We discuss the cone theorem for quasi-log schemes and the Mori hyperbolicity. In particular, we establish that the log canonical divisor of a Mori hyperbolic projective normal pair is nef if it is nef when restricted to the non-lc locus. This answers Svaldi's question completely. We also treat the uniruledness of the degenerate locus of an extremal contraction morphism for quasi-log schemes. Furthermore, we prove that every fiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlc locus.
The specialized Vámos polynomial is a hyperbolic polynomial of degree four in four variables with the property that none of its powers admits a definite determinantal representation. We will use a heuristical method to prove that its hyperbolicity cone is a spectrahedron.
Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow.
Recently, the authors and de Wolff introduced the imaginary projection of a polynomial $f\in\mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. In this article, we study the relation between the imaginary projections and hyperbolicity cones, where the latter ones are only defined …
We survey our recent papers (some being joint ones) about the relation between the geometry of a compact Kähler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We give applications of LMMP to positivity of log canonical divisor of a Mori / Brody / Lang hyperbolic (quasi-) projective (singular) variety.
In this paper we develop a technique for discovering (non-effective) irrational rays at the boundary of the Mori cone for linear systems on a general blowup of the plane, and give examples of such irrational rays.
We propose a generalization of SHGH Conjectures to a smooth projective surface Y: the so called Segre Problem. The study of linear systems on Y can be translated in terms of the Mori cone of the blow up $X = Bl_r Y$ at $r$ general points. Generalizing a result by de Fernex, we prove that if Segre Problem holds true, then a part of the Mori cone of $X$ does coincide with a part of the positive cone of $X$.
We determine the Mori cone of holomorphic symplectic varieties deformation equivalent to the punctual Hilbert scheme on a K3 surface. Our description is given in terms of Markman's extended Hodge lattice.
We use a family of algebraic foliations given by A. Lins Neto to provide new evidences to a conjecture, related to the Harbourne-Hirschowitz's one and implying the Nagata's conjecture, which concerns the structure of the Mori cone of blow-ups of $\mathbb{P}^2$ at very general points.
In this paper we prove that certain linear systems (and all their multiples) of plane curves with general base points and zero--self intersection are empty, thus exhibiting further examples of rays at the boundary of the Mori cone of a general blow--up of the plane.
The fundamental property of Fano varieties with mild singularities is that they have a finite polyhedral Mori cone. Thus, it is very interesting to ask: What we can say about algebraic varieties with a finite polyhedral Mori cone? I give a review of known results. All of them were obtained applying methods which were originated in the theory of discrete groups generated by reflections in hyperbolic spaces with a fundamental chamber of finite volume. This is my report on Fano conference in Torino, October 2002
We discuss some properties of the extremal rays of the cone of effective curves of surfaces that are obtained by blowing up the projective plane at points in very general position. The main motivation is to rectify an incorrect interpretation, in terms of the geometry of this cone, of the Segre--Harbourne--Gimigliano--Hirschowitz conjecture. Even though the arguments are based on elementary computations, the point of view leads to observe some other properties of the Mori cone, among which the fact that in the case of ten or more points the cone is not countably generated.
We prove that if $m \ge n-3$ then every $S_m$-invariant F-nef divisor on the moduli space of stable $n$-pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine the Mori cone of the moduli spaces of stable curves of small genus with few marked points.
The Kleiman-Mori cone plays important roles in the birational geometry. In this paper, we construct complete varieties whose Kleiman-Mori cones have interesting properties. First, we construct a simple and explicit example of complete non-projective singular varieties for which Kleiman's ampleness criterion does not hold. More precisely, we construct a complete non-projective toric variety $X$ and a line bundle $L$ on $X$ such that $L$ is positive on $\bar {NE}(X)\setminus \{0\}$. Next, we construct complete singular varieties $X$ with $NE(X)=N_1(X)\simeq \mathbb R^k$ for any $k$. These explic…
In this paper we prove that no multiple of the linear system of plane curves of degree $d\geq 4$ with a point of multiplicity $d-m$ (with $2 \leq m \leq d$) and $m(2d-m)$ simple general points is effective.
