Description
In this work, we provide a complete classification of the Feynman-integral geometries at two-loop order in four-dimensional Quantum Field Theory with standard quadratic propagators. We do this by considering a finite basis of integrals in the 't Hooft-Veltman scheme and analysing the leading singularities of the integrals for generic kinematic values through the loop-by-loop Baikov representation. Aside from the Riemann sphere, we find elliptic curves, hyperelliptic curves of genus 2 and 3 as well as K3 surfaces, and in addition a smooth and non-degenerate Del Pezzo surface of degree 2, resulting in a curve of geometric genus 3. These geometries determine the space of functions relevant for Quantum Field Theories at two-loop order, including in the Standard Model.