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$\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties

Louisa F. Bröring

TL;DR

This work advances refined enumerative geometry by computing the compactly supported $"A^1$-Euler characteristic for low symmetric powers and for split toric varieties, and by relating these invariants to the Pajwani–Pál power structure on the Grothendieck–Witt ring. The authors introduce a $G$-equivariant quadratic Euler characteristic valued in $ ext{GW}^G(k)$, extend summation and blow-up techniques to equivariant settings, and use these tools to derive explicit formulas for $ ext{Sym}^2X$ and $ ext{Sym}^3X$ (characteristic constraints as stated). They show compatibility with Pajwani–Pál power-structure predictions for $n=2,3$ and verify this in characteristic zero; they also demonstrate that split toric varieties satisfy the same predictions. Overall, the paper provides concrete, computable refinements of classical symmetric-power counts, with broad implications for refined enumerative geometry over arbitrary fields.

Abstract

For a smooth, projective scheme $X$ over a field $k$ or any variety $X$ if $k$ has characteristic zero, we compute the compactly supported $\mathbb{A}^1$-Euler characteristic of $\operatorname{Sym}^2(X)$ if $\operatorname{char}(k) \ne 2$ and of $\operatorname{Sym}^3(X)$ if $\operatorname{char}(k) \ne 2,3$. We do so by extending the definition of a $G$-equivariant quadratic Euler characteristic first studied by Pajwani-Pál to arbitrary characteristic and by studying its relation to the $\mathbb{A}^1$-Euler characteristic of quotients. As an application, we show that the compactly supported $\mathbb{A}^1$-Euler characteristic of $\operatorname{Sym}^n(X)$ agrees with the prediction from the power structure constructed by Pajwani-Pál for $n = 2,3$. Furthermore, we compute the compactly supported $\mathbb{A}^1$-Euler characteristic of split toric varieties and show that the compactly supported $\mathbb{A}^1$-Euler characteristic of all of their symmetric powers agrees with the prediction from the power structure constructed by Pajwani-Pál.

$\mathbb{A}^1$-Euler Characteristic of Low Symmetric Powers and Split Toric Varieties

TL;DR

This work advances refined enumerative geometry by computing the compactly supported -Euler characteristic for low symmetric powers and for split toric varieties, and by relating these invariants to the Pajwani–Pál power structure on the Grothendieck–Witt ring. The authors introduce a -equivariant quadratic Euler characteristic valued in , extend summation and blow-up techniques to equivariant settings, and use these tools to derive explicit formulas for and (characteristic constraints as stated). They show compatibility with Pajwani–Pál power-structure predictions for and verify this in characteristic zero; they also demonstrate that split toric varieties satisfy the same predictions. Overall, the paper provides concrete, computable refinements of classical symmetric-power counts, with broad implications for refined enumerative geometry over arbitrary fields.

Abstract

For a smooth, projective scheme over a field or any variety if has characteristic zero, we compute the compactly supported -Euler characteristic of if and of if . We do so by extending the definition of a -equivariant quadratic Euler characteristic first studied by Pajwani-Pál to arbitrary characteristic and by studying its relation to the -Euler characteristic of quotients. As an application, we show that the compactly supported -Euler characteristic of agrees with the prediction from the power structure constructed by Pajwani-Pál for . Furthermore, we compute the compactly supported -Euler characteristic of split toric varieties and show that the compactly supported -Euler characteristic of all of their symmetric powers agrees with the prediction from the power structure constructed by Pajwani-Pál.
Paper Structure (11 sections, 67 theorems, 188 equations)

This paper contains 11 sections, 67 theorems, 188 equations.

Table of Contents

  1. The $\mathbb{A}^1$-Euler Characteristic
  2. Power Structures and Symmetric Powers
  3. $\mathbb{A}^1$-Euler Characteristic of a Split Toric Variety and its Symmetric Powers
  4. Results on the Smoothness of Quotients
  5. The Equivariant Quadratic Euler Characteristic
  6. Construction
  7. Properties
  8. Application to the $\mathbb{A}^1$-Euler Characteristic of Quotients
  9. Explicit Computation for Sym2
  10. Explicit Computation for Sym3
  11. Comparison with the Power Structure Prediction

Key Result

Theorem 1

Let $X$ be a connected, smooth, projective scheme over $k$. Express the $\mathbb{A}^1$-Euler characteristic of $X$ as $\chi_c(X/k) = \beta -mH$ with $\beta = \sum_{i=1}^n\langle \alpha_i\rangle$ and $m\ge 0$. Then if $\dim X$ is odd, we have in $\mathop{\mathrm{GW}}\nolimits(k)$ If $\dim X$ is even, we have in $\mathop{\mathrm{GW}}\nolimits(k)$.

Theorems & Definitions (172)

  • Theorem : see Theorem \ref{['thm:qec-sym2']}
  • Theorem : see Theorem \ref{['thm:sym3']} and Remark \ref{['rem:sym3-formula-indep']}
  • Theorem : see Theorem \ref{['thm:smooth-quotient-euler-char']}
  • Theorem : see Theorem \ref{['thm:compatibility-a2-smooth-projective']}
  • Theorem : see Theorem \ref{['thm:sym3-comparison']}
  • Theorem : see Theorem \ref{['thm:sym23-compatibility-char-zero']}
  • Proposition : see Proposition \ref{['prop:euler-char-toric']}
  • Proposition : see Proposition \ref{['prop:split-toric-symmetrisable']}
  • Definition 1.1
  • Definition 1.2
  • ...and 162 more