arXiv subjects

Louisa F. Bröring

Publications and source records attributed to Louisa F. Bröring.

3 recordsLinked to original sources

arXiv2026

Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings

For an $n$-fold geometrically cyclic branched covering $Y$ of a smooth, projective scheme $X$ branched at a smooth closed subscheme $Z\subset X$ with $n \in k^\times$, we compute the quadratic Euler characteristic of $Y$ in terms of certain Euler classes on $X$ and $Z$ using the quadratic Riemann-Hurwitz formula of Levine. When $n$ is odd, we relate the quadratic Euler characteristic of $Y$ to the quadratic Euler characteristics of $X$ and $Z$, obtaining similar formulae to the situation in topology. As an application, we compute the quadratic Euler characteristic of geometrically cyclic branched double coverings of $\mathbb{P}^2$.

math.AG↗

arXiv2026

The $\mathbb{A}^1$-Euler characteristic of symmetric powers

The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an integer, which carries a lot of information, but is difficult to compute in practice. In this survey, we discuss a conjectural way for computing the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a variety in terms of the $\mathbb{A}^1$-Euler characteristic of the variety itself formulated using the theory of power structures. We discuss evidence towards the conjecture so far, techniques to approach it, and applications.

math.AG↗

arXiv2026

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

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.

math.AG↗