Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

Derived categories of Fano threefolds

We consider the structure of the derived categories of coherent sheaves on Fano threefolds with Picard number 1 and describe a strange relation between derived categories of different threefolds. In the appendix we discuss how the ring of algebraic cycles of a smooth projective variety is related to the Grothendieck group of its derived category. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 264, pp. 116–128. In memory of V.A. Iskovskikh     

Fano threefolds in positive characteristic

We show that the birational classification in positive characteristicof smooth Fano threefolds X with Picard number 1 is the same as incharacteristic zero. In particular, there are no exotic such Fanos; asa consequence of the classification, X is shown to be liftable withoutramification to characteristic zero and to contain a line. The maintechniques employed are those of Ekedahl and of Mori and Takeuchi.     

Linear systems on Fano threefolds II

In this note we classify (1) a polarized smooth Fano threefold (X,H) for which |H| fails to be very ample and (2) a nef divisor N on a smooth Fano threefold with Bs|N|.Mathematics Subject Classification (2000): 14C20, 14J45, 14E25The author was supported in part by KOSEF Grant #R14 -2002-007-01001-0(2002).     

Iterated vanishing cycles

If A ^? is a bounded, constructible complex of sheaves on a complex analytic space X, and ${f : X \rightarrow \mathbb{C}}$ and ${g : X \rightarrow \mathbb{C}}$ are complex analytic functions, then the iterated vanishing cycles φ _g [?1](φ _f [?1]A ^?) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ _g [?1]φ _f [?1]A ^?) _x in terms of relative polar curves, algebra, and Morse modules of A ^?.     

Vector fields on smooth threefolds vanishing on complete intersections

A result of J. Wahl shows that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducible and reduced curve which is the scheme theoretic intersection of two ample divisors, X is isomorphic to the 3-dimensional projective space or the 3-dimensional quadric. Received: 24 April 2001     

Stable Ulrich bundles on Fano threefolds with Picard number 2

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of ${\mathbb{P}}^3$, Q (smooth quadric in ${\mathbb{P}}^4$), $V_3$ (smooth cubic in ${\mathbb{P}}^4$) or $V_4$ (complete intersection of two quadrics in ${\mathbb{P}}^5$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in ${\mathbb{P}}^3$. Also, we prove that there exist stable rank two Ulrich bundles with $c_1=3H$ on a generic member of this deformation class.     

Even and odd instanton bundles on Fano threefolds of Picard number one

We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K _X . We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in ${\mathbb{P}^5}$ , relating instanton bundles on X to vector bundles of higher rank on a the curve of genus 2 naturally associated with X.     

Cubic threefolds,Fano surfaces and the monodromy of the Gauss map

The Tannakian formalism allows to attach to any subvariety of an abelian variety an algebraic group in a natural way. The arising groups are closely related to moduli questions such as the Schottky problem, but in general they are still poorly understood. In this note we show that for the theta divisor on the intermediate Jacobian of a cubic threefold, the Tannaka group is exceptional of type E₆. This is the first known exceptional case, and it suggests a surprising connection with the monodromy of the Gauss map.     

On the number of singular points of terminal factorial Fano threefolds

Mathematical Notes -     

On the existence of almost Fano threefolds with del Pezzo fibrations

By Jahnke–Peternell–Radloff and Takeuchi, almost Fano threefolds with del Pezzo fibrations were classified. Among them, there exist 10 classes such that the existence of members of these was not proved. In this paper, we construct such examples belonging to each of 10 classes.