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We provide a classification of developable cubic hypersurfaces in P 4 . Using the correspondence between forms of degree 3 on P 4 and Artinian Gorenstein K -algebras, given by Macaulay-Matlis duality, we describe the locus in GOR ( 1 , 5 , 5 , 1 ) corresponding to those algebras which satisfy the Strong Lefschetz property.

Let X : = Spec ( R ) be an affine Noetherian scheme, and M ⊂ N be a pair of finitely generated R-modules. Denote their Rees algebras by R ( M ) and R ( N ) . Let N n be the nth homogeneous component of R ( N ) and let M n be the image of the nth homogeneous component of R ( M ) in N n . Denote by M n ‾ be the integral closure of M n in N n . We prove that Ass X ( N n / M n ‾ ) and Ass X ( N n / M n ) are asymptotically stable, generalizing known results for the case where M is an ideal or where N is a free module. Suppose either that M and N are free at the generic point of each irreducible component of X or N is contained in a free R-module. When X is universally catenary, we prove a generalization of a classical result due to McAdam and obtain a geometric classification of the points appearing in Ass X ( N n / M n ‾ ) . Notably, we show that if x ∈ Ass X ( N n / M n ‾ ) for some n, then x is the generic point of a codimension-one component of the nonfree locus of N / M or x is a generic point of an irreducible set in X where the fiber dimension Proj ( R ( M ) ) → X jumps. We prove a converse to this result without requiring X to be universally catenary. Our approach is geometric in spirit. Also, we recover, strengthen, and prove a sort of converse of an important result of Kleiman and Thorup about integral dependence of modules.

We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra A we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms l give maximal rank for all the multiplication maps × l : [ A ] i ⟶ [ A ] i + 1 . This is a Zariski open set and its complement is the non-Lefschetz locus. For monomial complete intersections, we completely describe the non-Lefschetz locus. For general complete intersections of codimension three and four we prove that the non-Lefschetz locus has the expected codimension, which in particular means that it is empty in a large family of examples. For general Gorenstein algebras of codimension three with a given Hilbert function, we prove that the non-Lefschetz locus has the expected codimension if the first difference of the Hilbert function is of decreasing type. For completeness we also give a full description of the non-Lefschetz locus for artinian algebras of codimension two.

Let , be a reductive Lie algebra over an algebraically closed field F with char F = p > 0 . Suppose G satisfies Jantzen's standard assumptions. Then the structure of Z, the center of the enveloping algebra , is described by (the extended) Veldkamp's theorem. We examine here the deviation of Z from this theorem, in case , or and p | n . It is shown that Veldkamp's description is valid for . This implies that Friedlander–Parshall–Donkin decomposition theorem for holds in case p is good for a semi-simple simply connected G (excluding, if p = 2 , A 1 -factors of G). In case or we prove a fiber product theorem for a polynomial extension of Z. However Veldkamp's description mostly fails for and . In particular Z is not Cohen–Macaulay if n > 4 , in both cases. Contrary to a result of Kac–Weisfeiler, we show for an odd prime p that and do not generate . We also show for that the codimension of the non-Azumaya locus of Z is at least 2 (if n ≥ 3 ), and exceeds 2 if n > 4 . This refutes a conjecture of Brown–Goodearl. We then show that Z is factorial (excluding ), thus confirming a conjecture of Premet–Tange. We also verify Humphreys conjecture on the parametrization of blocks, in case p is good for G.

We construct canonical non-vanishing global sections of powers of the Hodge bundle on each Ekedahl–Oort stratum of a Hodge type Shimura variety. In particular we recover the quasi-affineness of the Ekedahl–Oort strata. In the projective case, this gives a very short proof of non-emptiness of Ekedahl–Oort strata. It follows that the Newton strata are also nonempty, by a result of S. Nie. From the canonicity of our construction, we deduce the fact that the μ -ordinary locus is determined by the Ekedahl–Oort strata of its image under any embedding of Shimura varieties.

Let X be an equidimensional scheme of finite type over a perfect field k . Under these conditions, the multiplicity along points of X defines an upper semi-continuous function, say mult X : X → N , which stratifies X into its locally closed level sets. We study this stratification, and the behavior of the multiplicity when blowing up at regular equimultiple centers. We also discuss a natural compatibility of these two concepts when X is replaced with its underlying reduced scheme. The main result in this paper is to show that, given a variety X , there is a well defined Rees algebra over X , naturally attached to maximum value of the multiplicity.

