1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem. Quadratic Forms.- 7. Real Spectrum.- 8. Nash Functions.- 9. Stratifications.- 10. Real Places.- 11. Topology of Real Algebraic Varieties.- 12. Algebraic Vector Bundles.- 13. Polynomial or Regular Mappings with Values in Spheres.- 14. Algebraic Models of C? Manifolds.- 15. Witt Rings in Real Algebraic Geometry.- Index of Notation.

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Real Algebraic Geometry

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

Preliminaries Positive polynomials and sums of square Krivine's Positivstellensatz The moment problem Non-compact case Archimedean $T$-modules Schmudgen's Positivstellensatz Putinar's question Weak isotropy of quadratic forms Scheiderer's local-global principle Semidefinite programming and optimization Appendix 1: Tarski-Seidenberg theorem Appendix 2: Algebraic sets Bibiography

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Positive polynomials and sums of squares

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

Enriques Surfaces I

Topics in algebraic geometry Topics in theta functions Geometric applications of the functions of genus two Geometric applications of the functions of genus three Geometric aspects of the abelian modular functions of genus four Theta relations of genus four References.

Algebraic geometry and theta functions

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Point sets and allied Cremona groups. III

Part Ii of this account was devoted to a study of the invariants of a set Pk of n points in Sk under the group of permutations of the points. The set as a projective figure was mapped by a point P of a space lk(n-k-2) in which the permutation group appeared as a Cremona group Gnf!. In this part we shall consider the effect upon the set pk of certain Cremona transformations C in Sk These transformations C are of a special charatcer when k > 2 described hereafter by the term regular. They are determined essentially by their fundamental points alone and in all important particulars are entirely analogous to the ternary Cremona transformations. These regular Cremona transformations form the regular Cremona group in Sk. If one set of fundamental points of a regular transformation C be placed at P' there is determined a new set Pnk congruent to P' under C. The totality of sets P,' congruent in some order to a given set Pk is mapped in fk(n-k-2) by an aggregate of points P' which form a conjugate set under the extended group Gn, k of P'. This group Gn, k in " contains Gw! as a subgroup and in general is infinite and discontinuous. The major part of this article is devoted to a study of this group. In ? 5 a group gn, k of linear transformations which is isomorphic with Gn, k iS introduced. The new group brings to light properties both of Gn, k and of regular transformations in Sk. An interesting result is a determination of all types of regular transformations with a single symmetrical set of fundamental points. MIost of the types are well knob n but some are novel. In ? 6 another group en, k of linear transformations, also isomorphic with Gn, k, iS defined, which is particularly effectixe for a discussion of the infinite groups Gg,2, G8, 3, and G9,5. The close relation between the associated sets P7 and Q"-k-2 which appeared

Arthur B. Coble

Papers

Algebraic geometry and theta functions

Point sets and allied Cremona groups. III

Point Sets and Allied Cremona Groups.

The Ten Nodes of the Rational Sextic and of the Cayley Symmetroid

Point Sets and Allied Cremona Groups (Part III)