Using the $\sharp$-minimal model program of uniruled varieties we show that for any pair $(X, \H)$ consisting of a reduced and irreducible variety $X$ of dimension $k \geq 3$ and a globally generated big line bundle $\H$ on $X$ with $d:= \H^k$ and $n:= h^0(X, \H)-1$ such that $d<2(n-k)-4$, then $X$ is uniruled of $\H$-degree one, except if $(k,d,n)=(3,27,19)$ and a ${\sharp}$-minimal model of $(X, \H)$ is $(\PP^3,Ø_{\PP^3}(3))$. We also show that the bound is optimal for threefolds.
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.
A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness, both properties being in fact equivalent to the negativity of the logarithmic Kodaira dimension. Here we show in contrast that starting from dimension three, there exists smooth affine varieties which are affine-uniruled but not affine-ruled.
We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer's results, implies that varieties of the above type have strongly semistable tangent bundles with respect to any polarization.
A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised K3 surfaces. Finally we prove that the moduli space of Kummer surfaces associated to (1,21)-polarised abelian surfaces is uniruled.
Beauville asked if a compact Kähler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.
In this note we characterize polarized parallel transport operators on irreducible holomorphic symplectic varieties which are deformations of generalized Kummer varieties. We then apply such characterization to show the existence of ample uniruled divisors on these varieties and derive some interesting consequences on their Chow group of 0-cycles.
We develop criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology, which is a Hamiltonian Floer theory for some open symplectic manifolds including affine varieties. Symplectic cohomology is hard to compute, in general. However, certain vanishing and invariance properties of symplectic cohomology can be used to prove that our criteria for finding uniruled subvarieties hold in some cases. We provide applications of the criteria in birational geometry of log pairs in the direction of …
We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are obtained for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree > 1, is rational.
We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension $n$ implies the existence of good minimal models for projective log canonical pairs in dimension $n$.
We construct projectors in the ring of correspondences of a complex uniruled 3-fold $X$ which lift the Kuenneth components of the diagonal in singular cohomology and have other properties which were conjectured by J. Murre. Such Murre decompositions have been already obtained for curves, surfaces, abelian varieties and varieties with cell decompositions by the work of Manin, Shermenev, Beauville, Murre et.al.. In particular they define a natural filtration on the Chow groups of $X$ which was conjectured by Bloch and Beilinson. To do this we use Mori theory and construct projectors in the situa…
We prove that any ample class on a primitive symplectic variety that is locally trivial deformation of O'Grady's singular 6 dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn, Mongardi and Pacienza, extending their result for primitive symplectic varieties of this deformation type.
We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. We also show how to recover special cases by extending a technique introduced by Takayama.
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, When can a uniruled variety be smooth?, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, Systems of quadrics with tangential defects, Recognizing uniruled varieties, Recognizing intersections of quadrics, Recogniz…
A group $G$ is called Jordan if there is a positive integer $J=J_G$ such that every finite subgroup $\mathcal{B}$ of $G$ contains a commutative subgroup $\mathcal{A}\subset \mathcal{B}$ such that $\mathcal{A}$ is normal in $\mathcal{B}$ and the index $[\mathcal{B}:\mathcal{A}] \le J$ (V.L. Popov). In this paper we deal with Jordaness properties of the groups $Bir(X)$ of birational automorphisms of irreducible smooth projective varieties $X$ over an algebraically closed field of characteristic zero. It is known (Yu. Prokhorov - C. Shramov) that $Bir(X)$ is Jordan if $X$ is non-uniruled. On the …
This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in Complex Geometry", supported by the DFG. To start, we discuss existence results for rational curves on foliated manifolds. We construct the tangent morphism, define the variety of minimal rational tangents (VMRT) and name a number of results that show that many of the geometry properies of a uniruled variety are encoded in the projective geometry of the VMRT. The results are then applied in two differen…
Suppose that A and B are symplectomorphic smooth affine varieties. If A is acylic of dimension 2 then B has the same log Kodaira dimension as A. If the dimension of A is 3, has log Kodaira dimension 2 and satisfies some other conditions then B cannot be of log general type. We also show that if A and B are symplectomorphic affine varieties of any dimension then any compactification of A by a projective variety is uniruled if and only if any such compactification of B is uniruled.
Let $Y$ be a smooth cubic fourfold, $F$ be its Fano variety of lines and $Z$ be its associated LLSvS variety, parametrizing families of twisted cubics and some of their degenerations. In this short note, we show that the divisor of singular cubic surfaces on $Z$ has two irreducible components, one of which coincides with the uniruled branch divisor of a resolution of the Voisin map $F\times F \dashrightarrow Z$.
