相关论文: A Tropical Count of Binodal Cubic Surfaces
Using a quadratic version of the Bott residue theorem, we give a quadratic refinement of the count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.
We study the surface of Gauss double points associated to a very general quartic surface and the natural morphisms associated to it.
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…
We show that the number of non-trivial rational points of height at most $B$, that lie on the cubic surface $x_1x_2x_3=x_4(x_1+x_2+x_3)^2$, has order of magnitude $B(\log B)^6$. This agrees with the Manin conjecture.
We construct three sequences of regular surfaces of general type with unbounded numerical invariants whose canonical map is 2-to-1 onto a canonically embedded surface. Only sporadic examples of surfaces with these properties were previously…
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using…
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…
Let six points $1, ...6$ lie in general position in the real projective plane and consider the pencil of nodal cubics based at these points, with node at one of them, say 1. This pencil has five reducible cubics. We call combinatorial cubic…
We show that normal K3 surfaces with ten cusps exist in and only in characteristic 3. We determine these K3 surfaces according to the degrees of the polarizations. Explicit examples are given.
Which surfaces can be realized with two-dimensional faces of the five-dimensional cube (the penteract)? How can we visualize them? In recent work, Aveni, Govc, and Roldan, show that there exist 2690 connected closed cubical surfaces up to…
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational…
In arXiv:1505.04338(4), G. Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through certain conjugation invariant set of points on the toric boundary of the surface. Such a set consists of real…
In this paper we further develop the theory of geometric tropicalization due to Hacking, Keel and Tevelev and we describe tropical methods for implicitization of surfaces. More precisely, we enrich this theory with a combinatorial formula…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
We study Coble surfaces in characteristic 2, in particular, singularities of their canonical coverings. As an application we classify Coble surfaces with finite automorphism group in characteristic 2. There are exactly 9 types of such…
The aim of this note is to give a formula expressing the trace form associated with the 27 lines of a cubic surface.
This note contains preliminary calculation of topological types or real Enriques surfaces. We realize 59 topological types of real Enriques surfaces (Theorem 6) and show that all other topological types belong to the list of 21 topological…
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as…
We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…