Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

We present a geometric approach to the theory of Painleve equations based on rational surfaces Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type We classify all such surfaces X To each X, there corresponds a root subsystem of E

 (1)

 8 inside the Picard lattice of X We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces We show that the translation part of the affine Weyl group gives rise to discrete Painleve equations, and that the above action constitutes their group of symmetries by Backlund transformations The six Painleve differential equations appear as degenerate cases of this construction In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure

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Rational surfaces associated with affine root systems and geometry of the Painlevé equations

We describe recent work of Klyachko, Totaro, Knutson, and Tao that characterizes eigenvalues of sums of Hermitian matrices and decomposition of tensor products of representations of GLn(C). We explain related applications to invariant factors of products of matrices, intersections in Grassmann varieties, and singular values of sums and products of arbitrary matrices. Recent breakthroughs, primarily by A. Klyachko, B. Totaro, A. Knutson, and T. Tao, with contributions by P. Belkale and C. Woodward, have led to complete solutions of several old problems involving the various notions in the title. Our aim here is to describe this work and especially to show how these solutions are derived from it. Along the way, we will see that these problems are also related to other areas of mathematics, including geometric invariant theory, symplectic geometry, and combinatorics. In addition, we present some related applications to singular values of arbitrary matrices. Although many of the theorems we state here have not appeared elsewhere, their proofs are mostly “soft” algebra based on the hard geometric or combinatorial work of others. Indeed, this paper emphasizes concrete elementary arguments. We do give some new examples and counterexamples and raise some new open questions. We have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before. However, there is a very large literature, particularly for linear algebra problems about eigenvalues, singular values, and invariant factors. We have listed only a few of these articles, from whose bibliographies, we hope, an interested reader can trace the history; we apologize to the many whose work is not cited directly. We begin in the first five sections by describing each of the problems, together with some of their early histories, and we state as theorems the new solutions to these problems. In Section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs. Then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians. Sections 7, 8, 9, and 10 also contain variations and generalizations of some of the theorems stated in the first five sections, as well as attributions of the theorems to their authors. One of our fascinations with this subject, even now that we have proofs of the theorems, is the challenge to understand in a deeper way why all these subjects are Received by the editors in July 1999 and in revised form January 3, 2000. 2000 Mathematics Subject Classification. Primary 15A42, 22E46, 14M15; Secondary 05E15, 13F10, 14C17, 15A18, 47B07. The author was partly supported by NSF Grant #DMS9970435. c ©2000 American Mathematical Society

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Eigenvalues, invariant factors, highest weights, and Schubert calculus

We introduce a certain compactification of the space of projective configurations i.e. orbits of the group $PGL(k)$ on the space of $n$ - tuples of points in $P^{k-1}$ in general position. This compactification differs considerably from Mumford's geometric invariant theory quotient. It is obtained by considering limit position (in the Chow variety) of the closures of generic orbits. The same result will be obtained if we study orbits of the maximal torus on the Grassmannian $G(k,n)$. We study in detail the closures of the torus orbits and their "visible contours" which are Veronese varieties in the Grassmannian. For points on $P^1$ our construction gives the Grothemdieck - Knudsen moduli space of stable $n$ -punctured curves of genus 0. The "Chow quotient" interpretation of this space permits us to represent it as a blow up of a projective space.

Chow quotients of Grassmannians. I

This paper completes the classic and modern results on classification of conjugacy classes of finite subgroups of the group of birational automorphisms of the complex projective plane.

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Finite Subgroups of the Plane Cremona Group

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David Ortland

Papers

Point sets in projective spaces and theta functions