We provide a complete description of realizable period representations for meromorphic differentials on Riemann surfaces with prescribed orders of zeros and poles, hyperelliptic structure and spin parity.

This paper contains two results on Hodge loci in $\mathsf{M}_{g}$. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibers and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in $\mathsf{M}_{g}$. It is proved that the image under the period map of a divisor in $\mathsf{M}_{g}$ is not contained in a proper totally geodesic subvariety of $\mathsf{A}_{g}$. It follows that a Hodge locus in $\mathsf{M}_{g}$ has codimension at least 2.

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 curves which parameterize sections of the $k$th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$, then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

We present a new connection between the Hele-Shaw flow, also known as two-dimensional Laplacian growth, and the theory of holomorphic discs with boundary contained in a totally real submanifold. Using this, we prove short-time existence and uniqueness of the Hele-Shaw flow with varying permeability both when starting from a single point and also when starting from a smooth Jordan domain. Applying the same ideas, we prove that the moduli space of smooth quadrature domains is a smooth manifold whose dimension we also calculate, and we give a local existence theorem for the inverse potential problem in the plane.

We give another proof of Vojta's $1\,+\,\varepsilon $ conjecture.

Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

In the moduli space of smooth and complex irreducible projective curves of genus g, let be the locus of curves that do not satisfy the Gieseker-Petri theorem. Let be the subvariety of GPg formed by curves C of genus g with a pencil g1d=(V, L∈G1d(C) free of base points for which the Petri map μV:V⊗H0(C,KC⊗L−1)→H0(C,KC) is not injective. For g≥8, we construct in this work a family of irreducible plane curves of genus g with moduli

Let $C$ be a genus 2 algebraic curve defined by an equation of the form $y^2\,{=}\,x(x^2\,{-}\,1)(x\,{-} a)(x\,{-}\,1/a)$. As is well known, the five accessory parameters for such an equation can all be expressed in terms of $a$ and the accessory parameter $\frak b$ corresponding to $a$. The main result of the paper is that if $a'\,{=}\,\sqrt{1\,{-}\,{a}^2}$, which in general yields a non-isomorphic curve $C'$, then $\acb'a'({a'}^2\,{-}\,1) {=}\,{-}\frac38\,{-}\,\acb\,a\,({a}^2\,{-}\,1)$.

This is proven by it being shown how the uniformizing function from the unit disk to $C'$ can be explicitly described in terms of the uniformizing function for $C$.

We classify the periodic monodromies which are realized as the monodromies of hyperelliptic families.

We propose certain obstructions for the existence of hyperelliptic splitting families of degenerations of curves. Moreover we determine the complete system of hyperelliptic atomic fibers of genus 3.