We show that, at a fake exponent of an $A$-hypergeometric system and for an ordered negative support family, intrinsic perturbation within $\ker_{\mathbb{Z}}(A)$ can produce a strictly smaller coefficient space than ambient perturbation, which answers a question of Okuyama--Saito. We measure the failure by an intrinsic-perturbation obstruction module, a quotient of two colon ideals whose graded dual is the ambient coefficient space modulo the intrinsic one, and we realize this module by two right-exact sequences involving $\operatorname{Tor}_1$ and present it finitely by two antichains. We give configurations of lattice rank 1 and 2 with nonzero obstruction, including one with an ordered distinguished collection consisting of all negative supports attained only by lattice shifts of nonnegative weight, and thereby settle the case left open by Okuyama--Saito. We give, for every integer $q\geq 1$, a configuration of lattice rank 2 with a connected column matroid and with a normal affine semigroup generated by the reduced Gr\"obner basis vectors, whose obstruction module has dimension $q^2$, so this dimension is unbounded at fixed lattice rank.
For a homogeneous $A$-hypergeometric system, let $K_v$ be the intersection of the negative support family for a fake exponent $v$, and let $C_w$ be the set of indices of variables that do not occur in any minimal monomial generator of $\operatorname{in}_w(I_A)$. We prove that $K_v=\operatorname{nsupp}(v)\cap C_w$ holds...
We study the canonical formal solution space of a homogeneous $A$-hypergeometric system in a generic direction. We present as a finite-dimensional cokernel the quotient of that space by the span of the canonical series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered...
We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first c...
Lucas Chen, Joshua Im, Steven J. Miller et al.· 0 citations
Let $(R,\mathfrak m,k)$ be a Noetherian local ring, let $M$ be a nonzero finitely generated $R$-module, and let $J_1,\ldots,J_s$ be arbitrary proper ideals of positive height on $M$. We introduce the notion of directional mixed multiplicities, obtained from the multigraded quotients $J^{\mathbf u}M/J^{\mathbf u+\mathbf...
We compare two mixed Hodge structures attached to a unipotent variation $\mathbb V$ of mixed Hodge--Tate structures on $X=\mathbb P^1\setminus{0,1,\infty}$: the cohomology of its extension by $*$ at $0$ and by $!$ at $\infty$, and the vanishing cycles at $1$. The semi-holonomy isomorphism constructed in [arXiv:2604.263...
Let $\mathcal{A}\to S$ be an abelian scheme over a normal quasi-projective variety over a number field, and let $\mathcal{X}\subseteq\mathcal{A}$ be a positive-dimensional subvariety. For every integer $t>0$, we prove that, outside the $t$-degeneracy locus of $\mathcal{X}$, at most finitely many points lie in flat grou...
Fabrizio Barroero, L. Capuano, Tang-Li Ge et al.· 0 citations
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