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The integral Chow ring of weighted blow-ups

Veronica Arena, Stephen Obinna, Dan Abramovich

TL;DR

This work delivers an integral Chow-ring description for weighted blow-ups by developing a weighted projective-bundle calculus, a generalized splitting principle, and a Gysin formula capturing the pushforward along the weighted exceptional divisor. The authors derive an exact, computable presentation $A^*(\tilde{Y})\cong\frac{(A^*(X)[t])\cdot t\oplus A^*(Y)}{((P(t)-P(0))\alpha,-i_*(\alpha))}$ with $P(t)=c_{top}^{\mathbb{G}_m}(\mathcal{N}_XY)(t)$ and $[\tilde{X}]=-t$, and extend these constructions to quotient stacks. They also give a practical computation for $A^*(\bar{\mathcal{M}}_{1,2})$ using the weighted-blow-up model of the moduli space, and prove that the framework adapts to quotient-stack settings, preserving the integral structure. Overall, the results generalize Fulton’s classical blow-up theory to the weighted setting, enabling explicit integral Chow rings in a broad stack-theoretic context with concrete examples and broad applications to intersection theory on stacks.

Abstract

We give a formula for the integral Chow rings of weighted blow-ups. Along the way, we also compute the integral Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the appendix we compute the Chern class of a weighted blow-up.

The integral Chow ring of weighted blow-ups

TL;DR

This work delivers an integral Chow-ring description for weighted blow-ups by developing a weighted projective-bundle calculus, a generalized splitting principle, and a Gysin formula capturing the pushforward along the weighted exceptional divisor. The authors derive an exact, computable presentation with and , and extend these constructions to quotient stacks. They also give a practical computation for using the weighted-blow-up model of the moduli space, and prove that the framework adapts to quotient-stack settings, preserving the integral structure. Overall, the results generalize Fulton’s classical blow-up theory to the weighted setting, enabling explicit integral Chow rings in a broad stack-theoretic context with concrete examples and broad applications to intersection theory on stacks.

Abstract

We give a formula for the integral Chow rings of weighted blow-ups. Along the way, we also compute the integral Chow rings of weighted projective stack bundles, a formula for the Gysin homomorphism of a weighted blow-up, and a generalization of the splitting principle. In addition, in the appendix we compute the Chern class of a weighted blow-up.
Paper Structure (30 sections, 38 theorems, 66 equations)

This paper contains 30 sections, 38 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Equivariant Intersection Theory
  4. Chow groups of weighted projective stack bundles
  5. Equivariant Chern classes of a weighted line bundle
  6. Equivariant Chern classes of homogeneous bundles
  7. Chern classes of weighted affine bundles
  8. The Chow ring of a weighted projective stack bundle
  9. The splitting principle
  10. Structure groups
  11. Lemmas on classifying spaces
  12. The Splitting Principle
  13. The Gysin homomorphism induced by a weighted blow-up
  14. Notation
  15. Specialization to the weighted normal cone
  16. ...and 15 more sections

Key Result

Theorem 1

Let $X$, $Y$, $\tilde{X}$, $\tilde{Y}, f$ be as above, then we have the following exact sequence of Chow groups. Further, if we use rational coefficients, then this becomes a split short exact sequence with $g_*$ left inverse to $(f^!, -i_*)$.

Theorems & Definitions (82)

  • Theorem : \ref{['sequence']} Key sequence
  • Theorem : \ref{['main theorem 4']} Chow ring of a weighted blow-up
  • Theorem : \ref{['WeighedProjectiveBundleFormula']} Weighted projective bundle formula
  • Theorem : \ref{['Gysin homomorphism for weighted blow-up']} Weighted key formula
  • Theorem : \ref{['splitting principle']} The splitting principle
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 72 more