Related papers: Moduli spaces of quadratic maps: arithmetic and ge…

In this paper, we prove that the moduli space $\overline{M}_{X}(\nu)$ of $H$-Gieseker semistable sheaves on a smooth cubic threefold $X$ with Chern character $\nu=(4,-H,-\frac{5}{6}H^{2},\frac{1}{6}H^{3})$ is non-empty, smooth and…

Guo and the second author have shown that the closure $[I]$ in the Drury-Arveson space of a homogeneous principal ideal $I$ in $\mathbb{C}[z_1,...,z_n]$ is essentially normal. In this note, the authors extend this result to the closure of…

We study the GIT quotient of the symplectic grassmannian parametrizing lagrangian subspaces of \bigwedge^3{\mathbb C}^6 by the natural action of SL_6, call it M. This is a compactification of the moduli space of smooth double EPW-sextics.…

In the moduli space of polynomials of degree 3 with marked critical points c_1 and c_2, let C_{1,n} be the locus of maps for which c_1 has period n and let C_{2,m} be the locus of maps for which c_2 has period m. A consequence of Thurston's…

We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by…

Let $f \in Q(z)$ be a polynomial or rational function of degree 2. A special case of Morton and Silverman's Dynamical Uniform Boundedness Conjecture states that the number of rational preperiodic points of $f$ is bounded above by an…

For the moduli stack $\mathcal{M}_{g,n/\mathbb{F}_p}$ of smooth curves over $\text{Spec}~\mathbb{F}_p$ with the function field $K$, we show that if $g\geq3$, then the only $K$-rational points of the generic curve over $K$ are its $n$…

We consider Nahm's equations on a bounded open interval with first order poles at the ends. By imposing further boundary conditions, extended moduli spaces are identified with spaces of rational maps from $\mathbb{C}\mathrm{P}^1$ to…

We consider the space $\mathcal M$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\ch{n},$ up to its holomorphic isometry group ${\rm PU}(n,1).$ One of the important problems in complex hyperbolic…

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve…

The dynamics of all quadratic Newton maps of rational functions are completely described. The Julia set of such a map is found to be either a Jordan curve or totally disconnected. It is proved that no Newton map with degree at least three…

The notion of relatedness loci in the parabolic slices Per_1(e^{2\pi i p/q}) in moduli space of quadratic rational maps is introduced. They are counterparts of the disconnectedness or escape locus in the slice of quadratic polynomials. A…

In this paper we study quadratic points on the non-split Cartan modular curves $X_{ns}(p)$, for $p = 7, 11,$ and $13$. Recently, Siksek proved that all quadratic points on $X_{ns}(7)$ arise as pullbacks of rational points on $X_{ns}^+(7)$.…

As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic…

Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and…

This work is a PhD thesis. First we provide some general context on wonderful varieties and moduli spaces of rational curves. Working over complex numbers we prove that the moduli space of rational curves with no marked points on the…

We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn…

Deformation K-theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2-periodic above the rational cohomological dimension of G…

We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these…

Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5…