Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve.
We give a short proof of a Grothendieck-Lefschetz Theorem for equivariant Picard groups of nonsingular varieties with the action of an affine algebraic group.
We propose a strengthening of the Grothendieck--Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities. Version 2: Main conjecture changed, following recent results of Bhatt and de Jong. Small changes to the proofs.
The monodromy action in the homology (generally with twisted coefficients) of complements of stratified complex analytic varieties depending on parameters is studied. For a wide class of local degenerations of such families (stratified Morse singularities) local monodromy and variation operators are reduced to similar operators acting in the transversal slices of corresponding strata. In particular, these results imply a main part of F. Pham's generalized Picard--Lefschetz formulae.
We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi-Yau 3-fold $X$ with Picard number one whose invariants are $$(H_X^3, c_2 (X) \cdot H_X, {e} (X)) = (8, 44, -88),$$ where $H_X$, $e(X)$ and $c_2(X)$ are an ample generator of $\Pic(X)$, the topological Euler characteristic number and the second Chern class of $X$ respectively.
We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we discuss classification results for 4-dimensional complex smooth projective varieties admitting some particular fiber type contractions.
It is conjectured that a Fano manifold of Picard number 1 which is not a projective space admits no endomorphisms of degree bigger than 1. Beauville confirmed this for hypersurfaces of projective space. We study this problem for hypersurfaces of an arbitrary Fano manifold of Picard number 1.
In this paper, we classify Fano fourfolds whose the second exterior power of tangent bundles are numerically effective with Picard number greater than one.
This article settles the question of existence of smooth weak Fano threefolds of Picard number two with small anti-canonical map and previously classified numerical invariants obtained by blowing up certain curves on smooth Fano threefolds of Picard number 1 with the exception of 12 numerical cases.
We classify smooth Fano manifolds X with the Picard number $ρ_X \geq 3$ such that there exists an extremal ray which has a birational contraction that maps a divisor to a point.
Using a construction due to C. Casagrande and further developed by the author, we prove that the Picard number of a non-smooth Fano 3-fold with isolated factorial canonical singularities, is at most 6.
This note is a short survey on the Lefschetz defect, an invariant of smooth Fano varieties that has been recently introduced; it is related to the Picard number rho(X) of X, and to the Picard number of prime divisors in X. We explain the definition of the Lefschetz defect delta(X) and its origin, the known results - mainly on the case delta(X)>1, and several examples, especially in dimensions 3 and 4. In particular, we explain how the results on the Lefschetz defect allow to recover the classification of Fano 3-folds with rho(X)>4, due to Mori and Mukai. We also review the known examples of Fa…
We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases. In the Fano case, we prove that the generalized Mukai conjecture holds.
We calculate the Picard group of generic (very general) spectral varieties living in the total space of a very ample line bundle over an algebraically closed field $k$ of odd characteristics or characteristic 0. We follow the strategy of Ravindra and Srinivas [RS06,RS09] via formal Picard groups. As an application, we calculate the generic fibers of Hitchin systems over a smooth quintic surface of Picard number 1.
A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration.
We prove a strengthening of the Grothendieck-Lefschetz hyperplane theorem for local Picard groups conjectured by Kollár. Our approach, which relies on acyclicity results for absolute integral closures, also leads to a restriction theorem for higher rank bundles on projective varieties in positive characteristic.
In this paper, we investigate the minimal length of chains of minimal rational curves needed to join two general points on a Fano manifold of Picard number 1. In particular, we give a sharp bound of the length by a fundamental argument. As an application, we compute the length for Fano manifolds of dimension < 8.
In this paper we study the geometry of mildly singular Fano varieties on which there is an effective prime divisor of Picard number one. Afterwards, we address the case of toric varieties. Finally, we treat the lifting of extremal contractions to universal covering spaces in codimension 1.
We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \ge (\dim X +3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.
We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions.
We prove the conjectural Bogomolov-Gieseker type inequality for tilt slope stable objects on each Fano threefold X of Picard number one. Based on the previous works on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D^b(X). We also get a new stronger bound for the Chern characters of slope semistable sheaves on X.
The Picard number of a Fano manifold X obtained by blowing up a curve in a smooth projective variety is known to be at most 5, in any dimension greater than or equal to 4. We show that the Picard number attains to the maximal if and only if X is the blow-up of the projective space whose center consists of two points, the strict transform of the line joining them and a linear subspace or a quadric of codimension 2. This result is obtained as a consequence of a classification of special types of Fano manifolds.
