Seshadri constants and Fano manifolds

In memory of Meeyoung Kim In this paper, we give a lower bound of Seshadri constants on smooth Fano varieties. More precisely, we show that on a smooth Fano manifold of dimension n whose anticanonical system is base point free, Seshadri constants of ample divisors are bounded from below by one over n–2. As a corollary we recover the earlier result on Fano threefolds. Mathematics Subject Classification (2000):14J45, 14N30.The author was supported in part by KOSEF Grant R14-2002-007-01001-0(2002).     

Ample Vector Bundles with Small g − q

Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ^{⊕ (dim Z?1)} is ≤ h ¹( _X ) + 2.     

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).     

Threefolds in ℙ5 with One Apparent Quadruple Point

Abstract

In this article we classify all the smooth threefolds in ?⁵ with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ? ⁿ with one apparent (n ? 1)-point and with irreducible family of (n ? 1)-secant lines.     

Ruled threefolds homeomorphic to ℙ2 × ℙ1 and their divisors

We study a class of ruled threefolds, namely, manifolds X with a projection p:X→ℙ², such that each fiber is isomorphic to ℙ¹, and which are homeomorphic to ℙ²×ℙ¹; and we characterize ample and very ample line bundles on such threefolds. This paper was written with the financial support of M.P.I. The author is a member of G.N.S.A.G.A. of the C.N.R.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

Towards the classification of weak Fano threefolds with ρ = 2

In this paper, examples of type II Sarkisov links between smooth complex projective Fano threefolds with Picard number one are provided. To show examples of these links, we study smooth weak Fano threefolds X with Picard number two and with a divisorial extremal ray. We assume that the pluri-anticanonical morphism of X contracts only a finite number of curves. The numerical classification of these particular smooth weak Fano threefolds is completed and the geometric existence of some numerical cases is proven.     

On threefolds covered by lines

A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, ifX is such a threefold, let Σ denote the Fano scheme of lines onX and μ the number of lines contained inX and passing through a general point ofX. Assume that Σ is generically reduced. Then μ ≤ 6. Moreover,X is birationally a scroll over a surface (μ = 1), orX is a quadric bundle, orX belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with −K _X = 2H _X . This work has been done in the framework of the activities of EAGER. Both authors have been supported by funds of MURST, Progetto Geometria Algebrica, Algebra Commutativa e Aspetti Computazionali. The paper has been widely improved while the second author was guest at the University of Oslo. The financial support from both the University of Oslo and the Norwegian Research Council is gratefully aknowledged.     

Semistability of Logarithmic Cotangent Bundle on Some Projective Manifolds

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kähler–Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf Ω _X (log D) on X. We prove semistability of this sheaf, when the log canonical sheaf K _X  + D is ample or trivial, or when ?K _X  ? D is ample, i.e., when X is a log Fano n-fold of dimension n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.     

On (2, 4) complete intersection threefolds that contain an Enriques surface

We study nodal complete intersection threefolds of type (2, 4) in \({\mathbb {P}}^5\) which contain an Enriques surface in its Fano embedding. We completely determine Calabi–Yau birational models of a generic such threefold. These models have Hodge numbers \(h^{11}=2,h^{12}=32\). We also describe Calabi–Yau varieties with Hodge numbers \((h^{11},h^{12})\) equal to (2, 26), (23, 5) and (31, 1). The last two pairs of Hodge numbers are, to the best of our knowledge, new.     

Bethe–Sommerfeld Conjecture for Pseudodifferential Perturbation

We consider a periodic pseudodifferential operator H = (? Δ) ^l  + A (l > 0) in R ^d which satisfies the following conditions: (i) the symbol of H is smooth in x, and (ii) the perturbation A has order smaller than 2l ? 1. Under these assumptions, we prove that the spectrum of H contains a half-line.     

On the Unit Groups of Commutative Modular Group Algebras of KT-Groups

ABSTRACT

In this article, we prove that the inner projection of a projective curve with higher linear syzygies has also higher linear syzygies. Specifically, if a very ample line bundle ? on a smooth projective curve X satisfies property N _p for p  ≥  1 and H ¹ (? ^? 2) =  0 , then ?( ?  q ) satisfies property N _{p ? 1} for any point q  ∈  X . We also give simple proofs of well-known theorems about syzygies and raise some questions related to the line bundles of degree 2 g which do not satisfy property N ₁ .     

The Dimension of the Hilbert Scheme of Special Threefolds #

The Hilbert scheme of 3-folds in ? ⁿ , n ≥  6 , that are scrolls over ? ² or over a smooth quadric surface Q  ? ? ³ or that are quadric or cubic fibrations over ? ¹ is studied. All known such threefolds of degree 7  ≤ d ≤  11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.     

Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? _n+1-cover if and only if n is even and G? (C ₂) ⁿ , where by a ? _m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p).     

Non-annulation effective et positivité locale des fibrés en droites amples adjoints

We show how to use effective non-vanishing to prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. We prove the effective non-vanishing conjecture of Ionescu–Kawamata in dimension 3 in the case of line bundles of “high” volume.     

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

Abstract

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H ₁, H ₂ ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.     

On c -Normal Maximal and Minimal Subgroups of Sylow Subgroups of Finite Groups. II

Abstract

A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H _G  = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are:

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ?.

Theorem Let ? be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ?. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ?.     

ON MAŞEK'S CRITERION ON VERY AMPLENESS

ABSTRACT

In this note, we give a new proof of Ma?ek's criterion on very ampleness of adjoint linear systems on surfaces and prove very ampleness for certain polarized threefolds.