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Creators/Authors contains: "Auel, Asher"

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  1. ; (, International Mathematics Research Notices)
    Abstract The Brill–Noether loci $$\mathcal{M}^{r}_{g,d}$$ parameterize curves of genus $$g$$ admitting a linear system of rank $$r$$ and degree $$d$$. When the Brill–Noether number is negative, they are proper subvarieties of the moduli space of genus $$g$$ curves. We explain a strategy for distinguishing Brill–Noether loci by studying the lifting of linear systems on curves in polarized K3 surfaces, which motivates a conjecture identifying the maximal Brill–Noether loci. Via an analysis of the stability of Lazarsfeld–Mukai bundles, we obtain new lifting results for line bundles of type $$g^{3}_{d}$$ that suffice to prove the maximal Brill–Noether loci conjecture in genus $$3$$–$19$, $22$, $23$, and infinitely many cases. 
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    Free, publicly-accessible full text available October 1, 2026
  2. ; ; ; (, Mathematics of Computation)
    We compute a complete set of isomorphism classes of cubic fourfolds over F 2 \mathbb {F}_2 . Using this, we are able to compile statistics about various invariants of cubic fourfolds, including their counts of points, lines, and planes; all zeta functions of the smooth cubic fourfolds over F 2 \mathbb {F}_2 ; and their Newton polygons. One particular outcome is the number of smooth cubic fourfolds over F 2 \mathbb {F}_2 , which we fit into the asymptotic framework of discriminant complements. Another motivation is the realization problem for zeta functions of K 3 K3 surfaces. We present a refinement to the standard method of orbit enumeration that leverages filtrations and gives a significant speedup. In the case of cubic fourfolds, the relevant filtration is determined by Waring representation and the method brings the problem into the computationally tractable range. 
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    Free, publicly-accessible full text available July 1, 2026
  3. ; (, Algebra & Number Theory)
    We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in 2^n variables over function fields of transcendence degree n at least 2 over an algebraically closed field of characteristic not 2. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified cohomology of products of elliptic curves over discretely valued fields. 
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  4. ; ; (, The American Mathematical Monthly)
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  5. ; ; ; ; (, Inventiones mathematicae)
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