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

We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to…

We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients to the monomial parameterization of the toric variety, one can change the ML…

In this paper we study the problem of existence of orbifold Kaehler-Einstein metrics on del Pezzo surfaces of degree 1 with Du Val singular points. Moreover we compute global log canonical thresholds of del Pezzo surfaces of degree 1 with…

We settle a conjecture by Coons and Sullivant stating that the maximum likelihood (ML) degree of a facial submodel of a toric model is at most the ML degree of the model itself. We discuss the impact on the ML degree from observing zeros in…

We give a classification of toric log del Pezzo surfaces with two or three singular points.

We give upper bounds for the number of rational points of bounded anti-canonical height on del Pezzo surfaces of degree at most five over any global field whose characteristic is not equal to two or three. For number fields these results…

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index $\ell$. This upper bound turns out to be a quadratic polynomial in…

We construct first examples of singular del Pezzo surfaces with Zariski dense exceptional sets in Manin's conjecture, varying in degrees $1, 2$ and $3$. The obstructions arise from accumulating quasi-\'etale covers. We classify all…

We give a complete classification of del Pezzo surfaces with quotient singularities and Picard rank 1 which admit a Q-Gorenstein smoothing. There are 14 infinite families of toric examples. The surfaces in each family correspond to…

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree $1$ and elliptic fibrations. It turns out that del Pezzo surfaces with positive…

The Manin-Peyre conjecture is established for a split singular quintic del Pezzo surface with singularity type $\mathbf{A}_2$ and two split singular quartic del Pezzo surfaces with singularity types $\mathbf{A}_3+\mathbf{A}_1$ and…

In algebraic statistics, the maximum likelihood degree of a statistical model refers to the number of solutions (counted with multiplicity) of the score equations over the complex field. In this paper, the maximum likelihood degree of the…

This paper focuses on the classification of all toric log Del Pezzo surfaces with exactly one singularity up to isomorphism, and on the description of how they are embedded as intersections of finitely many quadrics into suitable projective…

We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and…

Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is…

In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type…

We study the problem of maximum likelihood estimation for $3$-dimensional linear spaces of $3\times 3$ symmetric matrices from the point of view of algebraic statistics where we view these nets of conics as linear concentration or linear…

Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.

We characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type $\mathbf{A}_3$ over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these…

Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for discrete algebraic statistical models are reflected in the geometry of the $\textit{likelihood correspondence}$, a variety…