On the geometry of C^3/D_27 and del Pezzo surfaces

We clarify some aspects of the geometry of a resolution of the orbifold X = C3/D_27, the noncompact complex manifold underlying the brane quiver standard model recently proposed by Verlinde and Wijnholt. We explicitly realize a map between X and the total space of the canonical bundle over a degree 1 quasi del Pezzo surface, thus defining a desingularization of X. Our analysis relys essentially on the relationship existing between the normalizer group of D_27 and the Hessian group and on the study of the behaviour of the Hesse pencil of plane cubic curves under the quotient.Comment: 23 pages, 5 figures, 2 tables. JHEP style. Added references. Corrected typos. Revised introduction, results unchanged

Compactifications of the moduli space of plane quartics and two lines

We study the moduli space of triples (C,L₁,L₂) consisting of quartic curves C and lines L₁ and L₂. Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the period map of certain lattice polarized K3 surfaces. The GIT construction depends on two parameters t₁ and t₂ which correspond to the choice of a linearization. For t₁=t₂=1 we describe the GIT moduli explicitly and relate it to the construction via K3 surfaces

A note on Severi varieties of nodal curves on Enriques surfaces

accepted for publication in the Springer Indam Series volume on the occasion of the conference Birational Geometry and Moduli Spaces held in Rome, june 2018Let |L| be a linear system on a smooth complex Enriques surface S whose general member is a smooth and irreducible curve of genus p, with L 2 > 0, and let V |L |,δ (S) be the Severi variety of irreducible δ-nodal curves in |L|. We denote by π : X → S the universal covering of S. In this note we compute the dimensions of the irreducible components V of V |L |,δ (S). In particular we prove that, if C is the curve corresponding to a general element [C] of V, then the codimension of V in |L| is δ if π −1 (C) is irreducible in X and it is δ − 1 if π −1 (C) consists of two irreducible components