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Motivated by the question of the impact of selective advantage in populations with skewed reproduction mechanims, we study a Moran model with selection. We assume that there are two types of individuals, where the reproductive success of one type is larger than the other. The higher reproductive success may stem from either more frequent reproduction, or from larger numbers of offspring, and is encoded in a measure Λ for each of the two types. Our approach consists of constructing a Λ-asymmetric Moran model in which individuals of the two populations compete, rather than considering a Moran model for each population. Under certain conditions, that we call the "partial order of adaptation", we can couple these measures. This allows us to construct the central object of this paper, the Λ−asymmetric ancestral selection graph, leading to a pathwise duality of the forward in time Λ-asymmetric Moran model with its ancestral process. Interestingly, the construction also provides a connection to the theory of optimal transport. We apply the ancestral selection graph in order to obtain scaling limits of the forward and backward processes, and note that the frequency process converges to the solution of an SDE with discontinous paths. Finally, we derive a Griffiths representation for the generator of the SDE and use it to find a semi-explicit formula for the probability of fixation of the less beneficial of the two types.
We prove that the projectivized strata of differentials are not contained in pointed Brill-Noether divisors, with only a few exceptions. For a generic element in a stratum of differentials, we show that many of the associated pointed Brill-Noether loci are of expected dimension. We use our results to study the Auel-Haburcak Conjecture: We obtain new non-containments between maximal Brill-Noether loci in Mg. Our results regarding quadratic differentials imply that the quadratic strata in genus 6 are uniruled.
For genus g=r(r+1)2+1, we prove that via the forgetful map, the universal Prym-Brill-Noether locus Rrg has a unique irreducible component dominating the moduli space Rg of Prym curves.
For genus g=2i≥4 and the length g−1 partition μ=(4,2,…,2,−2,…,−2) of 0, we compute the first coefficients of the class of D¯¯¯¯(μ) in PicQ(R¯¯¯¯g), where D(μ) is the divisor consisting of pairs [C,η]∈Rg with η≅OC(2x1+x2+⋯+xi−1−xi−⋯−x2i−1) for some points x1,…,x2i−1 on C. We further provide several enumerative results that will be used for this computation.