Related papers: Picard numbers of quintic surfaces
We study the groups of automorphisms of rational algebraic surfaces that admit a relatively minimal pencil of curves of arithmetic genus one over an algebraically closed field of arbitrary characteristic. In particular, we classify such…
Using results of our papers [19], [20] and [21] about classification of degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups, we classify Picard lattices of Kahlerian K3 surfaces. By classification we understand…
Motivated by a problem originating in string theory, we study elliptic fibrations on K3 surfaces with large Picard number modulo isomorphism. We give methods to determine upper bounds for the number of inequivalent K3 surfaces sharing the…
We study lines on smooth cubic surfaces over the field of $p$-adic numbers, from a theoretical and computational point of view. Segre showed that the possible counts of such lines are $0,1,2,3,5,7,9,15$ or $27$. We show that each of these…
In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to…
Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the…
Toric log Del Pezzo surfaces with Picard number 1 have been completely classified whenever their index is $\le 2$: In this paper we extend the classification for those having index 3: We prove that, up to isomorphism, there are exactly 18…
Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has…
Upper and lower bounds, of the expected order of magnitude, are obtained for the number of rational points of bounded height on any quartic del Pezzo surface over $\mathbb{Q}$ that contains a conic defined over $\mathbb{Q}$.
We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This…
We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo…
This article is a part of a series aimed at classifying normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic, up to an isomorphism. The key invariant guiding our classification is the…
A new class of examples of surfaces with maximal Picard number is constructed. These carry pencils of genus two or three curves such their Jacobian fibrations are isogenous to fibre products of elliptic modular surfaces.
P. Stiller computed the Picard numbers of several families of elliptic surfaces, the rank of the N\'eron-Severi groups of these surfaces. However he did not give the generators of these groups. In this paper we give $\mathbb Q$-bases of…
We show 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 $20$, and that if equality is attained, then the…
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm…
In this paper we discuss the number of Enriques quotients of a fixed K3 surface. We prove the finiteness and unboundedness of the number. We also show an example of Kummer surface of product type where we can successfully classify all the…
The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the quintic, is that of the symmetric group S_5. Induced by its five-dimensional linear permutation representation is a three-dimensional…
We give a counting formula for the twisted Fourier-Mukai partners of a projective K3 surface. As an application, we describe all twisted Fourier-Mukai partners of a projective K3 surface of Picard number 1.
We study the geometry of the K3 surfaces $X$ with a finite number automorphisms and Picard number $\geq 3$. We describe these surfaces classified by Nikulin and Vinberg as double covers of simpler surfaces or embedded in a projective space.…