Title: Extensions of projective varieties

Abstract: Algebraic geometry studies geometric shapes defined by polynomial equations. Extendability asks whether a given shape can be realized as a slice of a higher-dimensional one. For example, a plane passing through the center of a sphere cuts out a circle. Viewed in reverse, the sphere is an extension of the circle. More generally, we ask which algebraic shapes arise by intersecting a higher-dimensional algebraic shape with a hyperplane, the analogue of a plane in higher dimensions. The challenge is to find extensions beyond the cone obtained by joining every point of the original shape to a new vertex.

Understanding which extensions exist, and how many are possible, connects this question about slices to the classification of algebraic shapes. Extendability is closely related to certain numerical invariants arising from cohomology. In this talk, I will describe an approach to bounding these invariants by passing to suitable limits called ribbons, which are infinitesimally thickened algebraic shapes. Calculations on these limits provide bounds for nearby smooth varieties. I will explain how this approach sheds light on the extendability of K3 surfaces, a fundamental class of algebraic surfaces, and leads to new proofs of classification results for Fano threefolds, an important class of three-dimensional algebraic varieties. The talk will begin with concrete examples and emphasize the geometric ideas behind the method.