Related papers: Global generation of test ideals in mixed characte…

The generalized test ideals introduced in [HY] are related to multiplier ideals via reduction to characteristic p. In addition, they satisfy many of the subtle properties of the multiplier ideals, which in characteristic zero follow via…

Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $|K_X + L|$.…

Let $X$ be an integral scheme of finite type over a complete DVR of mixed characteristic. We provide a definition of a test ideal which agrees with the multiplier ideal after inverting $p$, is computed from a sufficiently large alteration,…

This paper presents a novel approach to constructing finite generating sets for infinitely generated ideals. By integrating algebraic and computational techniques, we provide a method to identify finite generators, demonstrated through…

We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global $F$-regularity to mixed characteristic and identify certain stable…

Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer…

Gradient-based methods are widely used to solve various optimization problems, however, they are either constrained by local optima dilemmas, simple convex constraints, and continuous differentiability requirements, or limited to…

We train deep generative models on datasets of reflexive polytopes. This enables us to compare how well the models have picked up on various global properties of generated samples. Our datasets are complete in the sense that every single…

Let $X$ be a smooth projective variety over an algebraically closed field $\mathbb{K}$ with arbitrary characteristic. Suppose $L$ is an ample and globally generated line bundle. By Castelnuovo--Mumford regularity, we show that $K_X \otimes…

The diffusion probabilistic generative models are widely used to generate high-quality data. Though they can synthetic data that does not exist in the training set, the rationale behind such generalization is still unexplored. In this…

In this work, we propose a global model selection criterion to estimate the graph of conditional dependencies of a random vector based on a finite sample. By global criterion, we mean optimizing a function over the entire set of possible…

Statistical evaluation aims to estimate the generalization performance of a model using held-out i.i.d.\ test data sampled from the ground-truth distribution. In supervised learning settings such as classification, performance metrics such…

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about…

We study higher jumping numbers and generalized test ideals associated to determinantal ideals over a field of positive characteristic. We work in positive characteristic and give a complete characterization of both families for ideals…

For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$.…

We consider ideals in a polynomial ring generated by collections of power sum polynomials, and obtain conditions under which these define complete intersection rings, normal domains, and unique factorization domains. We also settle a key…

We provide a simulation algorithm to simulate from a (multivariate) characteristic function, which is only accessible in a black-box format. The method is based on a generative neural network, whose loss function exploits a specific…

We consider several notions of genericity appearing in algebraic geometry and commutative algebra. Special emphasis is put on various stability notions which are defined in a combinatorial manner and for which a number of equivalent…

This paper studies the multiplicative ideal structure of commutative rings in which every finitely generated ideal is quasi-projective. Section 2 provides some preliminaries on quasi-projective modules over commutative rings. Section 3…

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We…