The Sarkisov Program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If X and Y are terminal Q-factorial projective varieties endowed with a structure of Mori fibre space, any birational map between them can be decomposed into a finite number of elementary Sarkisov links. This decomposition is not unique in general, and any two distinct decompositions define a relation in the Sarkisov Program. This paper shows that relations in the Sarkisov Program are generated by some elementary relations. Roughly speaking,…
In this paper we investigate complex uniruled varieties $X$ whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point $x\in X$ form a linear subspace of $T_xX$. As an application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wiśniewski's characterizations of ${\mathbb P}^n$.
We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certa…
1-flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification resu…
We show that if the variety of minimal rational tangents (VMRT) of a uniruled projective manifold at a general point is projectively equivalent to that of a symplectic or an odd-symplectic Grassmannian, the germ of a general minimal rational curve is biholomorphic to the germ of a general line in a presymplectic Grassmannian. As an application, we characterize symplectic and odd-symplectic Grassmannians, among Fano manifolds of Picard number 1, by their VMRT at a general point and prove their rigidity under global Kähler deformation. Analogous results for $G/P$ associated with a long root were…
We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.
We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties.
The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomo…
The Castelnuovo-Mumford regularity of varieties of degree r and dimension n in the r-dimensional projective space that have an extremal secant line, is at least d-r+n+1. We classify these varieties and show that their regularity is exactly d-r+n+1, as predicted by the regularity conjecture.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
We classify complex projective varieties of dimension $2r \geq 8$ swept out by a family of codimension two grassmannians of lines $\mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $\G(1,r)$ or the grassmannian $\mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics.
We approach non-divisorial base loci of big and nef line bundles on irreducible symplectic varieties. While for K3 surfaces, only divisorial base loci can occur, nothing was known about the behaviour of non-divisorial base loci for more general irreducible symplectic varieties. We determine the base loci of all big and nef line bundles on the Hilbert scheme of two points on very general K3 surfaces of genus two and on their birational models. Remarkably, we find an ample line bundle with a non-trivial base locus in codimension two. We deduce that, generically in the moduli spaces of polarized …
In this paper, we construct compactifications of $SL_2(\mathbb{C})$-character varieties of $n$-punctured projective line and study the boundary divisor of the compactifications. This study is motivated by the conjecture for the configuration of the boundary divisor, due to C. Simpson. We verify the conjecture for a few examples.
We prove that up to automorphisms a line admits a unique embedding into the regular part of of a simplicial toric variety of dimension n>=4 over an algebraically closed field of characteristic zero which is smooth in codimension 2.
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.
In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.
In this paper we give for any integer l > 2 a numerical criterion ensuring the existence of a chain of length l of lines through two general points of an irreducible variety X in P^N, involving the degrees and the number of homogeneous polynomials defining X. We show that our criterion is sharp.
We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using celebrated results by Ein and Zak.
A conjecture of Batyrev and Manin relates arithmetic properties of varieties with ample anticanonical class to geometric invariants; in particular, counting functions defined by metrized ample line bundles and the corresponding asymptotics of rational points of bounded height are interpreted in terms of cones of effective divisors and certain thresholds with respect to these cones. This framework leads to the notion of balanced line bundles, whose counting functions, conjecturally, capture generic distributions of rational points. We investigate balanced line bundles in the context of the Mini…
Over a family of varieties with singular special fiber, the relative Picard functor (i.e. the moduli space of line bundles) may fail to be compact. We propose a stability condition for line bundles on reducible varieties that is aimed at compactifying it. This stability condition generalizes the notion of `balanced multidegree' used by Caporaso in compactifying the relative Picard functor over families of curves. Unlike the latter, it is defined `asymptotically'; an important theme of this paper is that although line bundles on higher-dimensional varieties are more complicated than those on cu…
We prove that the Picard group of a regular simply connected variety over an algebraically closed field of arbitrary characteristic is finitely generated. The main difficulty to overcome is the unavailability of resolution of singularities. From this we deduce that in positive characteristic there exist no nontrivial stratified line bundles on such a variety, and we present a complex analog.
The first part of this note contains a review of basic properties of the variety of lines contained in an embedded projective variety and passing through a general point. In particular we provide a detailed proof that for varieties defined by quadratic equations the base locus of the projective second fundamental form at a general point coincides, as a scheme, with the variety of lines. The second part concerns the problem of extending embedded projective manifolds, using the geometry of the variety of lines. Some applications to the case of homogeneous manifolds are included.
In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert--Samuel formula and the arithmetic Hilbert--Samuel formula.
In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, called varieties of lines. We also describe their ACM property from combinatorial algebra point of view.
We characterize the double Veronese embedding of P^n as the only variety that, under certain general conditions, can be isomorphically projected from the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.
In this paper we give top most and least indices of non vanishing cohomology modules of line bundles in the Kac Moody setting in terms of Weyl group combinatorics. We also prove a surjection Theorem.
