부산대학교 도서관

This is a modern account of (part of) a book by A. Coble [{\it Algebraic geometry and theta functions}, Amer. Math. Soc. NewYork, 1929, Jbuch {\bf 55}, 808].The theme is the theory of invariants of finite ordered sets inprojective space, with special regard to low dimensions, and itsreincarnation in other forms of life in the realm of algebraicgeometry.\parThe book falls naturally into three parts. A first part deals with thegeneralities of the invariant theory of sets as above. Blowing up theprojective space at the points of such a set yields what is called ageneralized del Pezzo variety. A second part of the book is devoted tothe study of these varieties. It includes: the root systems related tothem, their Weyl groups and the so-called Cremona representations ofthese. The last part is where theta functions come in, in the form oftwo very classical themes. One of these is the relationship betweenordered sets of $2g+2$ points on the projective line and hyperellipticcurves of genus $g$ together with a level-$2$ structure on them. Theother one is a similar relationship between ordered sets of 7 pointson the projective plane and curves of genus 3 with a level-2 structureon them. This last story is a very beautiful one, a witness to theexuberant botany of classical algebraic geometry.