# Intersections of Quadrics: 25 years later

# Intersections of Quadrics: 25 years later

Consider $F:\R^n\rightarrow\mathbb{R}^2$ given by two quadratic forms and let $V=F^{-1}(0)$ and $Z=V\cap S^{n-1}$. In January 1984 I began to study the topology of the generic $Z$ when the quadratic forms are simultaneously diagonalizable. By the end of the year I had an answer, but only around 1986-87 I wrote the details of a proof that left out some cases. Last year I proved the remaining cases and also the non-diagonalizable ones (joint work with Vinicio G\'omez), thus determining the topological type of all the generic $Z$. (Also the \textit{diffeomorphism} type, except for 3 four-dimensional cases). I will describe the proof, with emphasis on some topological constructions and the algebraic topology involved.In the diagonal case $Z$ turns out to be a \textit{generalized moment-angle complex} as defined by A. Bahri, M. Bendersky, F. Cohen and S. Gitler. This framework gave new clues for the study of the varieties defined by $k>2$ quadratic forms, for which a complete answer seems out of reach, but one can describe some large, natural families (joint work with Sam, inspired by ideas of F. Bosio and L. Meersseman), also in the non-diagonalizable case (where $Z$ is \textit{not} a moment-angle complex). In the talk I will only mention how the work for $k>2$ gave also clues for the new results reported above for $k=2$.