Related papers: Rationality of singular cubic threefolds over $\ma…
It is unknown whether smooth cubic threefolds have an (integral Chow-theoretic) decomposition of the diagonal, or whether they are stably rational or not in general. As a first step towards making progress on these questions, we compute the…
We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted…
The present paper deals with lines contained in a smooth complex cubic threefold. It is well-known that the set of lines of the second type on a cubic threefold is a curve on its Fano surface. Here we give a description of the singularities…
We provide explicit equations and parameterizations of some new rational Gushel-Mukai fourfolds of special type.
A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely…
We study rationality properties of smooth complete intersections of three quadrics in $\mathbb{P}^7$. We exhibit a smooth family of such intersections with both rational and non-rational fibers.
A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area.…
We prove modularity for a huge class of rigid Calabi-Yau threefolds over $\Q$. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular.
We study 3-manifolds in $\mathbb{R}^5$ with corank $1$ singularities. At the singular point we define the curvature locus using the first and second fundamental forms, which contains all the local second order geometrical information about…
We show a special case of Steenbrink vanishing for rational singularities in positive characteristic. As a consequence, we prove that strongly $F$-regular threefolds and klt threefolds in characteristic $p>41$ satisfy Steenbrink vanishing.
The works of Hassett and Kuznetsov identify countably many divisors $C_d$ in the open subset of $\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizing all cubic 4-folds and conjecture that the cubics corresponding to…
We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of…
We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
Parametric Cartan theory of exterior differential systems, and explicit cohomology of projective manifolds reveal united rationality features of differential algebraic geometry.
We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…
We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.
We study degree of irrationality of quasismooth anticanonically embedded weighted Fano 3-fold hypersurfaces that have terminal singularities.
The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a Q-factorial…
We explore some qualitative dynamics in the neighborhood of the $3-dimensional$ two-fold symmetric singularity. We study the existence of an one-parameter family of regular (pseudo) periodic orbits of such systems near a reversible two-fold…
We construct singular quartic double fivefolds whose Kuznetsov component admits a crepant categorical resolution of singularities by a twisted Calabi--Yau threefold. We also construct rational specializations of these fivefolds where such a…