arXiv subjects
Publications and source records attributed to Eduardo Dias.
We prove that the moduli space of numerical Godeaux surfaces with torsion group $\mathbb{Z}/2$ is irreducible and unirational of dimension 8. Moreover, we show that the topological fundamental group of these surfaces is also $\mathbb{Z}/2$. Our approach is based on the explicit construction of equations for all universal covers of numerical Godeaux surfaces with torsion group $\mathbb{Z}/2$.
We propose a message passing neural network architecture designed to be equivariant to column and row permutations of a matrix. We illustrate its advantages over traditional architectures like multi-layer perceptrons (MLPs), convolutional neural networks (CNNs) and even Transformers, on the combinatorial optimization task of recovering a set of deleted entries of a Hadamard matrix. We argue that this is a powerful application of the principles of Geometric Deep Learning to fundamental mathematics, and a potential stepping stone toward more insights on the Hadamard conjecture using Machine Learning techniques.
We compute equations for Coughlan's family of Godeaux surfaces with torsion $\mathbb Z/2$, which we call $\mathbb Z/2$-Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify non-rational KSBA degenerations $W$ of $\mathbb Z/2$-Godeaux surfaces with one Wahl singularity, showing that $W$ is birational to particular either Enriques surfaces, or $D_{2,n}$ elliptic surfaces, with $n=3,4$ or $6$. We present examples for all possibilities in the first case, and for $n=3,4$ in the second.
Let $Y$ be an algebraic variety, $\mathcal{F}$ a locally free sheaf of $\mathcal{O}_Y$-modules, and $\mathcal{R}(\mathcal{F})$ the $\mathcal{O}_Y$-algebra $\operatorname{Sym}^\bullet \mathcal{F}$. In this paper we study local properties of sheaves of $\mathcal{O}_{\mathcal{R}(\mathcal{F})}$-ideals $\mathcal{I}$ such that $\mathcal{R}(\mathcal{F}))/\mathcal{I}$ is an algebraic cover of $Y$. Following the work of Miranda for triple covers, for $\mathcal{Q}$ a direct summand of $\mathcal{R}(\mathcal{F})$, we say that a morphism $Φ\colon \mathcal{Q}\rightarrow\mathcal{R}(\mathcal{F})/\langle\mathcal{Q}\rangle$ is a covering homomorphism if it induces such an ideal. As an application we study in detail the case of Gorenstein covering maps of degree $6$ for which the direct image of $φ_*\mathcal{O}_X$ admits an orthogonal decomposition. These are deformation of $S_3$-Galois branch covers.
Let $S$ be an abelian surface over an algebraically closed field $k$ with characteristic different from $2$ and $3$, and $\mathcal{L}$ a symmetric ample line bundle defining a polarisation of type $(1,3)$. Then the linear system $|\mathcal{L}|$ defines a covering map $φ\colon S\rightarrow \mathbb{P}^2$ of degree $6$. Furthermore, if $|\mathcal{L}|$ is base point free, then $φ_*\mathcal{O}_S = \mathcal{O}_{\mathbb{P}^2} \oplus Ω^1_{\mathbb{P}^2} \oplus Ω^1_{\mathbb{P}^2}\oplus\mathcal{O}_{\mathbb{P}^2}(-3)$. Using this decomposition, in this paper we construct the graded coordinate ring of $(S,\mathcal{L},θ)$, where $θ\colon G(\mathcal{L})\xrightarrow{\sim} H(1,3)$ is a level structure of canonical type. As a corollary we prove that the moduli space of such triples is rational.
Knowledge of human presence and interaction in a vehicle is of growing interest to vehicle manufacturers for design and safety purposes. We present a framework to perform the tasks of occupant detection and occupant classification for automatic child locks and airbag suppression. It operates for all passenger seats, using a single overhead camera. A transfer learning technique is introduced to make full use of training data from all seats whilst still maintaining some control over the bias, necessary for a system designed to penalize certain misclassifications more than others. An evaluation is performed on a challenging dataset with both weighted and unweighted classifiers, demonstrating the effectiveness of the transfer process.
We define weighted projective Reed-Muller codes over a subset of weighted projective space over a finite field. We focus on the case when the set X is a projective weighted torus. We show that the vanishing ideal of X is a lattice ideal and relate it with the lattice ideal of a minimal presentation of the semigroup algebra of Q, the numerical semigroup generated by the weights of the projective space. We compute the index of regularity of the vanishing ideal as function of the weights and the Frobenius number of Q. We compute the basic parameters of weighted projective Reed-Muller codes over a 1-dimensional weighted torus and prove they are maximum distance separable codes.