Pseudo-effective Cone Research Articles

On Pseudo-Effective Cones of Projective Bundles and Volume Function

On Pseudo-Effective Cones of Projective Bundles and Volume Function

Effective cone of the blowup of the symmetric product of a curve

Let C C be a smooth curve of genus g ≥ 1 g \geq 1 and let C ( 2 ) C^{(2)} be its second symmetric product. In this note we prove that if C C is very general, then the blowup of C ( 2 ) C^{(2)} at a very general point has nonpolyhedral pseudo-effective cone. The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.

Blown-up toric surfaces with non-polyhedral effective cone

Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone. As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space M ¯ 0 , n \overline{M}_{0,n} of stable rational curves is not polyhedral for n ≥ 10 n\geq 10 . These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density.

Slope semistability and positive cones of Grassmann bundles

Let [Formula: see text] be a vector bundle of rank [Formula: see text] on a smooth complex projective variety [Formula: see text]. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle Gr[Formula: see text] parametrizing [Formula: see text]-dimensional subspaces of the fibers of [Formula: see text], where [Formula: see text] rk([Formula: see text]), under assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. In particular, we show that nef cone and the pseudoeffective cone of Gr[Formula: see text] coincide if and only if nef cone and pseudoeffective cone of [Formula: see text] coincide under the assumption that [Formula: see text] is a slope semistable bundle on [Formula: see text] with [Formula: see text](End([Formula: see text]))=0. We also discuss about the nefness and ampleness of the universal quotient bundle [Formula: see text] on Gr[Formula: see text].

On the geometry of elliptic pairs

An elliptic pair(X,C) is a projective rational surface X with log terminal singularities, and an irreducible curve C contained in the smooth locus of X, with arithmetic genus 1 and self-intersection 0. They are a useful tool for determining whether the pseudo-effective cone of X is polyhedral [1], and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where X is the blow-up of a projective toric surface at the identity element of the torus. In this paper, we classify all toric elliptic pairs of Picard number two. Strikingly, it turns out that there are only three of these. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of P2 at nine points on a nodal cubic, in characteristic p. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes p of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes p of positive density.

Boucksom–Zariski and Weyl Chambers on Irreducible Holomorphic Symplectic Manifolds

Abstract Inspired by the work of Bauer, Küronya, and Szemberg, we provide for the big cone of a projective irreducible holomorphic symplectic (IHS) manifold a decomposition into chambers (which we describe in detail), in each of which the support of the negative part of the divisorial Zariski decomposition is constant. We see how the obtained decomposition of the big cone allows to describe the volume function. Moreover, similarly to the case of surfaces, we see that the big cone of a projective IHS manifold admits a decomposition into simple Weyl chambers, which we compare to that induced by the divisorial Zariski decomposition. To conclude, we determine the structure of the pseudo-effective cone.

NORMALISED TANGENT BUNDLE, VARIETIES WITH SMALL CODEGREE AND PSEUDOEFFECTIVE THRESHOLD

AbstractWe propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number $1$ .

Chow groups and pseudoffective cones of complexity-one T-varieties

We show that the pseudoeffective cone of k-cycles on a complete complexity-one T-variety is rational polyhedral for any k, generated by classes of T-invariant subvarieties. When X is also rational, we give a presentation of the Chow groups of X in terms of generators and relations, coming from the combinatorial data defining X as a T-variety.

Betti numbers and pseudoeffective cones in 2-Fano varieties

Abstract The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

Effective cones of cycles on products of projective bundles over curves

Let [Formula: see text] where [Formula: see text] is a smooth complex projective curve and let [Formula: see text], [Formula: see text] be two vector bundles over [Formula: see text]. In this paper, we compute the pseudo effective cones of higher co-dimension cycles on [Formula: see text].

Pseudo-effective cones of projective bundles and weak Zariski decomposition

In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over a smooth complex projective variety $X$, where $E$ is a semistable bundle on $X$ with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get the equality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) = \overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffective cone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we show the equality of nef and pseudoeffective cones of divisors of projective bundles over some special varieties. In particular, we show that weak Zariski decomposition exists on these projective bundles. We also show that a semistable bundle $E$ of rank $r \geq 2$ with $c_2\bigl(End(E)\bigr) = 0$ on a smooth complex projective variety of Picard number 1 is $k$-homogeneous i.e. $\overline{Eff}^k\bigl(\mathbb{P}_X(E)\bigr) = Nef^k\bigl(\mathbb{P}_{X}(E)\bigr)$ for all $1 \leq k < r$. Finally, we show that weak Zariski decomposition exists for a fibre product $\mathbb{P}_C(E)\times_C\mathbb{P}(E')$ over a smooth projective curve $C$.

Effective cycles on the symmetric product of a curve, II: the Abel–Jacobi faces

In this paper, which is a sequel of [BKLV17], we study the convex-geometric properties of the cone of pseudoeffective n -cycles in the symmetric product C_d of a smooth curve C . We introduce and study the Abel–Jacobi faces, related to the contractibility properties of the Abel–Jacobi morphism and to classical Brill–Noether varieties. We investigate when Abel–Jacobi faces are non-trivial, and we prove that for d sufficiently large (with respect to the genus of C ) they form a maximal chain of perfect faces of the tautological pseudoeffective cone (which coincides with the pseudoeffective cone if C is a very general curve).

