Oriented Cohomology Rings of Some Moduli Spaces via Blowups

TL;DR

This paper develops blowup formulas for oriented cohomology theories, computes explicit cohomology rings for specific moduli spaces, and connects these results to classical enumerative geometry problems.

Contribution

It introduces a blowup formula for non-$A^1$-invariant theories and provides explicit presentations of cohomology rings for various moduli spaces, extending known results.

Findings

Derived an additive blowup formula for motivic spectra.

Computed cohomology rings for del Pezzo surfaces and specific blowups.

Connected cohomology computations to classical enumerative geometry solutions.

Abstract

Oriented cohomology theories provide a general framework to perform intersection-theory-type calculus. The Chow ring, algebraic $K$-theory, and Levine--Morel's algebraic cobordism are all instances of such theories satisfying $\mathbb A^1$-invariance. Topological Hochschild homology, topological cyclic homology, and Hodge cohomology are important examples of theories without $\mathbb A^1$-invariance. In this paper, we prove an additive blowup formula for oriented cohomology theories in the non-$\mathbb A^1$-invariant category of motivic spectra, developed by Annala, Hoyois, and Iwasa. Then, we specialize to $\mathbb A^1$-invariant theories and give presentations of oriented cohomology rings of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces, the blowup of $\mathbb P^3$ along the twisted cubic, and the blowup of $\mathbb P^5$ along the Veronese surface, which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to classical enumerative geometry problems, such as Steiner's $3264$ conics, using arbitrary oriented cohomology theories. Finally, we give a presentation of oriented cohomology rings of $\overline M_{0,n}$, which generalizes Keel's presentation of the Chow ring.