Related papers: On large configurations of lines on quartic surfac…
We describe, for various degenerations $S\to \Delta$ of quartic $K3$ surfaces over the complex unit disk (e.g., to the union of four general planes, and to a general Kummer surface), the limits as $t\in \Delta^*$ tends to 0 of the Severi…
We first review some topics in the classical computational geometry of lines, in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of the set of lines in R^3 interacting with $n$ objects of fixed description…
Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p other than 2,3) of degree greater than 84d^2 contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds…
We study the geometry of quintic threefolds $X\subset \mathbb{P}^4$ with only ordinary triple points as singularities. In particular, we show that if a quintic threefold $X$ has a reducible hyperplane section then $X$ has at most $10$…
Let $X$ be a cubic fourfold that has only simple singularities and does not contain a plane. We prove that the Fano variety of lines on $X$ has the same analytic type of singularity as the Hilbert scheme of two points on a surface with only…
An effective divisor D on a smooth (compact complex) surface X is called even, if its class $[D] \in H^2(X,\Z)$ is divisible by 2. D may be assumed reduced w.l.o.g. Then D being even is equivalent to the existence of a double cover $Y \to…
We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…
We give restrictions on the weak combinatorics of line arrangements with singular points of odd multiplicity using topological arguments on locally-flat spheres in 4-manifolds. As a corollary, we show that there is no line arrangement…
We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16-nodal quintic as a certain Fano variety of lines could be viewed as a…
We prove that a nodal quartic threefold $X$ containing no planes is $Q$-factorial provided that it has not more than 12 singular points, with the exception of a quartic with exactly 12 singularities containing a quadric surface. We give…
We establish Manin's conjecture for a quartic del Pezzo surface split over Q and having a singularity of type A_3 and containing exactly four lines. It is the first example of split singular quartic del Pezzo surface whose universal torsor…
Jordan showed that the incidence variety of a smooth cubic surface containing 27 lines has solvable Galois group over the incidence variety of a smooth cubic surface containing 3 skew lines. As noted by Harris, it follows that for any…
Examples of algebraic surfaces of general type with maximal Picard number are not abundant in the literature. Moreover, most known examples either possess low invariants, lie near the Noether line $K^2=2\chi-6$ or are somewhat scattered. A…
The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such…
We find upper bounds, sharp in most cases, on the number of real hyperplane sections of real smooth polarized $K3$-surfaces that split into lines. Most bounds coincide with their complex counterparts.
By a K3-surface with nine cusps I mean a surface with nine isolated double points A_2, but otherwise smooth, such that its minimal desingularisation is a K3-surface. It is shown, that such a surface admits a cyclic triple cover branched…
We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or…
We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length…
We give construction of singular K3 surfaces with discriminant 3 and 4 as double coverings over the projective plane. Focusing on the similarities in their branching loci, we can generalize this construction, and obtain a three dimensional…
A famous configuration of 27 lines on a non-singular cubic surface in $\mathbb P^3$ contains remarkable subconfigurations, and in particular the ones formed by six pairwise disjoint lines. We study such six-line configurations in the case…