The Evolution of Enumerative Geometry: A Narrative from Classical Problems to Enriched Invariants

2025/10/05 by Candace Bethea, Bethea, Candace, Thomas Brazelton +1
Mathematics · #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.2510.04275

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.

Citations

• Galois Groups of Symmetric Cubic Surfaces
• Stacks, Monodromy and Symmetric Cubic Surfaces
• A quadratic Abramovich-Bertram formula
• Quadratic Segre indices
• The equivariant degree and an enriched count of rational cubics
• Quadratically enriched binomial coefficients over a finite field
• Monodromy in the space of symmetric cubic surfaces with a line
• Bitangents of real algebraic curves: signed count and constructions
• Low degree motivic Donaldson-Thomas invariants of the three-dimensional projective space
• An enriched count of nodal orbits in an invariant pencil of conics
• Arithmetic Counts of Tropical Plane Curves and Their Properties
• Universally counting curves in Calabi--Yau threefolds
• Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves
• On Motivic and Arithmetic Refinements of Donaldson-Thomas invariants
• A quadratically enriched count of rational curves
• A signed count of 2-torsion points on real abelian varieties
• Equivariant enumerative geometry
• Quadratically Enriched Tropical Intersections
• Arithmetic inflection of superelliptic curves
• Probabilistic enumerative geometry over p-adic numbers: linear spaces on complete intersections
• A¹-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
• Toward an algebraic theory of Welschinger invariants
• Probabilistic Schubert Calculus
• Random fields and the enumerative geometry of lines on real and complex\n hypersurfaces
• Counting real rational curves on K3 surfaces
• Comparing Euler classes
• The open Gromov-Witten-Welschinger theory of blowups of the projective plane
• Intrinsic signs and lower bounds in real algebraic geometry
• A Proof of the Göttsche-Yau-Zaslow Formula
• The local Donaldson-Thomas theory of curves
• Symmetric obstruction theories and Hilbert schemes of points on threefolds
• The local Gromov-Witten theory of curves
• Gromov-Witten theory and Donaldson-Thomas theory, I
• Quantum K -theory, I: Foundations
• A Simple Proof that Rational Curves on K3 are Nodal
• Gromov-Witten invariants of blow-ups
• Euler number of the compactified Jacobian and multiplicity of rational curves
• Localization of virtual classes
• Counting rational curves on K3 surfaces
• The quantum cohomology of blow-ups of P² and enumerative geometry
• Counting plane curves of any genus
• Equivariant Gromov - Witten Invariants
• Stacks of Stable Maps and Gromov-Witten Invariants
• Enumeration of rational curves via torus actions

Related

• On the crossroads of enumerative geometry and geometric representation\n theory
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• Enumerative Real Algebraic Geometry
• Enumerative geometry and geometric representation theory
• Algebraic and geometric methods in enumerative combinatorics