lambda \) have rational singularities, are normal and locally complete intersections, and construct a desingularization \(R_\lambda \) of \(\mathcal F ^a_\lambda \). The varieties \(R_\lambda \) can be viewed as towers of successive \(\mathbb{P }^1\)-fibrations, thus providing an analogue of the classical Bott–Samelson–Demazure–Hansen desingularization. We prove that the varieties \(R_\lambda \) are Frobenius split. This gives us Frobenius splitting for the degenerate flag varieties and allows to prove the Borel–Weil type theorem for \(\mathcal F ^a_\lambda \). Using the Atiyah–Bott–Lefschetz formula for \(R_\lambda \), we compute the \(q\)-characters of the highest weight \(\mathfrak sl _n\)-modules.

Three dimensional canonical singularities in codimension two in positive characteristic

We investigate local structure of a three dimensional variety X defined over an algebraically closed field k of characteristic p>0 with at most canonical singularities. Under the assumption that p⩾3 and a general hyperplane cut of X has at most rational singularities, we show that local structure of X in codimension two is well understood in the level of local equations. Consequently, we find that i) any singularity of such a variety X in codimension two is compound Du Val, ii) it has a crepant resolution, iii) it is analytically a product of a rational double point and a nonsingular curve when p⩾3 with two exceptions in p=3, and iv) R1π⁎OX˜=R1π⁎KX˜=0 holds outside some finite points of X for any resolution of singularities π:X˜→X.

Inequalities on Bruhat graphs, R- and Kazhdan–Lusztig polynomials

From a combinatorial perspective, we establish three inequalities on coefficients of R- and Kazhdan–Lusztig polynomials for crystallographic Coxeter groups: (1) Nonnegativity of (q−1)-coefficients of R-polynomials, (2) a new criterion of rational singularities of Bruhat intervals by sum of quadratic coefficients of R-polynomials, (3) existence of a certain strict inequality (coefficientwise) of Kazhdan–Lusztig polynomials. Our main idea is to understand Deodharʼs inequality in a connection with a sum of R-polynomials and edges of Bruhat graphs.

Derived splinters in positive characteristic

AbstractThis paper introduces the notion of a derived splinter. Roughly speaking, a scheme is a derived splinter if it splits off from the coherent cohomology of any proper cover. Over a field of characteristic 0, this condition characterises rational singularities, as suggested by the work of Kovács. Our main theorem asserts that over a field of characteristic p, derived splinters are the same as (underived) splinters, i.e. schemes that split off from any finite cover. Using this result, we answer some questions of Karen Smith concerning the extension of Serre/Kodaira-type vanishing results beyond the class of ample line bundles in positive characteristic; these are purely projective geometric statements independent of singularity considerations. In fact, we can prove ‘up to finite cover’ analogues in characteristic p of many vanishing theorems known in characteristic 0. All these results fit naturally in the study of F-singularities, and are motivated by a desire to understand the direct summand conjecture.

The ℓ-adic dualizing complex on an excellent surface with rational singularities

In this article, we show that if X is an excellent surface with rational singularities, the constant sheaf ℚl is a dualizing complex. In coefficient ℤl, we also prove that the obstruction for ℤl to become a dualizing complex, lies on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of l-adic cohomology groups on such surfaces.

Singularities of Generalized Richardson Varieties

Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood–Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this article, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singularities. We also study in detail the singular loci of projection varieties in Type A Grassmannians. We use Kleiman's Transversality Theorem to determine the singular locus of any intersection variety in terms of the singular loci of Schubert varieties. This is a generalization of a criterion for any Richardson variety to be smooth in terms of the nonvanishing of certain cohomology classes which has been known by some experts in the field, but we don't believe has been published previously.

Springer fiber components in the two columns case for types $A$ and $D$ are normal

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 = 0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

One-fibered ideals in 2-dimensional rational singularities that can be desingularized by blowing up the unique maximal ideal

Abstract Let (R;m) be a 2-dimensional rational singularity with algebraically closed residue field and whose associated graded ring is an integrally closed domain. Göhner has shown that for every prime divisor v of R, there exists a unique one-fibered complete m-primary ideal A v in R with unique Rees valuation v and such that any complete m-primary ideal with unique Rees valuation v, is a power of A v. We show that for v ≠ ordR, A v is the inverse transform of a simple complete ideal in an immediate quadratic transform of R, if and only if the degree coefficient d(A v; v) is 1. We then give a criterion for R to be regular.