arXiv subjects

Sander Rieken

Publications and source records attributed to Sander Rieken.

4 recordsLinked to original sources

arXiv2016

Hyperbolic Geometry and Moduli of Real Curves of Genus Three

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to maximal real quartic curves we obtain a more explicit description. We construct a Coxeter diagram that encodes the geometry of this component.

math.AG↗

arXiv2016

Moduli of real pointed quartic curves

We describe a natural open stratum in the moduli space of smooth real pointed quartic curves in the projective plane. This stratum consists of real isomorphism classes of pairs $(C, p)$ with $p$ a real point on the curve $C$ such that the tangent line at $p$ intersects the curve in two distinct points besides $p$. We will prove that this stratum consists of $20$ connected components. Each of these components has a real toric structure defined by an involution in the Weyl group of type $E_7$.

math.AG↗

arXiv2014

An Odd Presentation of $W(\mathrm{E}_6)$

The Weyl group $W(\mathrm{E}_6)$ has an odd presentation due to Christopher Simons as factor group of the Coxeter group on the Petersen graph by deflation of the free hexagons. The goal of this paper is to give a geometric meaning for this presentation, coming from the action of $W(\mathrm{E}_6)$ on the moduli space of marked maximally real cubic surfaces and its natural tessellation as seen through the period map of Allcock, Carlson and Toledo.

math.AG↗

arXiv2014

Two Lorentzian Lattices

We prove a conjecture formulated by the first author, which in turn provides a good deal of evidence for the monstrous proposal of Daniel Allcock.

math.AG↗