Two theorems witnessing the abundance of geometrically trivial strongly minimal autonomous differential equations of arbitrary order are shown. The first one states that a generic algebraic vector field of degree $d\geq 2$ on the affine space of dimension $n \geq 2$ is strongly minimal and geometrically trivial. The second one states that if $X_0$ is the complement of a smooth hyperplane section $H$ of a smooth projective variety $X$ of dimension $n$, then for $d$ large enough, the system of differential equations associated with a generic vector field on $X_0$ with a pole of order at most $d$ along $H$ is strongly minimal and geometrically trivial.
This produces the first examples of meromorphic functions that are new in the sense of Painlevé and satisfy autonomous differential equations of order $n \geq 4$.
Let $X \subset \mathbb{C}^n$ be an algebraic variety, and let $Λ\subset \mathbb{C}^n$ be a discrete subgroup whose real and complex spans agree. We describe the topological closure of the image of $X$ in $\mathbb{C}^n / Λ$, thereby extending a result of Peterzil-Starchenko in the case when $Λ$ is cocompact. We also obtain a similar extension when $X\subset \mathbb{R}^n$ is definable in an o-minimal structure with no restrictions on $Λ$, and as an application prove the following conjecture of Gallinaro: for a closed semi-algebraic $X\subset \mathbb{C}^n$ (such as a complex algebraic variety) an…
We consider the covering map $π:\mathbb{C}^n\to \mathbb{T}$ of a compact complex torus. Given an algebraic variety $X\subseteq \mathbb{C}^n$ we describe the topological closure of $π(X)$ in $\mathbb T$. We obtain a similar description when $\mathbb{T}$ is a real torus and $X\subseteq \mathbb{R}^n$ is a set definable in an o-minimal structure over the reals.
We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.
We provide sharp cylindrical parametrizations of cylindrical cell decompositions by maps with bounded $C^{r}$ norm in the sharply o-minimal setting, thus generalizing and strengthening the Yomdin-Gromov Algebraic Lemma. We introduce forts, geometrical objects encoding the combinatorial structure of cylindrical cell decompositions in o-minimal geometry. Cylindical decompositions, refinements of such decompositions, and cylindrical parametrizations of such decomposition become morphisms in the category of forts. We formulate and prove the above results in the language of forts.
In this manuscript we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of nonminimality of Freitag and Moosa. Our techniques are sufficient to show that generic order $h$ differential equations with nonconstant coefficients are strongly minimal, answering a question of Poizat (1980).
We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period ma…
Let S be an algebraic space, A an S-abelian algebraic space, L an S-fiberwise numerically trivial invertible module on A, and L* the sheaf of regular sections of L considered as a G_m-torsor on A. We classify the S-minimal models of L* into two types.
We define a linear code $C_η(δ_T,δ_X)$ by evaluating polynomials of bidegree $(δ_T,δ_X)$ in the Cox ring on $\mathbb{F}_q$-rational points of the Hirzebruch surface of parameter $η$ on the finite field $\mathbb{F}_q$. We give explicit parameters of the code, notably using Gröbner bases. The minimum distance provides an upper bound of the number of $\mathbb{F}_q$-rational points of a non-filling curve on a Hirzebruch surface.
On the minimal algebraic complexity of the rank-one approximation problem for general inner products
Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学）；Laboratoire J.-A. Dieudonné Université Côte d’Azur（蔚蓝海岸大学J.-A. 迪厄多内实验室）；Universidade Estadual de Campinas (UNICAMP)（坎皮纳斯州立大学）；Instituto de Matemática, Estatística e Computação Científica (IMECC)（数学、统计和科学计算研究所）；Departamento de Matemática Centro de Ciências Exatas e da Natureza Universidade Federal de Pernambuco（伯南布哥联邦大学精确与自然科学学院数学系）；INRIA and CMAP, École Polytechnique, IP Paris, CNRS（法国国家信息与自动化研究所与巴黎理工学院CMAP，CNRS）；Department of Mathematics, KTH Royal Institute of Technology（瑞典皇家理工学院数学系）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学数学研究所）
In this note, we construct two minimal surfaces of general type with geometric genus p_g= 3, irregularity q = 0, self-intersection of the canonical divisor K^22 =20,24 such that their canonical map is of degree 20. In one of these surfaces, the canonical linear system has a non-trivial fixed part. These surfaces, to our knowledge, are the first examples of minimal surfaces of general type with canonical map of degree 20.
We introduce a class of strongly étale difference algebras, whose role in the study of difference equations is analogous to the role of étale algebras in the study of algebraic equations. We deduce an improved version of Babbitt's decomposition theorem and we present applications to difference algebraic groups and the compatibility problem.
A theorem of Elekes and Szabó recognizes algebraic groups among certain complex algebraic varieties with maximal size intersections with finite grids. We establish a generalization to relations of any arity and dimension, definable in: 1) stable structures with distal expansions (includes algebraically and differentially closed fields of characteristic $0$); and 2) $o$-minimal expansions of groups. Our methods provide explicit bounds on the power saving exponent in the non-group case. Ingredients of the proof include: a higher arity generalization of the abelian group configuration theorem in …
In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let $k$ be a field of characteristic zero and let $X$ be a smooth and proper $k((t))$-variety with semi-ample canonical divisor. We prove that the essential skeleton of $X$ coincides with the skeleton of any minimal $dlt$-model and that it is a strong deformation retract of the Berkovich analytification of $X$. As an application, we show that the essential skeleton of a Calabi-Yau variety over $k((t))$ is a pseudo-manifold.
In this paper, we consider the affine variety codes obtained evaluating the polynomials $by=a_kx^k+\dots+a_1x+a_0$, $b,a_i\in\mathbb{F}_{q^r}$, at the affine $\F_{q^r}$-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolutely irreducibility of certain algebraic varieties.
We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.
