On the classification of toric Fano 4-folds

The biregular classification of smoothd-dimensional toric Fano varieties is equivalent to the classification of special simplicial polyhedraP in ℝ ^d , the so-called Fano polyhedra, up to an isomorphism of the standard lattice . In this paper, we explain the complete biregular classification of all 4-dimensional smooth toric Fano varieties. The main result states that there exist exactly 123 different types of toric Fano 4-folds somorphism. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory Vol. 56. Algebraic Geometry-9, 1998.     

Note on Fano 3-folds

LetE be an ample vector bundle of rankr on a compact complex manifoldX of dimension 3 with detE=–K _x, andi(X) the index ofX. Then it is proved in this note thati(X)r unless (X,E)(¹ × ²,p*O(1) q*), wherep,q are the projections and is isomorphic toO(2) O(1) or the tangent bundleT of ². This result gives a counterexample to the conjecture formed by T. Peternell.     

On the birational geometry of Fano 4-folds

We study the birational geometry of a Fano fourfold X from the point of view of Mori dream spaces; more precisely, we study rational contractions of X. Here a rational contraction is a rational map $f: X \dashrightarrow Y$ , where Y is normal and projective, which factors as a finite sequence of flips, followed by a surjective morphism with connected fibers. Such f is called elementary if ρ _X ? ρ _Y  = 1, where ρ is the Picard number. We first give a characterization of non-movable prime divisors in X, when ρ _X  ≥ 6; this is related to the study of birational and divisorial elementary rational contractions of X. Then we study the rational contractions of fiber type on X which are elementary or, more generally, “quasi-elementary”. The main result is that ρ _X  ≤ 11 if X has an elementary rational contraction of fiber type, and ρ _X  ≤ 18 if X has a quasi-elementary rational contraction of fiber type.     

Remarks on abelian surfaces in nonsingular toric Fano 4-folds

In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. In consequence, we determine the possibilities of these embeddings, except for the remaining 18 nonsingular toric Fano 4-folds. Received: 12 July 2002     

Vector bundles on ruled Fano 3-folds

Let be a ruled Fano 3-fold. The goal of this paper is to compute the dimension, prove the irreducibility and smoothness and describe the structure of the moduli space M _L (2;c ₁,c ₂) of L-stable, rank 2 vector bundles E on X with certain Chern classes and for a suitable polarization L closely related to c ₂. More precisely, we will cover the study of some moduli spaces M _L (2;c ₁,c ₂) such that the generic point is given as a non-trivial extension of line bundles. This work nicely reflects the general philosophy that moduli spaces inherits a lot of geometrical properties of the underlying variety. Received: 16 February 1999 / Revised version: 2 July 1999     

Sarkisov links for index 1 Fano 3-folds in codimension 4

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.     

A note on the Picard number of singular Fano 3-folds

Using a construction due to C. Casagrande and further developed by the author (Int. Math. Res. Notices, 2012), we prove that the Picard number of a non-smooth Fano 3-fold with isolated factorial canonical singularities is at most 6.     

A remark on extremal stability of configuration spaces

We explicitly study the extremal stability of configuration spaces of complex projective spaces of any dimension, and show that the homology groups are vanish in extremal stable range. As a consequence, we give an affirmative answer of the question of Knudsen, Miller and Tosteson.     

A remark on sets having the Steinhaus property

A point set satisfies the Steinhaus property if no matter how it is placed on a plane, it covers exactly one integer lattice point. Whether or not such a set exists, is an open problem. Beck has proved [1] that any bounded set satisfying the Steinhaus property is not Lebesgue measurable. We show that any such set (bounded or not) must have empty interior. As a corollary, we deduce that closed sets do not have the Steinhaus property, fact noted by Sierpinski [3] under the additional assumption of boundedness.     

Three equivalent conjectures on the birational geometry of Fano 3-folds

I propose three equivalent conjectures on the birational geometry of Fano 3-folds. Roughly speaking, they suggest that ergodic, or chaotic, behaviour does not occur for Fano 3-folds.     

Rank 2 stable vector bundles on Fano 3-folds of index 2

Let X be a Fano 3-fold of the first kind with index 2. In this paper, we characterize the chern classes of rank 2 stable vector bundles on X and we find a bound for the least twist of a rank 2 reflexive sheaf on X which has a global section.     

A remark on the failure of multiplicity one for GSp(4)

We revisit a classical result of Howe and Piatetski-Shapiro on the failure of strong multiplicity one for GSp(4).     

Projecting 4-folds from G(1, 5) to G(1, 4)

 We study 4-dimensional subvarieties of the Grassmannian G(1,5) with singular locus of dimension at most 1 that can be isomorphically projected to G(1,4). Received: 26 July 2001     

A remark on the space of metrics having nontrivial harmonic spinors

Let M be a closed spin manifold of dimension n ≡ 3 mod 4. We give a simple proof of the fact that the space of metrics on M with invertible Dirac operator is either empty or it has infinitely many path components.     

A characterization of certain irreducible symplectic 4-folds

 We give a characterization of irreducible symplectic fourfolds which are given as Hilbert scheme of points on a K3 surface. Received: 26 June 2002 / Revised version: 9 September 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): Primary 14J32, Secondary 32Q20