Let $X$ be a smooth $n$-dimensional projective variety over an algebraically closed field $k$ such that $K_X$ is not nef. We give a characterization of non nef extremal rays of $X$ of maximal length (i.e of length $n-1$); in the case of $\ch(k) = 0$ we also characterize non nef rays of length $n-2$.
We compute the Mori cone of curves of the moduli space \M_{g,n} of stable n-pointed curves of genus g in the case when g and n are relatively small. For instance, we show that for g<14 every curve in \M_g is numerically equivalent to an effective sum of 1-strata (loci of curves with 3g-4 nodes). We also prove that the nef cone of \M_{0,6} is composed of 11 natural subcones all contained in the convex hull of boundary classes. We apply this result to classify the fibrations of the moduli space of rational curves with n<7 marked points.
We investigate nef and movable cones of hypersurfaces in Mori dream spaces. The first result is: Let $Z$ be a smooth Mori dream space of dimension at least four whose extremal contractions are of fiber type of relative dimension at least two and let $X$ be a smooth ample divisor in $Z$, then $X$ is a Mori dream space as well. The second result is: Let $Z$ be a Fano manifold of dimension at least four whose extremal contractions are of fiber type and let $X$ be a smooth anti-canonical hypersurface in $Z$, which is a smooth Calabi--Yau variety, then the unique minimal model of $X$ up to isomorph…
A Bott tower of height $r$ is a sequence of projective bundles $$X_r \overset{π_r}\longrightarrow X_{r-1} \overset{π_{r-1}}\longrightarrow \cdots \overset{π_2}\longrightarrow X_1=\mathbb P^1 \overset{π_1} \longrightarrow X_0=\{pt\}, $$ where $X_i=\mathbb P (\mathcal O_{X_{i-1}}\oplus \mathcal L_{i-1})$ for a line bundle $\mathcal L_{i-1}$ over $X_{i-1}$ for all $1\leq i\leq r$ and $\mathbb P(-)$ denotes the projectivization. These are smooth projective toric varieties and we refer to the top object $X_{r}$ also as a Bott tower. In this article, we study the Mori cone and numerically effective …
The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call such a variety a Mori Dream Space. There turn out to be many examples, including quasi-smooth projective toric (or more generally, spherical) varieties, many GIT quotients, and log Fano 3-folds. We characterize Mori dream spaces as GIT quotients of affine varieties by a torus in a manner generalizing Cox's construction of toric varieties as quotients of affine space. Via the quotient description, the…
In the present paper we prove that, on a hyperkähler manifold, walls of the kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.
Given a log canonical pair $(X, Δ)$, we show that $K_X+Δ$ is nef assuming there is no non-constant map from the affine line with values in the open strata of the stratification induced by the non-klt locus of $(X, Δ)$. This implies a generalization of the Cone Theorem where each $K_X+Δ$-negative extremal ray is spanned by a rational curve that is the closure of a copy of the affine line contained in one of the open strata of $\mathrm{Nklt}(X, Δ)$. Moreover, we give a criterion of Nakai type to determine when under the above condition $K_X+Δ$ is ample and we prove some partial results in the ca…
In this paper we study effective, nef and semiample cones of minimal surfaces of general type with $p_g=0.$ We provide examples of minimal surfaces of general type with $p_g=0, 2 \leq K^2 \leq 9$ which are Mori dream spaces. On these examples we also give explicit description of effective cones and all irreducible reduced curves of negative self-intersection. We also present non-minimal surfaces of general type with $p_g=0$ that are not Mori dream surfaces.
We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.
We denote by FPMC the class of all non-singular projective algebraic surfaces X over C with a finite polyhedral Mori cone NE(X)\subset NS(X)\otimes R. If rho(X)=rk NS(X)\ge 3, then the set Exc(X) of all exceptional curves on X\in FPMC is finite and generates NE(X). Let δ_E(X) be the maximum of (-E^2) and p_E(X) the maximum of p_a(E) respectively for E\in Exc(X). For fixed ρ\ge 3, δ_E and p_E we denote by FPMC_{ρ,δ_E,p_E} the class of all X\in FPMC such that ρ(X)=ρ, δ_E(X)=δ_E and p_E(X)=p_E. We prove that the class FPMC_{ρ,δ_E,p_E} is bounded: for any X\in FPMC_{ρ,δ_E,p_E} there exist an ample…
We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if -1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our result also has computational applications, since this sufficient condition can be checked with a single semidefinite program.
