부산대학교 도서관

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