Related papers: Exceptional Loci on $\bar M_{0,n}$ and Hypergraph …

An $(n,s,q)$-graph is an $n$-vertex multigraph where every set of $s$ vertices spans at most $q$ edges. In this paper, we determine the maximum product of the edge multiplicities in $(n,s,q)$-graphs if the congruence class of $q$ modulo…

Inside the projectivized $k$-th Hodge bundle, we construct a collection of divisors obtained by imposing vanishing at a Brill-Noether special point. We compute the classes of the closures of such divisors in two ways, using incidence…

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…

Let $H_{d,g}$ denote the Hilbert scheme of locally Cohen-Macaulay curves of degree $d$ and genus $g$ in projective three space. We show that, given a smooth irreducible curve $C$ of degree $d$ and genus $g$, there is a rational curve…

This paper shows that an analytic space $X$ has a unique maximal model through which every proper surjective morphism from a non-singular analytic space to $X$ factors. This is called the {\sl geometric minimal model} of $X$ and…

A result of Keel and McKernan states that a hypothetical counterexample to the F-conjecture must come from rigid curves on $\bar {M}_{0,n}$ that intersect the interior. We exhibit several ways of constructing rigid curves. In all our…

From Smyth's classification, modular compactifications of pointed smooth rational curves are indexed by combinatorial data, so-called extremal assignments. We explore their combinatorial structures and show that any extremal assignment is a…

We prove a formula for the ${\mathbb S}_n$-equivariant Euler characteristic of the moduli space of graphs $\mathcal{MG}_{g,n}$. Moreover, we prove that the rational ${\mathbb S}_n$-invariant cohomology of $\mathcal{MG}_{g,n}$ stabilizes for…

We construct a compact minitwistor space from a hyperelliptic curve with real structure and show that it yields a lot of new Lorentzian Einstein-Weyl spaces all of which are diffeomorphic to the 3-dimensional deSitter space. These…

In this note we analyse the scrollar invariants of $k:1$ covers of $\mathbb P^1$ that factor through the normalisation of a nodal curve in the $m$-th Hirzebruch surface $\mathbb F_m$. We then give an existence theorem for nodal curves in…

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on…

In the paper we discuss three different notions of extremal holomorphic mappings: weak $m$-extremals, $m$-extremals and $m$-complex geodesics. We discuss relations between them in general case and in the special cases of unit ball,…

We give lower bounds for the rank of the first homology group of the real points of the Deligne-Mumford-Knudsen compactification of stable n-pointed curves of genus 0,which coincides with the Chow quotient (RP^1)^n//PGL(2,R).The study has…

Let (S,H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c_1(E),H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether…

In this article, we study the topology and bifurcations of the moduli space $\mathcal{M}_3$ of cubic Newton maps. It's a subspace of the moduli space of cubic rational maps, carrying the Riemann orbifold structure $(\mathbb{\widehat{C}},…

Brill-Noether loci ${\mathcal M}^r_{g,d}$ are those subsets of the moduli space ${\mathcal M}_g$ determined by the existence of a linear series of degree $d$ and dimension $r$. By looking at non-singular curves in a neighborhood of a…

Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be an ample line bundle on $X$. Let $M_{X,H}(r;c_1, ..., c_{s})$ be the moduli space of $H$-stable vector bundles $E$ on $X$ of rank $r$ and Chern classes $c_i(E)=c_i$ for…

Extending results for space curves we establish bounds for the cohomology of a non-degenerate curve in projective $n$-space. As a consequence, for any given $n$ we determine all possible pairs $(d, g)$ where $d$ is the degree and $g$ is the…

We produce a flexible tool for contracting subcurves of logarithmic hyperelliptic curves, which is local around the subcurve and commutes with arbitrary base-change. As an application, we prove that hyperelliptic multiscale differentials…

We study a class of exceptional minimal surfaces in spheres for which all Hopf differentials are holomorphic. Extending results of Eschenburg and Tribuzy \cite{ET0}, we obtain a description of exceptional surfaces in terms of a set of…