Let $H_{a,b}^n$ denote the component of the Hilbert scheme whose general point parameterizes an $a$-plane union a $b$-plane meeting transversely in $\mathbf{P}^n$. We describe the effective and nef cones of $H_{a,b}^n$ and determine when the component is Fano. Moreover, we show that $H_{a,b}^n$ is a Mori dream space for all values of $a,b,n$.
In this paper, we study the divisor theory of the Simpson moduli space of semistable sheaves of dimension 1 on the projective plane. We prove that these spaces are all Mori dream spaces, and calculate their nef cones. We also study the effective cones of these spaces for most choices of numerical invariants. As a consequence, we work out precisely when two such spaces are isomorphic.
In this paper, we suggest a new approach to study minimal surfaces of general type with $p_g=0$ via their Cox rings, especially using the notion of combinatorially minimal Mori dream space introduced by Hausen. First, we study general properties of combinatorially minimal Mori dream surfaces. Then we discuss how to apply these ideas to the study of minimal surfaces of general type with $p_g=0$ which are very important but still mysterious objects. In our previous paper, we provided several examples of Mori dream surfaces of general type with $p_g=0$ and computed their effective cones explicitl…
The nef cone of a projective variety Y is an important and often elusive invariant. In this paper we construct two polyhedral lower bounds and one polyhedral upper bound for the nef cone of Y using an embedding of Y into a toric variety. The lower bounds generalize the combinatorial description of the nef cone of a Mori dream space, while the upper bound generalizes the F-conjecture for the nef cone of the moduli space \bar{M}_{0,n} to a wide class of varieties.
The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our …
Let X be a smooth Mori dream space of dimension at least 4. We show that, if X satisfies a suitable GIT condition which we call "small unstable locus", then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Neron-Severi spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space…
We establish a kind of subadjunction formula for quasi-log canonical pairs. As an application, we prove that a connected projective quasi-log canonical pair whose quasi-log canonical class is anti-ample is simply connected and rationally chain connected. We also supplement the cone theorem for quasi-log canonical pairs. More precisely, we prove that every negative extremal ray is spanned by a rational curve. Finally, we treat the notion of Mori hyperbolicity for quasi-log canonical pairs.
We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these results, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.
We study the birational geometry of the moduli space of complete $n$-quadrics $X$. We exhibit generators for Eff$(X)$ and Nef$(X)$, the cone of effective divisors and the cone of nef divisors, respectively. As a corollary X is a Mori Dream space. Furthermore, we run the MMP on $X$ and find a moduli interpretation for the models induced by the generators of the nef cone. In the case of complete quadric surfaces, we describe all the birational models of $X$ induced by the movable cone and find a moduli interpretation for some of these models.
Nagata solved Hilbert's 14-th problem in 1958 in the negative. The solution naturally lead him to a tantalizing conjecture that remains widely open after more than half a century of intense efforts. Using Nagata's theorem as starting point, and the conjecture, with its multiple variations, as motivation, we explore the important questions of finite generation for invariant rings, for support semigroups of multigraded algebras, and for Mori cones of divisors on blown up surfaces, and the rationality of Waldschimdt constants. Finally we suggest a connection between the Mori cone of the Zariski-R…
Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion f…
Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional proj…
In this paper, we study effective, nef and semiample cones of surfaces isogenous to a product of mixed type with $p_g=0$. In particular, we prove that all reducible fake quadrics are Mori dream surfaces.
We describe the set of Mori structures for a Fano 3-fold of index 2 and degree 1 (the double cone over the Veronese surface). In partiular, it is proved that such a Fano variety is not rational, the group of birational automorphisms coincides with the group of biregular automorphisms, and there are no structures of conic bundle.
We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini's lemma from algebraic geometry. Polyhedral cone…
Let $X$ be a projective toric surface of Picard number one blown up at a general point. We bring an infinite family of examples of such $X$ whose Kleiman-Mori cone of curves is not closed: there is no negative curve generating one of the two boundary rays of the cone. These examples are related to Nagata's conjecture and rationality of Seshadri constants.
