Related papers: On two rationality conjectures for cubic fourfolds

Koll\'ar's conjecture states that a complex projective surface $S$ with quotient singularities and with $H^2(S,\bbQ)\cong \bbQ$ should be rational if its smooth part $S^0$ is simply connected. We confirm the conjecture under the additional…

In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of…

The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism type. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. In Voisin's original proof of the global…

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.

F-theory/heterotic duality is formulated in the stable degeneration limit of a K3 fibration on the F-theory side. In this note, we analyze the structure of the stable degeneration limit. We discuss whether stable degeneration exists for…

We prove a formula expressing the motivic integral (\cite{ls}) of a K3 surface over $\bC((t))$ with semi-stable reduction in terms of the associated limit Hodge structure. Secondly, for every smooth variety over a non-archimedean field we…

Let X be a K3 surface with a polarization H with H^2=2rs. Assume that H.N(X)=Z for the Picard lattice N(X). The moduli space Y of sheaves over X with the Mukai vector (r,H,s) is again a K3 surface. We prove that Y\cong X, if there exists…

In this paper, we deal with the linear Weingarten factorable surfaces in the isotropic 3-space I^{3} satisfying the relation aK+bH=c, where K is the relative curvature and H the isotropic mean curvature, a,b,cR. We obtain a complete…

We show that the K-moduli spaces of log Fano pairs $(\mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$…

Self-rational maps of generic algebraic K3 surfaces are conjectured to be trivial. We relate this conjecture to a conjecture concerning the irreducibility of the universal Severi varieties parametrizing nodal curves of given genus and…

We construct geometric compactifications of the moduli space $F_{2d}$ of polarized K3 surfaces, in any degree $2d$. Our construction is via KSBA theory, by considering canonical choices of divisor $R\in |nL|$ on each polarized K3 surface…

Building on work of Segre and Koll'ar on cubic hypersurfaces, we construct over imperfect fields of characteristic p\geq 3 particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose…

We study links between algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH_0(X) into…

We give a canonical birational map between the moduli space of pfaffian vector bundles on a cubic surface and the space of complete pentahedra inscribed in the cubic surface. The universal situation is also considered, and we obtain a…

In this paper we show that the moduli space of nodal cubic surfaces is isomorphic to a quotient of a 4-dimensional complex ball by an arithmetic subgroup of the unitary group. This complex ball uniformization uses the periods of certain K3…

We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the root lattice A_3. Our proof is based on…

For a general cubic fourfold $X \subset \mathbb{P}^5$, we compute the Hodge numbers of the locus $S \subset F$ of lines of second type. We also give an upper bound of 6 for the degree of irrationality of the Fano scheme of lines of any…

Using a construction of Hassett--V\'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a…

We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of…

Given X a K3 surface, a mirror dual to X can be identified with a component of the moduli space of semistable sheaves on X. We consider fibrations by K3 surfaces over a one dimensional base that are Calabi-Yau and we charaterize the dual…