For $\rho \in (0,1]$, let $R(\rho) = \mathbb{Z}[[z]] \cap O(B(0,\rho))$ denote the ring of power series with integer Taylor coefficients converging on the open disk $B(0,\rho)$. We prove that these rings are pairwise non-isomorphic as abstract rings. Three ingredients drive the proof: the ideal $(z)$ is the unique principal ideal with quotient $\mathbb{Z}$, so any isomorphism sends $z$ to a generator $g$ of $(z)$; every isomorphism is substitution by $g$, because it respects the $(z)$-adic filtration; and a Hadamard gap series with a natural boundary at $|w| = \rho_1$ forces the image $g(B(0,\rho_2))$ into $B(0,\rho_1)$, after which the Schwarz lemma and integrality of coefficients force $g = \pm z$ and $\rho_1 = \rho_2$.
Let $X$ be a smooth complex affine variety of dimension $n$, and let $F=(f_1,\ldots,f_r)$ be a tuple of nonzero regular functions on $X$ such that $f:=\prod_{i=1}^r f_i$ is not invertible. We study the zero loci of the Bernstein-Sato ideals $B_F^{\mathbf a}$ for nonnegative integral shifts $\mathbf a$. For a fixed log...
This is a sequel to \cite{PaperI}, where it is shown that a discrete effective divisor on the open unit disk $\D$ is the zero divisor of a holomorphic function with integer Taylor coefficients if and only if it is invariant under complex conjugation. Building on that realization theorem and on the unit characterization...
Let $X$ be a closed, smooth, orientable, positive definite $4$-manifold with $H_1(X ; \mathbb{Z}) = 0$. Let $p$ be a prime and suppose that $G = \mathbb{Z}_p$ acts smoothly on $X$ (if $p=2$ we also require an assumption on how $G$ acts on $H^2(X ; \mathbb{Z})$). Using equivariant Yang--Mills theory, Hambleton--Lee and...
For $\gcd(a,b)=1$, we show that the moduli space of $m$-codimensional $\Bbbk[\![T^a,T^b]\!]$-submodules of $\Bbbk[\![T]\!]^n$ is paved by affine cells, by proving that each Bia\l ynicki-Birula stratum of a closed related moduli space with respect to the natural $\mathbb{G}_m$-action is an affine bundle over the fixed p...
Yifeng Huang, Ruo-Fan Jiang, A. Oblomkov· 4 citations· ⚡4
Let $k$ be an arbitrary field and let $S=k[x,y,z]$ be the polynomial ring with three variables $x,y,z$. We study integrally closed $(x,y,z)$-primary ideals of $S$ that are homogeneous for a positive weighted grading but need not be homogeneous for the standard grading. From a numerical semigroup $H$ of embedding dimens...
For $r>2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field o...
Hrishabh Mishra· 0 citations
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