CyberSec Research

In this paper, we establish a structure theorem for a compact K\"{a}hler manifold $X$ of rational dimension $\mathrm{rd}(X)\leq n-k$ under the mixed partially semi-positive curvature condition $\mathcal{S}_{a,b,k} \geq 0$, which is introduced as a unified framework for addressing two partially semi-positive curvature conditions -- namely, $k$-semi-positive Ricci curvature and semi-positive $k$-scalar curvature. As a main corollary, we show that a compact K\"{a}hler manifold $(X,g)$ with $k$-semi-positive Ricci curvature and $\mathrm{rd}(X)\leq n-k$ actually has semi-positive Ricci curvature and $\mathrm{rd}(X)\geq \nu(-K_X)$. Of independent interest, we also confirm the rational connectedness of compact K\"{a}hler manifolds with positive orthogonal Ricci curvature, among other results.

Edoukou, Ling and Xing in 2010, conjectured that in \mathbb{P}^n(\mathbb{F}_{q^2}), n \geq 3, the maximum number of common points of a non-degenerate Hermitian variety \mathcal{U}_n and a hypersurface of degree d is achieved only when the hypersurface is a union of d distinct hyperplanes meeting in a common linear space \Pi_{n-2} of codimension 2 such that \Pi_{n-2} \cap \mathcal{U}_n is a non-degenerate Hermitian variety. Furthermore, these d hyperplanes are tangent to \mathcal{U}_n if n is odd and non-tangent if n is even. In this paper, we show that the conjecture is true for d = 3 and q \geq 7.

Let $k$ be a $d$-local field such that the corresponding $1$-local field $k^{(d-1)}$ is a $p$-adic field and $C$ a curve over $k$. Let $K$ be the function field of $C$. We prove that for each $n,m \in \mathbf{N}$, and hypersurface $Z$ of $\mathbf{P}^n_K$ with degree $m$ such that $m^{d+1} \leq n$, the $(d+1)$-th Milnor $\mathrm{K}$-theory group is generated by the images norms of finite extension $L$ of $K$ such that $Z$ admits an $L$-point. Let $j \in \{1,\cdots , d\}$. When $C$ admits a point in an extension $l/k$ that is not $i$-ramified for every $i \in \{1, \cdots, d-j\}$ we generalise this result to hypersurfaces $Z$ of $\mathbf{P}_K^n$ with degree $m$ such that $m^{j+1} \leq n$. \par In order to prove these results we give a description of the Tate-Shafarevich group $\Sha^{d+2}(K,\mathbf{Q}/\mathbf{Z}(d+1))$ in terms of the combinatorics of the special fibre of certain models of the curve $C$.

This work explores the space of foliations on projective spaces over algebraically closed fields of positive characteristic, with a particular focus on the codimension one case. It describes how the irreducible components of these spaces varies with the characteristic of the base field in very low degrees and establishes an arbitrary characteristic version of Calvo-Andrade's stability of generic logarithmic $1$-forms under deformation.

We compute the complexity of del Pezzo surfaces with du Val singularities.

This paper has three main goals : (1) To give an axiomatic formulation of the construction of "reduced \v{C}ech complexes", complexes using fewer than the usual number of intersections but still computing cohomology of sheaves; (2) To give a construction of such a reduced \v{C}ech complex for every semi-proper toric variety $X$, such that every open used in the complex is torus stable, and such that the cell complex governing the reduced \v{C}ech complex has dimension the cohomological dimension of $X$; and (3) to give an algorithm to compute the higher direct images of line bundles relative to a toric fibration between smooth proper toric varieties.

In this note we show that the nonnegative part of a proper complex toric variety has the homeomorphism type of a sphere, and consequently that the nonnegative part has a natural structure of a cell complex. This extends previous results of Ehlers and Jurkiewicz. The proof also provides a simplicial decomposition of the nonnegative part, and a parameterization of each maximal simplex.

We prove a conjecture of Gorsky, Hogancamp, Mellit, and Nakagane in the Weyl group case. Namely, we show that the left and right adjoints of the parabolic induction functor between the associated Hecke categories of Soergel bimodules differ by the relative full twist.

In this paper we construct complex tori, denoted by $S_{\mathbb{B}_{1,p,q}}$, as quotients of tensor products of Cayley--Dickson algebras, denoted $\mathbb{B}_{1,p,q}=\mathbb{C}\otimes \mathbb{H}^{\otimes p}\otimes \mathbb{O}^{\otimes q}$, with their integral subrings. We then show that these complex tori have endomorphism rings of full rank and are isogenous to the direct sum of $2^{2p+3q}$ copies of an elliptic curve $E$ of $j$-invariant $1728$.

