Related papers: Rationality of singular cubic threefolds over $\ma…

We study unirationality and rationality of Fano threefolds of degree 18 over nonclosed fields.

Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by…

We investigate the rationality problem for $\mathbf{Q}$-Fano threefolds of Fano index $\ge 2$.

We describe those unipotent representations of a finite group of Lie type which are defined over the rational numbers.

We provide a rough classification of threefold exceptionally non-canonical cDV quotient singularities by studying their combinatorial behavior.

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type…

The point is to compare the mathematical meaning of the ``number of rational curves on a Calabi-Yau threefold'' to the meaning ascribed to the same notion by string theorists.

F. Campana had asked whether a certain threefold is rational. In arXiv:1310.3569v1 [mathAG], this variety was shown to be birational to a specific conic bundle and then to be unirational. We prove that this conic bundle is rational.

We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.

We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field…

We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…

This paper is a sequel to [arXiv:2403.18389]. We investigate the rationality problem for $\mathbf{Q}$-Fano threefolds of Fano index $\ge 3$.

Motivated by the question of rationality of cubic fourfolds, we show that a cubic X has an associated K3 surface in the sense of Hassett if and only if the variety F of lines on X is birational to a moduli space of sheaves on a K3 surface,…

Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as…

Two cubic equations and three auxiliary equations for edges and face diagonals of a rational perfect cuboid have been recently derived. They constitute a background for two inverse problems. The coefficients of cubic equations and the right…

We complete the study of rationality problem for hypersurfaces $X_t\subset \mathbb{P}^4$ of degree $4$ invariant under the action of the symmetric group $S_6$.

We review the state of the art of the classification of real uniruled threefolds

We prove that good quotients of algebraic varieties with 1-rational singularities also have 1-rational singularities. This refines a result of Boutot on rational singularities of good quotients.

We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian…