Related papers: Even Sets of Lines on Quartic Surfaces

Let $S$ and $T$ be reduced divisors on $\mathbb{P}^2$ which have no common components, and $\Delta=S+2\,T.$ We assume $\deg\Delta=6.$ Let $\pi:X\to\mathbb{P}^2$ be a normal triple cover with branch divisor $\Delta,$ i.e. $\pi$ is ramified…

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalization map of a…

Let S be a surface in complex projective 3-space, having only nodes as singularities. Suppose that S has degree 6. We show that the maximum number of nodes which S can have is 65. An abbreviated history of this is as follows. Basset showed…

Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under…

For a double solid $V\to P_3(C)$ branched over a surface $B\subset P_3(C)$ with only ordinary nodes as singularities, we give a set of generators of the divisor class group $Pic(\tilde{V}})$ in terms of contact surfaces of $B$ with only…

Let $X\subset\mathbb P^{n+1}$ be a smooth complex projective hypersurface. In this paper we show that, if the degree of $X$ is large enough, then there exist global sections of the bundle of invariant jet differentials of order $n$ on $X$,…

We give an explicit formula for the $27$ lines of a smooth cubic surface near the Fermat surface. Our formula involves convergent power series with coefficients in the extension of rational numbers with the sixth root of unity. Our main…

For a (positive definite and integral) quadratic form $f$, a quadratic form is said to be {\it an isolation of $f$ from its proper subforms} if it represents all proper subforms of $f$, but not $f$ itself. It was proved that the minimal…

Following an approach of Dolgachev, Pinkham and Demazure, we classified in math.AG/0210153 normal affine surfaces with hyperbolic $\C^{*}$-actions in terms of pairs of $\Q$-divisors $(D_+,D_-)$ on a smooth affine curve. In the present paper…

We show, in this first part, that the maximal number of singular points of a normal quartic surface $X \subset \mathbb{P}^3_K$ defined over an algebraically closed field $K$ of characteristic $2$ is at most $16$. We produce examples with…

We prove a bound on the number of lines on a smooth degree-d surface in three-dimensional projective space for $d \geq 3$. This bound improves a bound due to Segre and renders some of his arguments rigorous. It is the best known bound for…

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the…

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $\varphi$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in…

We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor,…

Let $G$ be a plane graph and let $C$ be a cycle in $G$. For each finite face of $G$, count the number of edges of $C$ the face contains. We call this the Slitherlink signature of $C$. The symmetric difference $A$ of two cycles with the same…

The objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds. It is shown that a hypersurface has normal crossings if and only if it is a free divisor, has a radical…

Since the works of Krasnov and Scheiderer, there has been an interest in studying effective totally real divisors on a curve X defined over a real closed field, i.e., effective divisors supported on the real locus. Scheiderer proved that,…

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…

The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in…

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…