Related papers: Singular cubic fourfolds containing a plane
There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…
It is the purpose of this paper to construct families of examples of nonsymplectic 4-manifolds which (up to sign) have just one Seiberg-Witten basic class.
We introduce the notion of a simultaneous categorical resolution of singularities, a categorical version of simultaneous resolutions of rational double points of surface degenerations. Furthermore, we suggest a construction of simultaneous…
We construct a family of plane curves as pull-backs of a conic for abelian coverings of P^2. If the conic is tangent to the ramification lines one obtains a family of curves of degree 2n with 3n singularities of type A_{n-1}. We calculate…
We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will…
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…
The notion of a quaternionic gerbe is presented as a new way of bundling algebraic structures over a four manifold. The structure groupoid of this fibration is described in some detail. The Euclidean conformal group R*SO(4) appears…
For a general cubic fourfold $X\subset\mathbb{P}^5$ with Fano scheme of lines $F$, we prove a number of properties of the universal family of lines $I\to F$ and various subloci. We first describe the moduli and ramification theory of the…
Recently, W. Barth and S. Rams discussed sextics with up to 30 $A_2$-singularities (also called cusps) and their connection to coding theory [math.AG/0403018]. In the present paper, we find a sextic with 35 cusps within a four-parameter…
Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six…
We construct explicit examples of cubic surfaces over $\bbQ$ such that the 27 lines are acted upon by the index two subgroup of the maximal possible Galois group. This is the simple group of order $25 920$. Our examples are given in…
For a smooth surface in $\mathbb{R}^3$ this article contains local study of certain affine equidistants, that is loci of points at a fixed ratio between points of contact of parallel tangent planes (but excluding ratios 0 and 1 where the…
We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface $S$, defined over a number field K such that S^[2](K) is not empty, then X has a model over K such…
We show that a complex planar curve homeomorphic to the projective line has at most four singular points. If it has exactly four then it has degree five and is unique up to a projective equivalence.
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either…
We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given…
In this paper we explore the intersection of the Hassett divisor $\mathcal C_8$, parametrizing smooth cubic fourfolds $X$ containing a plane $P$ with other divisors $\mathcal C_i$. Notably we study the irreducible components of the…
Under several geometric conditions imposed below, the existence of the discrete spectrum below the essential spectrum is shown for the Dirichlet Laplacian on the quantum layer built over a spherically symmetric hypersurface with a pole…
In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to…
We relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.