Related papers: On a Zeuthen-type problem

One can easily show that any meromorphic function on a complex closed Riemann surface can be represented as a composition of a birational map of this surface to CP^2 and a projection of the image curve from an appropriate point p in CP^2 to…

Finding the so-called characteristic numbers of the complex projective plane ${\mathbb C}P^2$ is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given $d$ and $g$ one has to find the number of…

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high…

Let $C$ be an irreducible projective plane curve in the complex projective space ${\mathbb{P}}^2$. The classification of such curves, up to the action of the automorphism group $PGL(3,{\mathbb{C}})$ on ${\mathbb{P}}^2$, is a very difficult…

In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin…

Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as…

Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex numbers. Let $H^0_{d,A}(Y)$ be the Hurwitz space which parametrizes coverings $p:X \to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal…

We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational…

In this paper, we study a problem that is in a sense a reversal of the Hurwitz counting problem. The Hurwitz problem asks: for a generic target -- $\mathbb P^1$ with a list of $n$ points $q_1,\dots,q_n\in \mathbb P^1$ -- and partitions…

Let E be the restriction of the null-correlation bundle on $\mathbb{P}^{3}$ to a hyperplane. In this article, we show that the projective bundle $\mathbb{P}(E)$ is isomorphic to a blow-up of a non-singular quadric in $\mathbb{P}^{4}$ along…

In this paper, we aim to provide an accessible survey to various formulae for calculating single Hurwitz numbers. Single Hurwitz numbers count certain classes of meromorphic functions on complex algebraic curves and have a rich geometric…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We present a study of real Hurwitz numbers enumerating a special kind of real meromorphic functions, which we call simple framed purely real functions. We deduce partial differential equations of cut-and-join type for generating functions…

Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the Eynard-Orantin topological recursion, an infinite-order differential equation called a…

In this paper we begin to study curves on a weighted projective plane with one trivial weight, ${\mathbb P}(1,m,n)$, by determining the genus of curves of Fermat type. These are curves defined by a ``homogeneous'' polynomial analagous to…

A degree $d$ genus $g$ cover of the complex projective line by a smooth curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. Which bundles are possible? Equivalently, which…

We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We…

In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if…

We consider Jordan curves of the form $\gamma=\cup_{j=1}^n \gamma_j$ on the Riemann sphere for which each $\gamma_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$. These Jordan curves are…

For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…