Abstract:  Motivated by an elementary question about quadratic residues, we shall discuss the geometry of what I call strongly degenerated intersections of quadrics. Let X be a complete intersection of m+1 quadrics in P^n. Consider all linear combinations Q_a of our quadrics. They are parametrized by P^m. Let H be given by the condition that Q_a is not smooth (then the determinant of the equation of Q_a is 0, and Q_a is a cone). We say that X is strongly degenerated if the determinant hypersurface H = H(X) is a union of hyperplanes. The main example m = 2, n = 4 generically gives us a smooth curve X of genus 5 that has a very specific geometry. For simplicity we shall mostly discuss curves, touching higher dimensions but slightly. The subject being new, there are much more questions than answers. I presuppose only the basic knowledge of projective and algebraic geometry. Light refreshments will be served at 15:00 at the faculty lounge on the 8th floor.