The Evolution of Enumerative Geometry: A Narrative from Classical Problems to Enriched Invariants
Candace Bethea, Thomas Brazelton
TL;DR
This survey traces the evolution of enumerative geometry from classical counts and the conservation-of-number principle to enriched invariants that take field, orientation, and symmetry into account. It connects Schubert calculus and Hilbert’s program to modern frameworks (Gromov–Witten, virtual fundamental classes, and Chow–Witt theory) and then to quadratically enriched counts valued in the Grothendieck–Witt ring, including $N_{X,d, ext{σ}}$-type curve counts whose rank and signature recover the complex and real invariants respectively. It highlights local index computations via EKL degrees, the integration of tropical methods, and emergent directions in random and equivariant enumerative geometry, illustrating a unified arithmetic-geometric paradigm with broad computational and conceptual reach. The significance lies in providing a cohesive path from classical problems to arithmetic and motivic refinements, broadening applicability to fields beyond $\mathbb{C}$ and enabling new connections to moduli, topology, and algebraic K-theory.
Abstract
Enumerative geometry, the art and science of counting geometric objects satisfying geometric conditions, has seen a resurgence of activity in recent years due to an influx of new techniques that allow for enriched computations. This paper offers a historical survey of enumerative geometry, starting with its classical origins and real counterparts, to new advances in quadratic enrichment. We include a brief survey of the paradigm shift initiated by Gromov-Witten theory, whose impact can be seen in recent results in quadratically enriched enumerative geometry. Finally, we conclude with a brief overview of emerging directions including random and equivariant enumerative geometry.