Papers

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 established there, we study the prime and maximal spectra of the rings $R = \mathbb{Z}[i][[z]] \cap O(\mathbb{D})$ and $R_{\mathbb{R}} = \mathbb{Z}[[z]] \cap O(\mathbb{D})$. We give a trichotomy for maximal ideals, classify the prime ideals lying outside the class $\mathfrak{P}_1$ of primes containing a unit-constant-term element, attach to each element of $\mathfrak{P}_1$ a complete analytic invariant, and construct an injection from ultrafilters on admissible divisors into $\mathfrak{P}_1$ by an ultraproduct device. We prove the point-evaluation ideals $P_a$ are prime for all $a \in \D$ and maximal for real $a$, and deduce a conditional Bézout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.

The holomorphic foliations of degree $d$ on $\mathbb{CP}^{2}$ with isolated singularities are determined by their singular subschemes, which correspond to elements in the Hilbert Scheme of $N=d^{2}+d+1$ points. In this paper, we analyze the affine varieties parametrized by a partition of $N=7$ that cover the Hilbert Scheme of seven points, with the property that the Hilbert function of the elements in these affine varieties corresponds to that of holomorphic foliation of degree $d=2$. Moreover, we present some results for degree $3$.

We show that the graded Betti numbers are not invariants of the tropicalisation of a homogeneous ideal, or the tropicalisation of a projective algebraic scheme over a valued field. We do this by producing a family of saturated ideals that all realise the same tropical ideal with one member having a distinct Betti table from the rest.

We introduce several conjectures which mainly deal with restrictions on the Betti tables of licci ideals. We focus on a series of questions that compare the number of generators of homogeneous licci ideals in polynomial rings to the maximal last shift in their graded free resolution. We prove these conjectures in a large number of cases.

Let $K$ be a complete nonarchimedean valued field with $v(K^\times)=\mathbf R$, and let $V=\mathcal O_K$. We prove that $K$ is spherically complete if and only if $V[[T]]$ is coherent, and that this is also equivalent to $V[[T]]$ being a GCD domain. If $K$ is perfect of characteristic $p$, the same characterization holds for the Witt vector ring $W(V)$. Thus, this settles the previously unresolved full-real-value-group case in the coherence problems for both formal power series and Witt vector rings. In particular, this result also gives affirmative answers to Questions~9 and~10 of Anderson–Kang–Park. The proof combines a coherence criterion for complete rings with a spherically complete valuation quotient and a uniform construction of non-finitely generated intersections of two principal ideals from an empty ball chain.

Given a scheme $X$ of finite type over the complex numbers, the Behrend function is a constructible function $\nu_X: X(\mathbb C) \rightarrow \mathbb Z $ introduced by Behrend in order to define enumerative invariants in Donaldson–Thomas theory. Even in simple cases, the Behrend function is very difficult to compute. In this article, we tackle the problem of computing the Behrend function of zero-dimensional schemes. We obtain a number of explicit formulas, in particular, for arbitrary zero-dimensional monomial schemes, thus providing vast generalizations of previous work of Graffeo–Ricolfi. Our main tools come from the theory of blowup algebras. Along the way, we establish results of independent interest related to the integer decomposition property, weighted Veronese subrings, and reduced fiber rings.

Let $R$ be a complete Cohen–Macaulay local ring and $\Lambda$ an $R$-order. We study the Krull–Gabriel dimension of the functor category $\mathrm{mod}\,\underline{\mathcal C}(\Lambda)$, where $\underline{\mathcal C}(\Lambda)$ is the stable category of maximal Cohen–Macaulay $\Lambda$-modules. This work is motivated by the non-existence theorem of Herzog and Krause for Krull–Gabriel dimension $1$ over artin algebras. We first prove that, if $\Lambda$ is Gorenstein and $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(\Lambda)\leq 1$, then $\Lambda$ is an isolated singularity. We then show that, if $\Lambda$ is an isolated singularity of uncountable Cohen–Macaulay representation type, then $\mathrm{KGdim}\,\mathrm{mod}\,\underline{\mathcal C}(\Lambda)\neq 1$.

Shimada recently showed that Frobenius–Witt cotangent complex, the animation of Saito's Frobenius–Witt differentials, vanishes on perfectoid rings, thus it serves as a candidate of “absolute” cotangent complex. In this article, we give a pullback description of their arithmetic extension, and as a consequence, we give a direct description of Frobenius–Witt cotangent complex, which leads to generalizations to derived rings and animated pre-log rings. This description allows us to compute Frobenius–Witt cotangent complex of derived $\delta$-rings as well. We also propose a version of Frobenius–Witt cotangent complex relative to derived $\delta$-rings, and establish a vanishing result for prisms, which generalizes vanishing of Frobenius–Witt cotangent complex of perfectoid rings. Finally, independently of previous considerations, we give a regularity criterion via Frobenius–Witt cotangent complex for $p$-local Noetherian (not necessarily local) rings without $F$-finiteness. We also record the flatness of Frobenius–Witt cotangent complex of valuation rings, with essential ideas due to ChatGPT-6 Astra.

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=\mu_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a transfer principle: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the total diagonal: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $\Delta_i$, the raw derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $\Delta_i$ extends holomorphically there, with $\Delta_i\phi=\partial_i^r\phi/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.

Let $(R,\mathfrak m)$ be a Noetherian local ring admitting a canonical module $\omega_R$. We prove that $R$ is quasi-Gorenstein if and only if $\operatorname{Fitt}_1^R(\omega_R)\cong\omega_R$. When $R$ is Cohen-Macaulay, the same condition characterizes the Gorenstein property, confirming an expectation of Eisenbud, Ficarra, Herzog, and Moradi and extending their result for canonical ideals in local Cohen–Macaulay domains of type at most two.

We provide a cluster algebraic construction of the cube root symbol letters that appear at leading order in the near-collinear expansion of the four-point energy correlator in $\mathcal{N}=4$ SYM theory.

In this paper, we study the homological shift ideals of edge ideals associated with weighted oriented graphs. For a weighted oriented graph $D$, let $HS_k(I(D))$ denote the $k^{th}$ homological shift ideal of its edge ideal $I(D)$. If $D$ is vertex-splittable, then we characterize that $HS_1(I(D))$ has linear quotients if and only if $D_6$, $D_7$, and $D_8$ are not induced subgraphs of $D$. Furthermore, we show that if $I(D)$ has linear quotients, then $\sqrt{HS_k(I(D))} = HS_k(I(G))$, for all $k\geq 1$, where $G$ is the underlying simple graph of $D$. We show that if $I(D)$ has homological linear quotients, then $I(G)$ also has homological linear quotients. If $D$ is a tree, then we establish the following characterization: \begin{align*} HS_k(I(D)) \text{ has linear quotients for all } k\geq 0 \iff ~ &G~ \text{is} \text{ a star graph or a broom graph} \\ &\text{ and }~ D \text{ is $D_i$-free, for } i=1,2,5,6,8. \end{align*}

Fix a generic weight vector and a fake exponent of a homogeneous $A$-hypergeometric system. Using all corresponding standard pairs, including embedded ones, we construct an Artinian quotient of the Stanley–Reisner ring of the link of the negative support. Its Hilbert series gives the graded dimensions of the orthogonal complement of the local fake indicial ideal and, under the Okuyama–Saito Frobenius condition, those of the leading logarithmic coefficient space of actual series solutions. The construction requires no Cohen–Macaulay hypothesis. When a top-dimensional standard pair occurs and the link is Cohen–Macaulay, the Hilbert series specializes to the $h$-polynomial of the link.