Related papers: Strata of $k$-differentials

A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification for the moduli space of…

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified…

We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit…

In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, i.e. spaces of translation surfaces. In the last decade, several of these have been constructed,…

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g \geq 0$ can have for $k \geq 3$. These local invariants include the orders of zeros and poles, as well as the $k$-residues at the poles. We…

We study the local invariants that a meromorphic $k$-differential on a Riemann surface of genus $g\geq0$ can have. These local invariants are the orders of zeros and poles, and the $k$-residues at the poles. We show that for a given pattern…

Double ramification loci, also known as strata of $0$-differentials, are algebraic subvarieties of the moduli space of smooth curves parametrizing Riemann surfaces such that there exists a rational function with prescribed ramification over…

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with…

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper we study affine related properties of strata of $k$-differentials on smooth…

Strata of $k$-differentials on smooth curves parameterize sections of the $k$-th power of the canonical bundle with prescribed orders of zeros and poles. Define the tautological ring of the projectivized strata using the $\kappa$ and $\psi$…

Flat surfaces that correspond to $k$-differentials on compact Riemann surfaces are of finite area provided there is no pole of order $k$ or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat…

For $g\geq2$, $j=1,\dots,g$ and $n\geq g+j$ we exhibit infinitely many new rigid and extremal effective codimension $j$ cycles in $\overline{\mathcal{M}}_{g,n}$ from the strata of quadratic differentials and projections of these strata…

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian…

Let $f\colon C \rightarrow \mathbb{P}^1$ be a degree $k$ genus $g$ cover. The stratification of line bundles $L \in \mathrm{Pic}^d(C)$ by the splitting type of $f_*L$ is a refinement of the stratification by Brill-Noether loci $W^r_d(C)$.…

Generalized strata of meromorphic differentials are loci in the usual strata of differentials, where certain sets of residues sum up to zero. They appear naturally in the boundary of the multi-scale compactification of the usual strata.…

Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for…

Let $(\Sigma,p)$ be a pointed Riemann surface of genus $g\geq 1$. For any integer $k\geq 1$, we parametrize the space of meromorphic quadratic differentials on $\Sigma$ with a pole of order $(k+2)$ at $p$, having a connected critical graph…

Given an algebra $A$ over a differential field $K$, we study derivations on $A$ that are compatible with the derivation on $K$. There is a universal object, which is a twisted version of the usual module of differentials, and we establish…

Let $C$ be a complex projective integral curve with planar singularities. In this note, we study numerical relations among its versal deformation space, moduli space of stable maps, and compactified Jacobian. In particular, we correct a…