Related papers: Dimension of the Moduli Space of curves with an in…

We study the moduli space of $J$-holomorphic subvarieties in a $4$-dimensional symplectic manifold. For an arbitrary tamed almost complex structure, we show that the moduli space of a sphere class is formed by a family of linear system…

Let M_2 be the moduli space that classifies genus 2 curves. If a curve C is defined over a field k, the corresponding moduli point P=[C] is defined over k. Mestre solved the converse problem for curves with Aut(C) isomorphic to C_2. Given a…

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb C$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves…

Using the moduli space of semiorthogonal decompositions in a smooth projective family, introduced by the second, the third and the fourth author, we propose a novel approach to indecomposability questions for derived categories. Modulo a…

A new genus $g=g(X,\ce)$ is defined for the pairs $(X,\ce)$ that consist of $n$-dimensional compact complex manifolds $X$ and ample vector bundles $\ce$ of rank $r$ less than $n$ on $X$. In case $r=n-1$, $g$ is equal to curve genus. Above…

When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their…

Consider genus g curves that admit degree d covers to an elliptic curve simply branched at 2g-2 points. Vary a branch point and the locus of such covers forms a one-parameter family W. We investigate the geometry of W by using admissible…

We describe the birational geometry of the moduli space S_g^{-} of odd spin curves (theta-characteristics) for all genera g. The odd spin moduli space is a uniruled variety for g<12, and of general type for g at least 12. Furthermore, for…

We compute the first, second, third, and fifth rational cohomology groups of the moduli space of stable n-pointed genus g curves, for all g and n, using (mostly) algebro-geometric techniques.

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…

Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety $X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of $\overline{k}$ of elements fixed by field automorphisms…

The new result is the unirationality of the moduli space of curves of genus 14. The method applies to lower genus.

We find a closed formula for the number $\operatorname{hyp}(g)$ of hyperelliptic curves of genus $g$ over a finite field $k=\mathbb{F}_q$ of odd characteristic. These numbers $\operatorname{hyp}(g)$ are expressed as a polynomial in $q$ with…

Let K be a field. Let A be a central simple algebra over K and let X be the associated Brauer-Severi variety over K. It has recently been asked if there exists a genus 1 curve C over K such that K(C) splits A. In other words, is there a…

We study genus $g$ coverings of full moduli dimension of degree $d=[\frac {g+3} 2]$. There is a homomorphism between the corresponding Hurwitz space $\H$ of such covers to the moduli space $\M_g$ of genus $g$ curves. In the case $g=3$,…

While the tangent space to an equisingular family of curves can be discribed by the sections of a twisted ideal sheaf, this is no longer true if we only prescribe the multiplicity which a singular point should have. However, it is still…

We consider the cell decomposition of the moduli space of real genus two curves with a marked point on the only real oval. The cells are enumerated by certain graphs with their weights describing the complex structure on a curve. We show…

As an introduction to the concept of "moduli space" we consider the moduli space of similarity classes of acute and right triangles in the plane. This has a map to the moduli space of elliptic curves which is onto and generically…

Let $\mathcal X_g$ be a genus $g\geq 2$ superelliptic curve, $F$ its field of moduli, and $K$ the minimal field of definition. In this short note we construct an equation of the curve $\mathcal X_g$ over its minimal field of definition $K$…

We define a proper moduli stack for degree $p$ covers $f:Y \to \cX$ where $\cX$ is a twisted stable curve in the sense of [5] and [4], and $Y$ is a stable curve which via $f$ is a torsor over $\cX$ under a finite flat group scheme $\cG \to…