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We prove a version of the Bhargava-Kane-Lenstra-Poonen-Rains heuristics forSelmer groups of quadratic twist families of abelian varieties over globalfunction fields. As a consequence, we derive a result towards the "minimalistconjecture" on Selmer ranks of abelian varieties in such families. Moreprecisely, we show that the probabilities predicted in these two conjecturesare correct to within an error term in the size of the constant field, $q$,which goes to $0$ as $q$ grows. Two key inputs are a new homological stabilitytheorem for a generalized version of Hurwitz spaces parameterizing covers ofpunctured Riemann surfaces of arbitrary genus, and an expression of averagesizes of Selmer groups in terms of the number of rational points on theseHurwitz spaces over finite fields.

Homological stability for generalized Hurwitz spaces and Selmer groups in quadratic twist families over function fields