Louisa Bröring (Duisburg-Essen): Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings

The quadratic Euler characteristic $\chi(X)$ of a smooth, projective scheme
$X$ over a field $k$ of characteristic not two is a refinement of the
topological Euler characteristic to quadratic forms, constructed using motivic
homotopy theory. For example, if $k\subset \mathbb{R}$, then rank of $\chi(X)$
is equal to the topological Euler characteristic of $X(\mathbb{C})$ and the
signature of $\chi(X)$ with respect to the given embedding is equal to the
topological Euler characteristic of $X(\mathbb{R})$. The quadratic Euler
characteristic plays an import role in the programme of $\mathbb{A}^1$-refined
enumerative geometry.

After briefly introducing the quadratic Euler characteristic, we present a
computation of the quadratic Euler characteristic of geometrically cyclic
branched coverings leveraging Levine’s quadratic Riemann-Hurwitz formula. An
$n$-fold geometrically cyclic branched covering is a morphism $f\colon Y \to
X$ between smooth, projective schemes together with a smooth, closed subscheme
$Z \subset X$ satisfying the following condition: there exists a line bundle
$L$ over $X$ and a section $s \colon X \to L^{\otimes n}$ such that $Z$ is the
zero locus of $s$ and $f$ is the pullback along $s$ of the map $L \to
L^{\otimes n}$ taking $n$-th powers.

As an application, we compute the quadratic Euler characteristic of branched
double covers of $\mathbb{P}^2$, which includes some K3 surfaces.

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