相关论文: A Septic with 99 real Nodes

In this work we give a method for constructing a one-parameter family of complete CMC-1 (i.e. constant mean curvature 1) surfaces in hyperbolic 3-space that correspond to a given complete minimal surface with finite total curvature in…

The parametric degree of a rational surface is the degree of the polynomials in the smallest possible proper parametrization. An example shows that the parametric degree is not a geometric but an arithmetic concept, in the sense that it…

We examine the maximum dimension of a linear system of plane cubic curves whose $\mathbb{F}_q$-members are all geometrically irreducible. Computational evidence suggests that such a system has a maximum (projective) dimension of $3$. As a…

We introduce an asymptotic notion of positivity in algebraic geometry that turns out to be related to some high-dimensional convex sets. The dimension of the convex sets grows with the number of birational operations. In the case of complex…

A general strategy is given for the classification of graphs of rational surface singularities. For each maximal rational double point configuration we investigate the possible multiplicities in the fundamental cycle. We classify completely…

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound…

It is well-known that 1-planar graphs have minimum degree at most 7, and not hard to see that some 1-planar graphs have minimum degree exactly 7. In this note we show that any such 1-planar graph has at least 24 vertices, and this is tight.

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we prove that…

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…

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…

The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of…

The projective linear group \(\pgl(\comp,4)\) acts on cubic surfaces, considered as points of $\mathbb{P}_{\mathbb{C}}^{19}$. We compute the degree of the $15$-dimensional projective variety given by the Zariski closure of the orbit of a…

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors…

Let X be a non singular projective surface. Given a semistable non isotrivial fibration f over a smooth rational curve with general fiber non hyperelliptic of genus g bigger than 3, we show that if the number s of singular fibers is 5, then…

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable…

For every p >= 5, we determine all Z_p-invariant nonsingular quartic surfaces in the three dimensional projective space over an algebraically closed field of characteristic zero. In some cases, we also determine their full projective…

The Severi variety V_{n,d} of a smooth projective surface S is defined as the subvariety of the linear system |O_S(n)|, which parametrizes curves with d nodes. We show that, for a general surface S of degree k in P^3 and for all n>k-1,…

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a…

It follows from classical restrictions on the topology of real algebraic varieties that the first Betti number of the real part of a real nonsingular sextic in $\mathbb{CP}^3$ can not exceed $94$. We construct a real nonsingular sextic $X$…

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…