Preface.- Introduction.- General Discriminants and Resultants.- Projective Dual Varieties and General Discriminants.- The Cayley Method of Studying Discriminants.- Associated Varieties and General Resultants.- Chow Varieties.- Toric Varieties.- Newton Polytopes and Chow Polytopes.- Triangulations and Secondary Polytopes.- A-Resultants and Chow Polytopes of Toric Varieties.- A-Discriminants.- Principal A-Discriminants.- Regular A-Determinants and A-Discriminants.- Classical Discriminants and Resultants.- Discriminants and Resultants for Polynomials in One Variable.- Discriminants and Resultants for Forms in Several Variables.- Hyperdeterminants.- Appendix A. Determinants.- Appendix B. A. Cayley: On the Theory of Elimination.- Bibliography.- Notes and References.- List of Notation.- Index

Discriminants, Resultants, and Multidimensional Determinants

Preface to the second edition Preface to the first edition Introduction Part I. General Theory: 1. Preliminaries 2. Families of sheaves 3. The Grauert-Mullich Theorem 4. Moduli spaces Part II. Sheaves on Surfaces: 5. Construction methods 6. Moduli spaces on K3 surfaces 7. Restriction of sheaves to curves 8. Line bundles on the moduli space 9. Irreducibility and smoothness 10. Symplectic structures 11. Birational properties Glossary of notations References Index.

/pdf/the-geometry-of-moduli-spaces-of-sheaves-44gematwk3.pdf

The geometry of moduli spaces of sheaves

Stable reflexive sheaves

We show that the resolution of moduli space of ideal instantons parameterizes the instantons on noncommutative ℝ4. This moduli space appears to be the Higgs branch of the theory of k

 D0-branes bound to N

 D4-branes by the expectation value of the B field. It also appears as a regularized version of the target space of supersymmetric quantum mechanics arising in the light cone description of (2,0) superconformal theories in six dimensions.

https://www.kitp.ucsb.edu/sites/default/files/kitp/preprints/1998030.pdf

Instantons on Noncommutative R 4 , and (2; 0) Superconformal Six Dimensional Theory

Polynomial invariants for smooth four-manifolds

Vector Bundles on Complex Projective Spaces

We introduce coupled Seiberg-Witten equations, and we prove, using a generalized vortex equation, that, for Kaehler surfaces, the moduli space of solutions of these equations can be identified with a moduli space of holomorphic stable pairs. In the rank 1 case, one recovers Witten's result identifying the space of irreducible monopoles with a moduli space of divisors. As application, we give a short proof of the fact that a rational surface cannot be diffeomorphic to a minimal surface of general type.

/pdf/the-coupled-seiberg-witten-equations-vortices-and-moduli-4xurbjml2o.pdf

The coupled seiberg-witten equations, vortices, and moduli spaces of stable pairs

Soit E un fibre algebrique de rang 2n sur P c 2n+1 . On dit, E est un fibre instanton mathematique de nombre quantique k, si E est simple, a un polynome de Chern C t (E)=(1/1−t 2 ) k et une cohomologie naturelle H 9 (E(l)) dans −2n−1≤l≤0. On demontre que l'ensemble MIp 2n+1 (k) des classes d'isomorphismes des fibres instanton mathematiques de nombre quantique k sur P c 2n+1 peuvent s'identifier a un quotient MI p2n+1 (k)=SK(2n+2k)/GL(k,C), ou SK(2n+2k) est la variete des modules de Kronecker non degenerees symetriques simples de rang 2n+2k

Mathematical instanton bundles on P2n+1.

A classical result due to Segre states that on a real cubic surface in P3 R there exist two kinds of real lines: elliptic and hyperbolic lines. These two kinds of real lines are defined in an intrinsic way, i.e., their definition does not depend on any choices of orientation data. Segre's classification of smooth real cubic surfaces also shows that any such surface contains at least 3 real lines. Starting from these remarks and inspired by the classical problem mentioned above, our article has the following goals: (a) We explain a general principle which leads to lower bounds in real algebraic geometry. (b) We explain the reason for the appearance of intrinsic signs in the classical problem treated by Segre, showing that the same phenomenon occurs in a large class of enumerative problems in real algebraic geometry. (c) We illustrate these principles in the enumerative problem for real lines in real hypersurfaces of degree 2m-3 in PmR.

/pdf/intrinsic-signs-and-lower-bounds-in-real-algebraic-geometry-32yp1l1jn9.pdf

Intrinsic signs and lower bounds in real algebraic geometry

In this article I describe construction methods for smooth subvarieties of codimension 3 in projective spaces or other ambient spaces. The methods include liaison of 3-folds in ℙ6, sections in smooth reflexive sheaves, and Pfaffians of twisted skew-symmetric vector bundle morphisms. I use these methods to construct new families of 3-folds in ℙ6, and new codimension 3 submanifolds in ℙ8 and ℙ9.

Christian Okonek

Papers

Vector Bundles on Complex Projective Spaces

The coupled seiberg-witten equations, vortices, and moduli spaces of stable pairs

Mathematical instanton bundles on P2n+1.

Intrinsic signs and lower bounds in real algebraic geometry

Notes on varieties of codimension 3 in ℙ N