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.     

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.     

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.     

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.     

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.     

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     

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 .     

Local and relative Gromov-Witten invariants of the projective plane

Let B be a line or a smooth conic in ². We give a recursive formula for certain linear combinations of the genus 0 relative Gromov-Witten invariants of (²,B). After change of sign, this is equivalent to the WDVV equation for the equivariant local Gromov-Witten invariants of ² in ᵊA(–B). We show that these invariants coincide up to sign. Mathematics Subject Classification (2000):14N35, 14N10     

Normal surfaces of degree 7 in ℙ n

Let Y be a surface of degree 7 of the complex projective space ⁿ , linearly normal and not contained in any hyperplane of ⁿ . The purpose of this paper is to describe the normal singular surfaces of degree 7 embedded in ⁿ , as above. The main contribution of this paper is to describe all normal singular projective surfaces with sectional genus 4.     

Automorphisms of a Generic Jacobian Kummer Surface

For a generic quartic surface F in ³ with sixteen nodes, we shall prove that every birational automorphism of F is induced by a Cremona transformation of ³.     

Syzygies of Veronese Embeddings

We prove that the Veronese embedding _O ⁿ (d): ^{n N} with n2, d3 does not satisfy property N _p (according to Green and Lazarsfeld) if p3d–2. We make the conjecture that also the converse holds. This is true for n=2 and for n=d=3.     

3-Mannigfaltigkeiten im ℙ5 und ihre zugehörigen Stabilen Garben

In this paper we begin to study 3-folds in a projective space of dimension 5. Using results from [9] we give a classification of all 3-folds in ⁵ , up to degree 6. There are only 3 different types of 3-folds in ⁵ of degree 6 which are not complete intersections. These manifolds can be represented as zero schemes of sections in certain (extremal) semistable reflexive sheaves of rank 2 on ⁵ . Finally we obtain examples of stable reflexive sheaves on ⁵ with homologieal dimension 1, which do not belong to the extremal sheaves [10].

---

---

Dies ist ein Teil meiner Habilitationsschrift     

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.     

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.     

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.     

Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps

Summary. Let u and u_V V be the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem u=f on , u|=0. For any m and any bounded polygonal domain , we provide a construction of a new sequence of finite dimensional subspaces V_n such that where f H^m–1() is arbitrary and C is a constant that depends only on and not on n (we do not assume u H^m+1()). The existence of such a sequence of subspaces was first proved in a ground–breaking paper by Babuka [8]. Our method is different; it is based on the homogeneity properties of Sobolev spaces with weights and the well–posedness of non-homogeneous Dirichlet problem in suitable Sobolev spaces with weights, for which we provide a new proof, and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary. Our results extended right away to domains whose boundaries have conical points. We also indicate some of the changes necessary to deal with domains with cusps. Our numerical computation are in agreement with our theoretical results.The authors were supported in part by the NSF grant DMS 02-09497. Victor Nistor was also partially supported by NSF grant DMS 02-00808.     

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     

A note on generic projections

Let X^N=²ⁿ_K be a subvariety of dimension n and P^N a generic point. If the tangent variety TanX is equal to ^N then for generic points x, y of X the projective tangent spaces t_xX and t_yX meet in one point P=P(x,y). The main result of this paper is that the rational map (x,y)P(x,y) is dominant. In other words, a generic point P is uniquely determined by the ramification locus R(_P) of the linear projection _P:X^N–1.This paper was supported by the DFG Schwerpunkt Global methods in complex analysis, MUR and the Research Group GNSAGA of INDAM. This investigation was also supported by the University of Bologna, funds for selected research topics     

Colouring finite incidence structures

One of our results: Let denote a finite projective plane of ordern. Colour its vertices by colours. If no colour appears more than twice on any of the lines, then n + 1. Explicit constructions show that this bound is sharp when is desarguesian.     

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