We investigate the viability of defining an intersection product on algebraic cycles on a singular algebraic variety by pushing forward intersection products formed on a resolution of singularities. For varieties with resolutions having a certain structure (including all varieties over a field of characteristic zero), we obtain a stratification which reflects the geometry of the centers and the exceptional divisors. This stratification is sufficiently fine that divisors can be intersected with r-cycles (for r ≥ 1 ), and 2-cycles can be intersected on a fourfold, provided their incidences with the strata are controlled. Similar pairings are defined on a variety with one-dimensional singular locus.

We propose an object called 'sepcanonical system' on a stable curve $X_0$ which is to serve as limiting object- distinct from other such limits introduced previously- for the canonical system, as a smooth curve degenerates to $X_0$. First for curves which cannot be separated by 2 or fewer nodes, the so-called '2-inseparable' curves, the sepcanonical system is just the sections of the dualizing sheaf, which is not very ample iff $X_0$ is a limit of smooth hyperelliptic curves (such $X_0$ are called 2-inseparable hyperelliptics). For general, 2-separable curves $X_0$ this assertion is false, leading us to introduce the sepcanonical system, which is a collection of linear systems on the '2-inseparable parts' of $X_0$, each associated to a different twisted limit of the canonical system, where the entire collection varies smoothly with $X_0$. To define sepcanonical system, we must endow the curve with extra structure called an 'azimuthal structure'. We show that the sepcanonical system is 'essentially very ample' unless the curve is a tree-like arrangement of 2-inseparable hyperelliptics. In a subsequent paper, we will show that the latter property is equivalent to the curve being a limit of smooth hyperelliptics, and will essentially give defining equation for the closure of the locus of smooth hyperelliptic curves in the moduli space of stable curves. The current version includes additional references to, among others, Catanese, Maino, Esteves and Caporaso., Comment: arXiv admin note: substantial text overlap with arXiv:1011.0406; to appear in J. Algebra

To a dominant morphism $X/S \to Y/S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms., Comment: 25 pages, the introduction and section 1 are a little updated. The proof of Th 2.1 clarified

We treat non-symplectic automorphisms on $K3$ surfaces which act trivially on the N\'{e}ron-Severi lattice. In this paper, we classify non-symplectic automorphisms of prime-power order, especially 2-power order on $K3$ surfaces, i.e., we describe their fixed locus., Comment: 15 pages

We study Ekedahl–Oort strata on the moduli space Ag of g-dimensional principally polarized abelian varieties in positive characteristic, and Kottwitz–Rapoport strata on its variants AJ with parahoric level structure. First, we show that every Ekedahl–Oort stratum is isomorphic to a parahoric Kottwitz–Rapoport stratum. Second, both supersingular Ekedahl–Oort strata and supersingular Kottwitz–Rapoport strata are isomorphic to disjoint unions of Deligne–Lusztig varieties (see Hoeve (2010) [10] and Görtz and Yu (2010) [5], resp.), and here we compare these isomorphisms. Finally we give an explicit description of Kottwitz–Rapoport strata contained in the supersingular locus in the general parahoric case.

Let $X$ be a smooth irreducible nondegenerate projective variety and let $X^*$ denote its dual variety. It is well known that $\sigma_2(X)^*$, the dual of the 2-secant variety of $X$, is a component of the singular locus of $X^*$. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of $X$, is a component of the sigular locus of $X^*$. In this paper we provide a sufficient condition for this component to be of maximal dimension and show how it can be used to determine which dual varieties of Grassmannians are normal. That last part may be compared to what has been done for hyperdeterminants by J. Weyman and A. Zelevinski (1996)., Comment: 14 pages, appeared in Journal of Algebra (2011)

We give a new proof of the linearity of the Segre locus, that is, the locus of points from which a variety is projected non-birationally. Our proof works in the case where the characteristic is zero or large enough. For small characteristics, we give an example of a variety whose Segre locus is non-linear. To show these results, we explicitly give a method to compute polynomials generating the defining ideal of the Segre locus, for a variety embedded in projective space, in arbitrary characteristic.

