相关论文: Enumeration of singular algebraic curves
We study the intersection numbers defined on twisted homology or cohomology groups that are associated with hypergeometric integrals corresponding to degenerate hyperplane arrangements in the projective $k$-space. We present formulas to…
In this note we compute some enumerative invariants of real and complex projective spaces by means of some enriched graphs called floor diagrams.
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $\delta$ nodes and…
We give a complete classification of complex Q-homology projective planes with isolated rational double point singularities and numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all…
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all…
A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a…
In this paper we obtain an explicit formula for the number of curves in two dimensional complex projective space, of degree d, passing through d(d+3)/2-(k+1) generic points and having one node and one codimension k singularity, where k is…
The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…
The embeddings of complex plane projective curves in the plane are a cornerstone of the topological study of algebraic varieties. In this work, we deal with the local and global aspects of these embeddings, with a special attention to its…
In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…
It is often the case that a Selmer group of an abelian variety and a group related to an ideal class group can both be naturally embedded into the same cohomology group. One hopes to compute one from the other by finding how close each is…
A spectral sequence is defined which converges to the \v{C}ech cohomology of the Euclidean hull of a tiling of the plane with Euclidean finite local complexity. The terms of the second page are determined by the so-called ePE homology and…
Properties of the recently reported homogeneous Hilbert curves are deduced and reported. The nature of the affine transformations involved in the construction of the Hilbert curves is explored. The analytical representation of proper and…
A $\textit{polygonal curve}$ is a collection of $m$ connected line segments specified as the linear interpolation of a list of points $\{p_0, p_1, \ldots, p_m\}$. These curves may be obtained by sampling points from an oriented curve in…
We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main…
We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main…
Given a diagram for a trisection of a 4-manifold $X$, we describe the homology and the intersection form of $X$ in terms of the three subgroups of $H_1(\Sigma;\mathbb{Z})$ generated by the three sets of curves and the intersection pairing…
We give explicit formulas for the number of distinct elliptic curves over a finite field, up to isomorphism, in the families of Legendre, Jacobi, Hessian and generalized Hessian curves.