Unstable Loci in Flag Varieties and Variation of Quotients

Abstract We consider the action of a semisimple subgroup $\hat{G}$ of a semisimple complex group $G$ on the flag variety $X=G/B$ and the linearizations of this action by line bundles $\mathcal L$ on $X$. We give an explicit description of the associated unstable locus in dependence of $\mathcal L$, as well as a formula for its (co)dimension. We observe that the codimension is equal to 1 on the regular boundary of the $\hat{G}$-ample cone and grows towards the interior in steps by 1, in a way that the line bundles with unstable locus of codimension at least $q$ form a convex polyhedral cone. We also give a description and a recursive algorithm for determining all GIT-classes in the $\hat{G}$-ample cone of $X$. As an application, we give conditions ensuring the existence of GIT-classes $C$ with an unstable locus of codimension at least two and which moreover yield geometric GIT quotients. Such quotients $Y_C$ reflect global information on $\hat{G}$-invariants. They are always Mori dream spaces, and the Mori chambers of the pseudoeffective cone $\overline{\textrm{Eff}}(Y_C)$ correspond to the GIT chambers of the $\hat{G}$-ample cone of $X$. Moreover, all rational contractions $f: Y_{C} \ \scriptsize{-}\scriptsize{-}{\scriptsize{-}\kern-5pt\scriptsize{&amp;gt;}}\ Y^{\prime}$ to normal projective varieties $Y^{\prime}$ are induced by GIT from linearizations of the action of $\hat{G}$ on $X$. In particular, this is shown to hold for a diagonal embedding $\hat{G} \hookrightarrow (\hat{G})^k$, with sufficiently large $k$.

Fibration and classification of smooth projective toric varieties of low Picard number

In this paper, we show that a smooth toric variety [Formula: see text] of Picard number [Formula: see text] always admits a nef primitive collection supported on a hyperplane admitting non-trivial intersection with the cone [Formula: see text] of numerically effective divisors and cutting a facet of the pseudo-effective cone [Formula: see text], that is [Formula: see text]. In particular, this means that [Formula: see text] admits non-trivial and non-big numerically effective divisors. Geometrically, this guarantees the existence of a fiber type contraction morphism over a smooth toric variety of dimension and Picard number lower than those of [Formula: see text], so giving rise to a classification of smooth and complete toric varieties with [Formula: see text]. Moreover, we revise and improve results of Oda–Miyake by exhibiting an extension of the above result to projective, toric, varieties of dimension [Formula: see text] and Picard number [Formula: see text], allowing us to classifying all these threefolds. We then improve results of Fujino–Sato, by presenting sharp (counter)examples of smooth, projective, toric varieties of any dimension [Formula: see text] and Picard number [Formula: see text] whose non-trivial nef divisors are big, that is [Formula: see text]. Producing those examples represents an important goal of computational techniques in definitely setting an open geometric problem. In particular, for [Formula: see text], the given example turns out to be a weak Fano toric fourfold of Picard number 4.

Balanced complexes and effective divisors on

Doran, Giansiracusa, and Jensen showed a bijection between homogeneous elements in the Cox ring of not divisible by any exceptional divisor section, and weighted pure-dimensional simplicial complexes satisfying a zero-tension condition. Motivated by the study of the monoid of effective divisors, the pseudoeffective cone and the Cox ring of we point out a simplification of the zero-tension condition and study the space of balanced complexes. We give examples of irreducible elements in the monoid of effective divisors of for large n. In the case of we classify all such irreducible elements arising from nonsingular complexes and give an example of how irreducibility can be shown in the singular case.Communicated by Lawrence Ein

Effective Divisors in the Projectivized Hodge Bundle

Abstract We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle ${\mathbb{P}}\overline{{\mathcal{H}}}_g$ over $\overline{{\mathcal{M}}}_g$, which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span extremal rays of the respective pseudoeffective cones.

Nef and pseudoeffective cones of product of projective bundles over a curve

Let X=P(E1)×CP(E2), where C is any smooth irreducible curve and E1, E2 are vector bundles over C. In this paper, we calculate the nef cone and pseudoeffective cone of X whenever E1 and E2 are semistable, and show that they coincide. We also calculate the nef cone and pseudoeffective cone of X in case both E1 and E2 have rank 2, and neither E1 nor E2 is semistable or at least one of them is semistable. As a result, in the case when both E1 and E2 are of rank 2 bundles on C, we get that nef cone and pseudoeffective cone of X coincide iff both E1 and E2 are semistable.

Correspondences between convex geometry and complex geometry

We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or K\"ahler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geometric inequalities for divisor classes. We also reinterpret several classical convex geometry results in the context of algebraic geometry: the Alexandrov body construction is the convex geometry version of divisorial Zariski decomposition; Minkowski's existence theorem is the convex geometry version of the duality between the pseudo-effective cone of divisors and the movable cone of curves.

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.

Asymptotic order-of-vanishing functions on the pseudoeffective cone

Asymptotic order-of-vanishing functions on the pseudoeffective cone