相关论文: The Resolution Property via Azumaya Algebras

Stack-theoretic blow-ups have proven to be efficient in resolving singularities over fields of characteristic zero. In this article, we move forward towards positive characteristic where new challenges arise. In particular, the dimension of…

We investigate two algebra of curves on a topological surface with punctures - the cluster algebra of surfaces defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. By establishing their…

We prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index (and volume if the ambient dimension is greater than three) in a Riemannian manifold…

We build two embedded resolution procedures of a quasi-ordinary singularity of complex analytic hypersurface, by using toric morphisms which depend only on the characteristic monomials associated to a quasi-ordinary projection of the…

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of…

For an Azumaya algebra $A$ which is free over its centre $R$, we prove that the $K$-theory of $A$ is isomorphic to $K$-theory of $R$ up to its rank torsion. We observe that a graded central simple algebra, graded by an abelian group, is a…

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of…

In this paper, we prove a similar result to the fundamental theorem of regular surfaces in classical differential geometry, which extends the classical theorem to the entire class of singular surfaces in Euclidean 3-space known as frontals.…

We prove a formality theorem for algebraic objects internal to smooth complex varieties that are not compact but whose mixed Hodge structure has a certain purity property.

We study the problem of whether a given finite algebra with finitely many basic operations contains a cube term; we give both structural and algorithmic results. We show that if such an algebra has a cube term then it has a cube term of…

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an \'etale topological realization of the stable motivic homotopy theory of smooth schemes over a base…

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum…

We introduce trim resolutions of complex algebraic varieties, a strengthening of the notion of small resolution. We prove that the characteristic cycle of the intersection cohomology sheaf of a variety admitting a trim resolution is…

We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study…

In this note, we prove that given a smooth proper family over a $p$-adic ring of integers, one gets a control of its crystalline torsion in terms of its \'{e}tale torsion, the cohomological degree, and the ramification. Our technical core…

We show, that for a morphism of schemes from X to Y, that is a finite modification in finitely many closed points, a cohomological Brauer class on Y is represented by an Azumaya algebra if its pullback to X is represented by an Azumaya…

We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $\Gamma$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$…

We introduce Liu algebras as Banach algebras which are 'locally affinoid', and define non-Archimedean Stein algebras as suitable inverse limits of these. We show that this gives rise to a complete functorial characterisation of…

We prove a relative GAGA theorem for perfect and pseudo-coherent complexes in non-archimedean analytic geometry, allowing bases given by Fredholm analytic rings, including those associated from affinoid perfectoid spaces. This answers a…

We prove that the number of dissections of a given polygon into triangles with fixed areas of faces is finite and that an equidissection is algebraic as long as the vertices of the original polygon have algebraic coordinates.