Related papers: A note on Maximum Likelihood Estimation for cubic …

We study global log canonical thresholds of del Pezzo surfaces.

In this article, we obtain an upper bound for the number of integral points on the del Pezzo surfaces of degree two.

We introduce and study the notion of $G$-coregularity of algebraic varieties endowed with an action of a finite group $G$. We compute $G$-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups…

Manin's conjecture is proved for a split del Pezzo surface of degree 5 with a singularity of type A_2.

We present a computational scheme that derives a global polynomial level set parametrisation for smooth closed surfaces from a regular surface-point set and prove its uniqueness. This enables us to approximate a broad class of smooth…

We report on our project to construct non-singular cubic surfaces over $\bbQ$ with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.

We present a new normal form for cubic surfaces that is well suited for p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27 trees in the tropicalization. The new normal form is a polynomial with eight terms,…

We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de…

We prove, via an "arithmetic surjectivity" approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for…

We explain a classical construction of a del Pezzo surface of degree d = 4 or 5 as a smooth order two congruence of lines in 3-space whose focal surface is a quartic surface $X_{20-d}$ with 20-d ordinary double points. We also show that…

We classify del Pezzo surfaces with Picard number is equal to one and with four log terminal singular points.

Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the…

Gaussian mixture models are central to classical statistics, widely used in the information sciences, and have a rich mathematical structure. We examine their maximum likelihood estimates through the lens of algebraic statistics. The MLE is…

We provide an estimate for the number of nontrivial integer points on the Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on the size of fundamental solutions for almost all $d$ in a certain class, based on a…

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We…

Let V be a plane smooth cubic curve over a finitely generated field k. The Mordell-Weil theorem for V states that there is a finite subset P \subset V(k) such that the whole V(k) can be obtained from P by drawing secants and tangents…

Let $S$ be a del Pezzo surface with at worst Du Val singularities of degree $2$ such that $S$ admits an $(-K_S)$-polar cylinder. In this article, we construct an $H$-polar cylinder for any ample $\mathbb{Q}$-divisor $H$ on $S$.

Algebraic statistics is a recently evolving field, where one would treat statistical models as algebraic objects and thereby use tools from computational commutative algebra and algebraic geometry in the analysis and computation of…

Algorithms working with linear algebraic groups often represent them via defining polynomial equations. One can always choose defining equations for an algebraic group to be of the degree at most the degree of the group as an algebraic…

This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the…