In this paper, we give the complete classification of full exceptional collections on smooth toric Fano threefolds and fourfolds with Picard rank two. To be precise, we give a partial answer to the conjecture in \cite{Kuz} and \cite{LYY}: we provide all the exceptional collections of maximal length and prove that they are in fact full exceptional collections.
Let X be an $n$-dimensional Fano manifold of Picard number 1. We study how many different ways X can compactify the complex vector group C^n equivariantly. Hassett and Tschinkel showed that when X = P^n with n \geq 2, there are many distinct ways that X can be realized as equivariant compactifications of C^n. Our result says that projective space is an exception: among Fano manifolds of Picard number 1 with smooth VMRT, projective space is the only one compactifying C^n equivariantly in more than one ways. This answers questions raised by Hassett-Tschinkel and Arzhantsev-Sharoyko.
A generalization of S. Mukai's conjecture says that if $X$ is a Fano $n$-fold with Picard number $ρ_X$ and pseudo-index $i_X$, then $ρ_X(i_X-1) \leq n$, with equality if and only if $X \cong (\mathbb{P}^{i_X-1})^{ρ_X}$. In this paper, we prove that this conjecture holds if $n=6$ and either $X$ admits a contraction of fiber type or $X$ admits no small contractions.
Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect. It is known that if delta(X) is at least 4, then X is isomorphic to a product SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the case where delta(X)=3. We show that there exists a smooth Fano variety T with dim T=dim X-2 such that X is obtained from T with two possible explicit constructions; in both cases there is a P^2-bundle Z over T such that X is the blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2 subvarieties. Then we apply the structure theorem to Fano 4-fo…
We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties …
K{ü}chle classified the Fano fourfolds that can be obtained as zero loci of global sections of homogeneous vector bundles on Grassmannians. Surprisingly, his classification exhibits two families of fourfolds with the same discrete invariants. Kuznetsov asked whether these two types of fourfolds are deformation equivalent. We show that the answer is positive in a very strong sense, since the two families are in fact the same! This phenomenon happens in higher dimension as well.
Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the codimension c of H in N_1(X) is at most 8. Moreover if c>2, then either X=SxY where S is a Del Pezzo surface, or c=3 and X has a flat fibration in Del Pezzo surfaces onto a Fano manifold Y, such that the difference of the Picard numbers of X and Y is 4. We give applications to Fano 4-folds, to Fano varieties with pseudo-index >1, and to surjective morphisms whose source is Fano, having some high-dimensional…
We study the topology of the space of affine hyperplanes $L \subset \CC^n$ which are in general position with respect to a given generic quadratic hypersurface $A$, and calculate the monodromy action of the fundamental group of this space on the relative homology groups $H_*(\CC^n, A \cup L)$ associated with such hyperplanes.
We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.
We introduce the notion of intrinsic Grassmannians which generalizes the well known weighted Grassmannians. An intrinsic Grassmannian is a normal projective variety whose Cox ring is defined by the Plücker ideal $I_{d,n}$ of the Grassmannian $\mathrm{Gr}(d,n)$. We give a complete classification of all smooth Fano intrinsic Grassmannians of type $(2,n)$ with Picard number two and prove an explicit formula to compute the total number of such varieties for an arbitrary $n$. We study their geometry and show that they satisfy Fujita's freeness conjecture.
We give a complete classification of smooth, complex projective Fano 4-folds of Picard number 3 having a prime divisor of Picard number 1. They form 28 distinct families, and we compute the main numerical invariants, study the base locus of the anticanonical divisor, and give a comment on rationality. Furthermore, we give an upper bound of both the dimension of the automorphism group and the dimension of the base of the Kuranishi family of a general member.
Let $f$ and $g$ be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of $fg$ to that of $f$ and $g$ separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu.
Let X be a Q-factorial Gorenstein Fano variety. Suppose that the singularities of X are canonical and that the locus where they are non-terminal has dimension zero. Let D be a prime divisor of X. We show that rho_X - rho_D < 9 (where rho is the Picard number). Moreover, if rho_X - rho_D > 3, there exists a finite morphism from X to S x Y, where S is a surface with rho_S at most 9. As an application we prove that, if X has dimension 3, then rho_X is at most 10.
Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number one. Assume that the VMRT at a general point of $X$ is smooth irreducible and non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$). It is proven that $\dim \mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is isomorphic to $\mathbb{P}^n, \mathbb{Q}^n$ or ${\rm Gr}(2,5)$. Furthermore, the equality $\dim \mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic to the 6-dimensional Lagrangian Grassmannian ${\rm Lag}(6)$ or a general hyperplane section of ${\rm Gr}(2,5)$.
