Complete Intersections of Quadrics

There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves. It was first noticed by Weil in the 50s and has since been a testbed for many theories: Hodge theory and motives in the 70s, derived categories in the 90s, Floer theory and mirror symmetry today. The two spaces are connected by some moduli problems with a very classical flavor, involving lots of lines on quadrics, or more fashionably by matrix factorizations. The story extends easily to intersections of three quadrics and double covers of P2, but going to four quadrics, the double covers becomes singular. I produce a non-Kaehler resolution of singularities with a clear geometric meaning, and relate its derived category to that of the intersection . As a special case I get a pair of derived-equivalent Calabi-Yau 3-folds. The example nicely illustrates the modern theory of flops and derived categories.