Let $\V$ be a mixed characteristic complete discrete valuation ring with perfect residue field $k$. We solve Berthelot's conjectures on the stability of the holonomicity over smooth projective formal $\V$-schemes. Then we build a category of complexes of arithmetic $\D$-modules over quasi-projective $k$-varieties with bounded, $F$-holonomic cohomology. We get its stability under Grothendieck's six operations.
Tautological systems developed in [6],[7] are Picard-Fuchs type systems to study period integrals of complete intersections in Fano varieties. We generalize tautological systems to local complete intersections, which are zero loci of global sections of vector bundles over Fano varieties. In particular, we obtain similar criterion as [6, 7] about holonomicity and regularity of the system. We also prove solution rank formulas and geometric realizations of solutions following the work of hypersurfaces [5, 4].
We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.
Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the systems of fundamental forms at general points of projective varieties. We study relations between the algebraic properties of symbol systems and the geometric properties of Euler-symmetric projective varieties. We describe also the relation between Eule…
We compute the Hilbert series of general weighted flag varieties and discuss a computer-aided method to determine their defining equations. We apply our results to weighted flag varieties coming from the Lie groups of type G_2 and GL(6), to construct some families of polarised projective varieties in codimensions 8 and 6, respectively.
We give a classification of embedded smooth projective varieties swept out by rational homogeneous varieties whose Picard number and codimension are one.
Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.
It is known that projective minimal models satisfy the celebrated Miyaoka-Yau inequalities. In this article, we extend these inequalities to the set of all smooth, projective and non-uniruled varieties.
We compute the divisor class group and the Picard group of projective varieties with Hibi rings as homogeneous coordinate rings. These varieties are precisely the toric varieties associated to order polytopes. We use tools from the theory of toric varieties to get a description of the two groups which only depends on combinatorial properties of the underlying poset.
Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.
The Lefschetz algebra $L^*(X)$ of a smooth complex projective variety $X$ is the subalgebra of the cohomology algebra of $X$ generated by divisor classes. We construct smooth complex projective varieties whose Lefschetz algebras do not satisfy analogues of the hard Lefschetz theorem and Poincaré duality.
In this note we introduce the concept of reflective projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, the Chern-Schwartz-MacPherson classes of the strata completely determine the local Euler obstructions and the polar degrees. We also propose an algorithm to compute the local Euler obstructions when such varieties are formed by group orbits. As examples we compute the local Euler obstructions of quadratic hypersurfaces and ordinary determinantal varieties to illustrate our method.
We study the equivariant cobordism rings for the action of a torus $T$ on smooth varieties over an algebraically closed field of characteristic zero. We prove a theorem describing the rational $T$-equivariant cobordism rings of smooth projective $G$-spherical varieties with the action of a maximal torus $T$ of $G$. As an application, we obtain explicit presentations for the rational equivariant cobordism rings of smooth projective horospherical varieties of Picard number one.
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo-Mumford regularity.
We give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with intersections by general effective Cartier divisors, we can compile a system of linear equations which determine the coefficients for the Segre class pushed forward to projective space. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space; to our knowledge, this is t…
We make explicit the equations of any projective $PGL_2(\C)$-variety defined by quadrics. We study their zero-locus and their relationship with the geometry of the Veronese curve.
We shortly describe the algorithms behind some of the functions provided by the Macaulay2 package MultiprojectiveVarieties, a package for multi-projective varieties and rational maps between them.
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of $G$-homogeneous varieties appear. These are varieties over which $G$ acts transitively with possibly non-reduced isotropy subgroup. In this paper we study these varieties which we call ${\it \mbox{projective pseudo-homogeneous varieties}}$ for $G$ inner type over $k$ and prove that their motives satisfy Rost nilpotence. We also find their motivic decompositions a…
We prove that the universal cover of a normal, projective variety X is quasi-projective if and only if a finite, étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.
We define partially ample subvarieties of projective varieties, generalizing Ottem's work on ample subvarieties, and show their ubiquity. As an application, we obtain a connectedness result for pre-images of subvarieties by morphisms, reminiscent to a problem posed by Fulton-Hansen.
We give an example of a pair of projective symplectic varieties in arbitrarily large dimensions which are D-equivalent, L-equivalent, and birationally inequivalent.
We provide a generalization of the algorithm of Eklund-Jost-Peterson for computing Segre classes of closed subschemes of projective k-space. The algorithm is here generalized to computing the Segre classes of closed subschemes of smooth projective toric varieties.
In this paper we construct a tilting sheaf for Severi-Brauer Varieties and Involution Varieties. This sheaf relates the derived category of each variety to the derived category of modules over a ring whose semisimple component consists of the Tits algebras of the corresponding linear algebraic group.
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
We give evidence for a uniformization-type conjecture, that any algebraic variety can be altered into a variety endowed with a tower of smooth fibrations of relative dimension one.
We list the irreducible reduced and not degenerate normal projective varieties $X\subset\mathbb{P}^N$ of dimension $n$ and degree five defined over an algebraically closed field $k$ of char$(k) = 0$. In the smooth case, or when $n = 2$, we give also classification results for any algebraically closed field $k$ of char$(k) \geq 0$.
In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds.
We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.
In this addendum we generalize some results of our article "Generically split projective homogeneous varieties", Duke Math. J. 152 (2010), no. 1, 155-173. More precisely, we remove all restrictions on the characteristic of the base field and complete our classification by the last missing case, namely $\PGO_{2n}^+$.
For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.
We give a new proof of a theorem by Pareschi, Popa and Schnell that the direct image of the canonical bundle of a smooth projective variety along a morphism to an abelian variety admits a Chen-Jiang decomposition, without using the theory of Hodge modules.
In this article we prove a result comparing rationality of algebraic cycles over the function field of a projective homogeneous variety under a linear algebraic group of type $F_4$ or $E_8$ and over the base field, which can be of any characteristic.
We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold P^1\timesP^2. In the course of the proof, we classify projective manifolds of dimension five swept out by planes.
