Degrees of Fano Quiver Moduli

RESUMO

This paper provides methods to compute degrees of Fano quiver moduli and gives explicit degree formulas for two classes of them. One such class is a series of toric Fano varieties that parametrize isomorphism classes of representations of the bipartite quiver $K_{(2,q)}^m$ with dimension vector $\underline{1}$, and the other class is the moduli space of q-point configurations on $\mathbb{P}^1$. In the former case, we obtain a formula comprised of sums of binomial coefficients in $q$ and $m$. Its special case $m = 1$ turns out to be the same as the central MacMahon numbers (OEIS entry A177043). In the latter case, we obtained a formula over $\mathbb{Q}$. These formulas are much more efficient than computing directly in Chow rings.