For a positive integer $N$, let $J_0(N)$ be the Jacobian of the modular curve $X_0(N)$. In this paper we completely determine the structure of the rational cuspidal subgroup of $J_0(N)$ when the largest perfect square dividing $N$ is either an odd prime power or a product of two odd prime powers. Indeed, we prove that the rational cuspidal divisor class group of $X_0(N)$ is the whole rational cuspidal subgroup of $J_0(N)$ for such an $N$, and the structure of the former group is already determined by the first author in [14].

We prove that the cohomology of the integral structure sheaf of a normal affinoid adic space over a non-archimedean field of characteristic zero is uniformly torsion. This result originated from a remark of Bartenwerfer around the 1980s and it partially answers a recent question of Hansen and Kedlaya (see also Problems 27 and 39 in the Non-Archimedean Scottish Book).

We prove that every topological/smooth $\T=(\C^{*})^{n}$-equivariant vector bundle over a topological toric manifold of dimension $2n$ is a topological/smooth Klyachko vector bundle in the sense of arXiv:2504.02205.

The intersection of deep learning and symbolic mathematics has seen rapid progress in recent years, exemplified by the work of Lample and Charton. They demonstrated that effective training of machine learning models for solving mathematical problems critically depends on high-quality, domain-specific datasets. In this paper, we address the computation of Gr\"obner basis using Transformers. While a dataset generation method tailored to Transformer-based Gr\"obner basis computation has previously been proposed, it lacked theoretical guarantees regarding the generality or quality of the generated datasets. In this work, we prove that datasets generated by the previously proposed algorithm are sufficiently general, enabling one to ensure that Transformers can learn a sufficiently diverse range of Gr\"obner bases. Moreover, we propose an extended and generalized algorithm to systematically construct datasets of ideal generators, further enhancing the training effectiveness of Transformer. Our results provide a rigorous geometric foundation for Transformers to address a mathematical problem, which is an answer to Lample and Charton's idea of training on diverse or representative inputs.

In this work, we investigate generalized Weierstrass semigroups in arbitrary Kummer extensions of function field $\mathbb{F}_q(x)$. We analyze their structure and properties, with a particular emphasis on their maximal elements. Explicit descriptions of the sets of absolute and relative maximal elements within these semigroups are provided. Additionally, we apply our results to function fields of the maximal curves $\mathcal{X}_{a,b,n,s}$ and $\mathcal{Y}_{n,s}$, which cannot be covered by the Hermitian curve, and the Beelen-Montanucci curve. Our results generalize and unify several earlier contributions in the theory of Weierstrass semigroups, providing new perspectives on the relationship between these semigroups and function fields.

We apply some recent progress on higher Du Bois singularities to study the $\mathbb{A}^1$-invariance of algebraic $K$-groups.

We prove that a stability condition on a K3 surface is determined by the masses of spherical objects up to a natural $\mathbb{C}$-action. This is motivated by the result of Huybrechts and the recent proposal of Bapat-Deopurkar-Licata on the construction of a compactification of a stability manifold. We also construct lax stability conditions in the sense of Broomhead-Pauksztello-Ploog-Woolf associated to spherical bundles.

Let $R:= \Bbbk[x_1,\ldots,x_{n}]$ be a polynomial ring over a field $\Bbbk$, $I \subset R$ be a homogeneous ideal with respect to a weight vector $\omega = (\omega_1,\ldots,\omega_n) \in (\mathbb{Z}^+)^n$, and denote by $d$ the Krull dimension of $R/I$. In this paper we study graded free resolutions of $R/I$ as $A$-module whenever $A :=\Bbbk[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. We exhibit a Schreyer-like method to compute a (non-necessarily minimal) graded free resolution of $R/I$ as $A$-module. When $R/I$ is a $3$-dimensional simplicial toric ring, we describe how to prune the previous resolution to obtain a minimal one. We finally provide an example of a $6$-dimensional simplicial toric ring whose Betti numbers, both as $R$-module and as $A$-module, depend on the characteristic of $\Bbbk$.

In this work, we provide a way of constructing new semiorthogonal decompositions using metric techniques (\`a la Neeman). Given a semiorthogonal decomposition on a category with a special kind of metric, which we call a compressible metric, we can construct new semiorthogonal decomposition on a category constructed from the given one using the aforementioned metric. In the algebro-geometric setting, this gives us a way of producing new semiorthogonal decompositions on various small triangulated categories associated to a scheme, if we are given one. In the general setting, the work is related to that of Sun-Zhang, while its applications to algebraic geometry are related to the work of Bondarko and Kuznetsov-Shinder.