Tootfinder

All rectangles exhibit canonical Ramsey property
Gennian Ge, Yang Shu, Zixiang Xu
https://arxiv.org/abs/2508.02465 https://arxiv.org/pdf/2508.02465

All rectangles exhibit canonical Ramsey property
In a seminal work, Cheng and Xu proved that for any positive integer \(r\), there exists an integer \(n_0\), independent of \(r\), such that every \(r\)-coloring of the \(n\)-dimensional Euclidean space \(\mathbb{E}^n\) with \(n \ge n_0\) contains either a monochromatic or a rainbow congruent copy of a square. This phenomenon of dimension-independence was later formalized as the canonical Ramsey property by Geheér, Sagdeev, and Tóth, who extended the result to all hypercubes, and to rectangle…

Variation of Kahler-Einstein metrics with mixed singularities
Xin Fu, Jiyuan Han, Yongpan Zou
https://arxiv.org/abs/2508.02212 https://arxiv.org/pdf/2508.0…

Variation of Kahler-Einstein metrics with mixed singularities
In this short note, we consider a fiberation f: (X, Delta) to Y between two compact Kahler manifolds with generic fiber of f being a smooth log canonical pair with ample canonical divisor, we prove that the current induced by variation of Kahler Einsteins with mixed cone and Poincare singularities is positive, hence generalize the result of Schumacher in the smooth case [22] and the result of Guenancia in the conic case [14]. As application, we prove the surjectivity of Albanese map for a smoot…

Quasi-canonical lifting of projective varieties in positive characteristic
Ryo Ishizuka, Kazuma Shimomoto
https://arxiv.org/abs/2506.01345 https://

Quasi-canonical lifting of projective varieties in positive characteristic
The main aim of this article is to give new classes of smooth projective varieties over characteristic $p>0$ that admit flat liftings over the Witt vectors together with additional data (logarithmic structure and the Frobenius morphism) by showing a descending property of such Frobenius liftability. We establish a refined form of the classical result due to Mehta-Srinivas on the existence of canonical liftings. For this purpose, we also establish a result on the algebraization of certain $p$-ad…

Linearly Distributive Fox Theorem
Rose Kudzman-Blais
https://arxiv.org/abs/2506.02180 https://arxiv.org/pdf/2506.02180

Linearly Distributive Fox Theorem
Linearly distributive categories (LDC), introduced by Cockett and Seely to model multiplicative linear logic, are categories equipped with two monoidal structures that interact via linear distributivities. A seminal result in monoidal category theory is the Fox theorem, which characterizes cartesian categories as symmetric monoidal categories whose objects are equipped with canonical comonoid structures. The aim of this work is to extend the Fox theorem to LDCs and characterize the subclass of …

Three-loop banana integrals with four unequal masses
Claude Duhr, Sara Maggio, Franziska Porkert, Cathrin Semper, Sven F. Stawinski
https://arxiv.org/abs/2507.23061 https://

Three-loop banana integrals with four unequal masses
We present a system of canonical differential equations satisfied by the three-loop banana integrals with four distinct non-zero masses in $D = 2-2\eps$ dimensions. Together with the initial condition in the small-mass limit, this provides all the ingredients to find analytic results for three-loop banana integrals in terms of iterated integrals to any desired order in the dimensional regulator. To obtain this result, we rely on recent advances in understanding the K3 geometry underlying these …

Configurational density of states of power-law potentials and the virial theorem in steady states
Sergio Davis
https://arxiv.org/abs/2507.22624 https://arx…

Configurational density of states of power-law potentials and the virial theorem in steady states
In this brief note, the configurational density of states of a system of particles interacting via power-law pair potentials is computed exactly. The result is consistent with a constant microcanonical heat capacity. The well-known form of the virial theorem for this class of systems is recovered using only the obtained configurational density of states, and shown to be valid beyond the canonical and microcanonical ensembles, in general steady states.

MMP for generalized foliated threefolds of rank one
Mengchu Li
https://arxiv.org/abs/2506.23708 https://arxiv.org/pdf/2506.23708

MMP for generalized foliated threefolds of rank one
We establish the minimal model program (MMP) for generalized foliated threefolds $(X, \mathcal{F}, B, \mathbf{M})$ of rank 1, extending the result of Cascini and Spicer in [CS20]. As an application of the generalized foliated MMP, we prove a base-point-free theorem for foliated triples on threefolds. We also prove the ACC for log canonical thresholds for generalized foliated threefolds of rank 1.

On local rings of finite syzygy representation type
Souvik Dey, Kaito Kimura, Jian Liu, Yuya Otake
https://arxiv.org/abs/2507.17097 https://arxiv.org/pdf/2…

On local rings of finite syzygy representation type
Let R be a commutative Noetherian local ring. We give a characterization of when the completion of R has an isolated singularity. This result simultaneously improves a theorem of Dao and Takahashi and a theorem of Bahlekeh, Hakimian, Salarian, and Takahashi. As an application, we strengthen the Auslander-Huneke-Leuschke-Wiegand theorem in the form refined by Dao and Takahashi. We further investigate the ascent and descent of finite and countable syzygy representation type along the canonical ma…

Liftings of ideals in positive characteristic to those in characteristic zero: Higher dimension
Shihoko Ishii
https://arxiv.org/abs/2506.23533 https://

Liftings of ideals in positive characteristic to those in characteristic zero: Higher dimension
We study a pair consisting of a smooth variety of arbitrary dimension over a field of positive characteristic and a multi-ideal with a real exponent. We prove that the set of log discrepancies for a fixed exponent is discrete. Additionally, we show that the set of log canonical thresholds (lcts) of multi-ideals on a smooth variety in positive characteristic is contained within the set of lcts of multi-ideals on a smooth variety over the complex number field. As a result, we find that the accumu…

A valuative approach to the anticanonical minimal model program
Sung Rak Choi, Sungwook Jang, Donghyeon Kim, Dae-Won Lee
https://arxiv.org/abs/2506.13637 h…

A valuative approach to the anticanonical minimal model program
In this paper, we show that the log canonical threshold of a potentially klt triple can be computed by a quasi-monomial valuation. The notion of potential triples provides a larger and more flexible framework to work with than that of generalized pairs. Our main result can be considered as an extension to the result of Xu on klt pairs. As an application of the main result, we show that we can run the MMP on any potentially klt triples and $-(K_X+Δ)$-MMP on the potentially klt pairs.

Cohen-Macaulay approximations and the $\text{SC}_r$-condition
Richard F. Bartels
https://arxiv.org/abs/2507.14424 https://arxiv.org/p…

Cohen-Macaulay approximations and the $\text{SC}_r$-condition
We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring $R$ satisfies the $\text{SC}_{2}$-condition if and only if $R$ is a UFD. For $r \geq 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the $\text{SC}_{r}$-condition, assuming that its first syzygy $Ω_{R}^{1}(M)$ s…

Ulrich ideals on rational triple points of dimension two
Kyosuke Maeda, Ken-ichi Yoshida
https://arxiv.org/abs/2507.12980 https://arx…

Ulrich ideals on rational triple points of dimension two
In this paper, we prove the canonical trace ideal trace(omega_A) is an Ulrich ideal for any two-dimensional rational triple point A. Using this, we classify all Ulrich ideals on rational triple points. Moreover, we show that if (A, m) is a two-dimensional quotient singularity with the multiplicity e \ge 4 then m is the unique Ulrich ideal of A. As a result, we can classify all Ulrich ideals of A if A is either a rational triple point or a quotient singularity.