We describe elementary transformations between minimal models of rational surfaces in terms of unprojections. These do not fit into the framework of Kustin-Miller unprojections as introduced by Papadakis and Reid, since we have to leave the world of projectively Gorenstein varieties. Also, our unprojections do not depend on the choice of the unprojection locus only, but need extra data corresponding to the choice of a divisor on this unprojection locus., Comment: 12 pages, only minor changes

Let f:C→B be a smoothing of a stable curve C and Sf∗ be the moduli space of theta characteristics on the smooth fibers of f. We describe the Néron model N(Sf∗), in terms of combinatorial invariants of the dual graph of C. Furthermore, we provide a modular description of N(Sf∗) and we construct an immersion ψf:N(Sf∗)↪JEσ, where JEσ is a suitable relative compactified Jacobian. We show that ψf factors through the locus of JEσ parametrizing locally free rank-1 sheaves.

The new intersection theorem is used to derive a criteria for flat descent in the setting of integral ring extensions. Applications, such as “purity of branch locus” for extensions of normal domains, are noted.

Given a normal surface singularity ( X , Q ) and a birational morphism to a non-singular surface π : X → S , we investigate the local geometry of the exceptional divisor L of π . We prove that the dimension of the tangent space to L at Q equals the number of exceptional components meeting at Q . Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterisation of minimal singularities is obtained in these terms.

We show that the F-signature of a strongly F-regular local ring of characteristic p exists in the case that the non- Q -Gorenstein locus is dimension 1, given that a certain bound on zeroth local cohomology modules holds. This bound is shown to hold for rings essentially of finite type over a field. This is the first case in which hypotheses sufficient to prove the existence of the F-signature are not readily sufficient to prove the implication that weak F-regularity implies strong F-regularity.

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a complete intersection of a linear covariant of quintics. The projective dual of this hypersurface can be identified with itself via an involution. It is shown that the Jacobian ideal of H is perfect of height two, and we describe its SL_2-equivariant minimal resolution. The last section develops a general formalism for evectants of covariants of binary forms, which is then used to calculate the evectant of H.

Given a Cayley–Hamilton smooth order A in a central simple algebra Σ, we determine the flat locus of the Brauer–Severi fibration of A. Moreover, we give a classification of all (reduced) central singularities where the flat locus differs from the Azumaya locus and show that the fibers over the flat, non-Azumaya points near these central singularities can be described as fibered products of graphs of projection maps. This generalizes an old result of Artin on the fibers of the Brauer–Severi fibration of a maximal order over a ramified point. Finally, we show these fibers are also toric quiver varieties and use this fact to compute their cohomology.

This paper studies the stable reduction of p-cyclic covers X→ P K 1 of the projective line over p-adic fields. So far, an algorithm to effectively determine the stable reduction of such covers is only known under additional hypothesis on the branch locus of the cover. Here, rather than restricting the type of cover, we consider the general case and obtain results on the structure of the special fiber Xk of the stable reduction of X. Special attention is payed to making all constructions effective. The central result is a formula computing the number of vanishing cycles on Xk. In particular, we give criteria for the special fiber of the stable reduction to be tree-like and for when X is a Mumford curve. Refining the analysis of vanishing cycles, we describe an algorithm that computes all the components of positive p-rank in the stable model.

We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X^G and the quotient surface Y = X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori's extremal ray theory and recent results of Alexeev and also Ambro on the existence of good anti-canonical divisors are used., with an Appendix by I. Dolgachev; to appear in Journal of Algebra

This paper begins by introducing and characterizing Buchsbaum–Rim sheaves on Z = Proj R, where R is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum–Rim sheaf B ϕ, i.e, we consider morphisms ψ: P → B ϕ of sheaves on Z dropping rank in the expected codimension, where H0*(Z, P ) is a free R-module. The main purpose of this paper is to study properties of schemes associated to the degeneracy locus S of ψ. It turns out that S is often not equidimensional. Let X denote the top-dimensional part of S. In this paper we measure the “difference” between X and S, compute their cohomology modules and describe ring-theoretic properties of their coordinate rings. Moreover, we produce graded free resolutions of X (and S) which are in general minimal. Among the applications we show how one can embed a subscheme into an arithmetically Gorenstein subscheme of the same dimension and prove that zero-loci of sections of the dual of a null correlation bundle are arithmetically Buchsbaum.

In this paper, we study linkage by a wider class of ideals than the complete intersections. We are most interested in how the Cohen–Macaulay property behaves along this more general notion of linkage. In particular, if idealsAandBare linked by a generically Gorenstein Cohen–Macaulay idealI, and ifAis a Cohen–Macaulay ideal, we give a criterion forBto be a Cohen–Macaulay ideal. WhenR/Bis not Cohen–Macaulay, we can give in many cases an easy description of the non–Cohen–Macaulay locus ofR/B, and also a criterion forR/Bto have almost maximal depth.