Related papers: The irregularity of cyclic multiple planes after Z…
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…
Given a family of complex affine planes, we show that it is trivial over a Zariski open subset of the base. The proof relies upon a relative version of the contraction theorem.
We obtain some recombination formulae for the spectra of (complex, reduced) plane curve singularities. As an application we prove: a generalization of Durfee's bound; a generalization of Givental's bound; the multiplicity of the curve…
The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which…
A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in…
We consider ``hyperideal'' circle patterns, i.e. patterns of disks appearing in the definition of the Delaunay decomposition associated to a set of disjoint disks, possibly with cone singularities at the center of those disks. Hyperideal…
G. Pareschi and M. Popa give criterions for global generations and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we generalize some of their…
Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus.…
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of…
In this paper we study the singularity formation for the geometric flow of complex curves $$z_t = -z_{xxx} + \frac{3}{2}\o z_{x} z_{xx}^2,$$ that was derived [R. E. Goldstein and D. M. Petrich, {\em Phys. Rev. Lett.}, 69 (1992), pp.…
The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane…
In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension…
In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and…
In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been…
We define a topological invariant of complex projective plane curves. As an application, we present new examples of arithmetic Zariski pairs.
We prove that a submaximal plane curve (i.e., an irreducible counterexample to Nagata's conjecture) with r singular points has sequence of multiplicities (m, n, ..., n) with m<sn for every integer with ((s-1)(s+2))^2 > 6.76(r-1).
Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum\limits_{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\Delta$, and $d=\left\lfloor \frac{\Delta m}{\Delta n-m}\right\rfloor$, we…
In this note, we introduce the notion of an unramified strongly cyclic covering for a cyclic curve, a class that has similar properties to, and contains, unramified double covers of hyperelliptic curves. We determine several of their basic…
We give a geometric approach to the relation between the irreducible components of the characteristic varieties of local systems on a plane curve arrangement complement and the associated pencils of plane curves discovered recently by M.…
An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if…