Related papers: Smooth models of singular $K3$-surfaces

We construct explicit examples of $K3$ surfaces over ${\mathbb Q}$ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces…

We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds…

We compactify the pure spinor formalism on a K3 surface. The pure spinor splits into a six-dimensional pure spinor, a projective superspace harmonic, and 6 non-covariant variables. A homological algebra argument reduces the calculation of…

We consider singular Q-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is C^1- or C*-ruled or the surface is up to isomorphism one of two…

Let $X\subset \P^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different…

We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…

A recent result shows that a general smooth plane quartic can be recovered from its 24 inflection lines and a single inflection point. Nevertheless, the question whether or not a smooth plane curve of degree at least 4 is determined by its…

This paper proposes a new notion of smoothness of algebras, termed differential smoothness, that combines the existence of a top form in a differential calculus over an algebra together with a strong version of the Poincar\'e duality…

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore…

The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic…

There are $75$ moduli spaces $F_S$ of K3 surfaces with a nonsymplectic involution. We give detailed descriptions of Kulikov models for one-parameter degenerations in $F_S$. In the $50$ cases where the fixed locus of the involution has a…

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this…

We develop an algorithm detecting real representatives in equisingular strata of projective models of $K3$-surfaces. We apply this algorithm to spatial quartics and find two new examples of real strata without real representatives. As a…

We classify K3 surfaces with a non-symplectic finite automorphism of high order. It is shown that such an automorphism cannot be of order 60, and for each of the orders 38, 44, 48, 50, 54 and 66, there exists a unique K3 surface with such…

We study singularities and geometric properties of surfaces given by the singular loci of normal congruence of frontals with pure-frontal singular points. These surfaces consist of the normal ruled surface and focal surfaces of the initial…

Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces…

We study curve singularities in a smooth surface relative to a smooth boundary curve. We consider the semiuniversal deformations and equisingular deformations of curves with a fixed local intersection number $w$ with the boundary, and prove…

For K\"ahler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal…

This paper further studies the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular…

For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we…