Designs and partial geometries over finite fields

Let be a finite field, and let (, B) be a nontrivial 2-(n, k, 1)-design over . Then each point induces a (k–1)-spread S on /. (, B) is said to be a geometric design if S is a geometric spread on / for each . In this paper, we prove that there are no geometric designs over any finite field .Research partially supported by NSF grant DMS-8703229.     

On Birational Maps and Jacobian Matrices

One is concerned with Cremona-like transformations, i.e., rational maps from ⁿ to ^m that are birational onto the image Y ^m and, moreover, the inverse map from Y to ⁿ lifts to ^m . We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ², i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ² are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ².Research partially supported by a DGICYT grant No. PB91-0731.     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation     

The Hilbert function of generic plane sections of curves of ℙ3

Summary A characteristic condition is given on a zero-dimensional differentiable 0-sequenceH={h _i}_i0,h ₁3, in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of ³, hence of points of ² with the uniform position property. In this way an answer is given to some question stated by Harris in [Ha₂].The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in ³ whose generic plane section has an assigned Hilbert function satisfying that condition.A.M.S. (1980) Subject classification. Primary 14C20, 14H99. Secondary 14C05     

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.     

Local Recognition Of Non-Incident Point-Hyperplane Graphs

Let be a projective space. By H() we denote the graph whose vertices are the non-incident point-hyperplane pairs of , two vertices (p,H) and (q,I) being adjacent if and only if p I and q H. In this paper we give a characterization of the graph H() (as well as of some related graphs) by its local structure. We apply this result by two characterizations of groups G with PSL _n ( )GPGL _n ( ), by properties of centralizers of some (generalized) reflections. Here is the (skew) field of coordinates of .     

The singularH 2,2-invariant quartic surfaces in ℙ3

AnH _2,2-invariant quartic surface in ₃ is a quartic surface in ₃ invariant under the Heisenberg groupH _2,2 of level (2, 2), the family ofH _2,2-invariant quartic surfaces is parametrized by ₄. For each ₄, the corresponding quartic surfaceX will be a Kummer surface, ifX is singular. The equation for { = 0} ₄ parametrizing all Kummer surfaces is well known. We find another more symmetric form (with respect to a 5-dimensional representation of the symmetric group S₆) for this equation.The aim of this note is to describe all singularH _2,2-invariant quartic surfaces in ₃.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

New examples of Kobayashi hyperbolic surfaces in ℂℙ3

We show that a generic small deformation of the union of two generic cones in ³ of degree 4 is Kobayashi hyperbolic. Hence we obtain new examples of hyperbolic surfaces in ³ of any given degree d 8.__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 90–94, 2005Original Russian Text Copyright © by B. Shiffman and M. ZaidenbergThe second authors research was supported in part by NSF grant No. DMS-0100474.Translated by B. Shiffman and M. Zaidenberg     

The truncated Witten genus

In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space Map(C_p,ⁿ⁺¹) is closely related to the truncated Witten genus of ⁿ. The spaces Map(C_p,ⁿ⁺¹) define a filtration of the space Map(S¹,ⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ⁿ.Acknowledgement This work was mainly done while the author was studying for a Ph.D. at Warwick University under the supervision of J. D. S. Jones. The author would like to thank John Jones for originally suggesting the problem and for his guidance whilst the author was at Warwick, to thank the EPSRC for funding his Ph.D., and to thank the referee for his detailed comments which helped greatly with the writing of this paper.     

Automorphismen und Modulraum galoisscher dreiblättriger Überlagerungen

The points of the modulus space M _g ² of compact hyperelliptic Riemann surfaces of genus^g g can be identified, with the PGL (2,)-orbits in the set of all subsets of ¹, which contain 2g+2 elements. The automorphism group Aut(X) of such a surface contains a canonical involution J, and Aut(X)/J is isomorphic to the finite subgroup of Aut(¹), whose elements permute the critical values of some twofold cover. In this paper we derive similar results for surfaces X, which admit a threefold Galois cover over ¹.

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Herrn Prof. Dr. Dr. h.c. Karl Stein in Dankbarkeit gewidmet     

Calculating cohomology groups of ℳ̄0,n (ℙ r ,d)

Here we investigate the rational cohomology of the moduli space ̄_0,n(^r,d) of degree d stable maps from n-pointed rational curves to ^r. We obtain partial results for small values of d with an inductive method inspired by a paper of Enrico Arbarello and Maurizio Cornalba.     

A sharp bound for the number of points where a rank 3 stable reflexive sheaf is not free

Let ³ be the projective space over an algebraically closed field k and let E be a rank 3 stable reflexive sheaf on ³ such that its restriction to a general plane is stable. The aim of this paper is to give a sharp bound of s= lengthExt ¹ (E,G)in terms of the Chern classes of E.     

Northcott's theorem on heights I. A general estimate

Given a pointP=(a _o:...:a _n) of projective space ⁿ = ⁿ (A) whereA is the field of algebraic numbers, letd(P) be its degree andH(P) its absolute multiplicative height. Northcott's Theorem says that givend, n andX, there are only finitely many pointsP ⁿ withd(P)d andH(P)X. We will show that there are at mostc(d, n)X ^d(d+n) such points.Supported in part by NSF grant DMS-9108581.     

Géométrie Orthogonale Non Symétrique et Congruences Quadratiques (Nonsymmetric Orthogonal Geometry and Quadratic Congruences)

This paper is devoted to the study of quadratic congruences, which appear in the theory of exceptional bundles on projective spaces. A quadratic congruence associates to each point P of a projective space a quadratic in containing P. We study several geometric constructions of quadratic congruences and try to classify them.     

Components of the Hilbert scheme of smooth space curves with the expected number of moduli

For any d and g such that the Brill-Noether number (d,g,3) is negative, d20 and gf(d), where f(d)=d^3/2/(6·2^1/2) + lower order terms, there exists a regular component of H _d,g ³ (the closure in Hilb₃ of the open set parametrizing smooth, connected, non-degenerate curves in ₃) with the expected number of moduli. Moreover, examples of the fact that such components are not unique are given.