The modular variety of hyperelliptic curves of genus three

The modular variety of non singular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi projective variety which admits several standard compactifications. The first one, X, comes from the realization of this variety as a sub-variety of the Siegel modular variety of level two and genus three .We will be to describe the equations of X in a suitable projective embedding and its Hilbert function. It will turn out that X is normal. A further model comes from geometric invariant theory using so-called semistable degenerated point configurations in (P^1)^8 . We denote this GIT-compactification by Y. The equations of this variety in a suitable projective embedding are known. This variety also can by identified with a Baily-Borel compactified ball-quotient. We will describe these results in some detail and obtain new proofs including some finer results for them. We have a birational map between Y and X . In this paper we use the fact that there are graded algebras (closely related to algebras of modular forms) A,B such that X=proj(A) and Y=proj(B). This homomorphism rests on the theory of Thomae (19th century), in which the thetanullwerte of hyperelliptic curves have been computed. Using the explicit equations for $A,B$ we can compute the base locus of the map from Y to X. Blowing up the base locus and the singularity of Y, we get a dominant, smooth model {\tilde Y}. We will see that {\tilde Y} is isomorphic to the compactification of families of marked projective lines (P^1,x_1,...,x_8), usually denoted by {\bar M_{0,8}}. There are several combinatorial similarities between the models X and Y. These similarities can be described best, if one uses the ball-model to describe Y.Comment: 39 page

Some Siegel threefolds with a Calabi-Yau model II

In the paper [FSM] we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties that admits a Calabi-Yau model in the following weak sense: there exists a desingularization in the category of complex spaces of the Satake compactification which admits a holomorphic three-form without zeros and whose first Betti number vanishes Basic for our method is the paper [GN] of van Geemen and Nygaard.Comment: 23 pages, no figure

Some ball quotients with a Calabi--Yau model

Recently we determined explicitly a Picard modular variety of general type. On the regular locus of this variety there are holomorphic three forms which have been constructed as Borcherds products. Resolutions of quotients of this variety, such that the zero divisors are in the branch locus, are candidates for Calabi-Yau manifolds. Here we treat one distinguished example for this. In fact we shall recover a known variety given by the equations $X_0X_1X_2=X_3X_4X_5, \,\, X_0^3+X_1^3+X_2^3=X_3^3+X_4^3+X_5^3.$ as a Picard modular variety. This variety has a projective small resolution which is a rigid Calabi-Yau manifold ($h^{12}=0$) with Euler number $72$

Crisis and Intergovernmental Retrenchment in the European Union? Framing the EU’s Answer to the COVID-19 Pandemic

The outbreak of the COVID-19 pandemic has placed severe pressure on the EU’s capacity to provide a timely and coordinated response capable of curbing the pandemic’s disastrous economic and social effects on EU member states. In this situation, the supranational institutions and their models of action are evidently under pressure, seeming incapable of leading the EU out of the stormy waters of the present crisis. The article frames the first months of management of the COVID-19 crisis at EU level as characterised by the limited increase in the level of steering capacity by supranational institutions, due to the reaffirmed centrality of the intergovernmental option. To explain this situation, the article considers the absence of the institutional capacity/legitimacy to extract resources from society(ies), and the subsequent impossibility of guaranteeing an effective and autonomous process of political (re)distribution, the key factors accounting for the weakness of vertical political integration in the response to the COVID-19 challenge. This explains why during the COVID-19 crisis as well, the pattern followed by the EU is rather similar to past patterns, thus confirming that this has fed retrenchment aimed at the enforcement of the intergovernmental model and the defence of the most sensitive core state powers against inference from supranational EU institutions

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus $g=3,4,5$. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.Comment: 21 page

The vanishing of two-point functions for three-loop superstring scattering amplitudes

In this paper we show that the two-point function for the three-loop chiral superstring measure ansatz proposed by Cacciatori, Dalla Piazza, and van Geemen vanishes. Our proof uses the reformulation of ansatz in terms of even cosets, theta functions, and specifically the theory of the $\Gamma_{00}$ linear system on Jacobians introduced by van Geemen and van der Geer. At the two-loop level, where the amplitudes were computed by D'Hoker and Phong, we give a new proof of the vanishing of the two-point function (which was proven by them). We also discuss the possible approaches to proving the vanishing of the two-point function for the proposed ansatz in higher genera

The modular variety of hyperelliptic curves of genus three

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus 3 is a 5-dimensional quasi-projective variety which admits several standard compactifications. The first one realizes this variety as a subvariety of the Siegel modular variety of level two and genus three. It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components. Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with 8 branch points. There are two important compactifications of this configuration space. The first one, Y, uses the semistable degenerated point configurations in (P(1))(8). This variety also can be identified with a Baily-Borel compactified ball-quotient Y = (B/Gamma[1 - i]) over bar. We will describe these results in some detail and obtain new proofs including some finer results for them. The other compactification uses the fact that families of marked projective lines can degenerate to stable marked curves of genus 0. We use the standard notation (M) over bar (0,8) for this compactification. We have a diagram [GRAPHICS] The horizontal arrow is only birational but not everywhere regular. In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) A, B such that X = proj(A), Y = proj(B)

From Manufacturing to Advanced Services. The (Uneven) Rise and Decline of Mediterranean City-Regions

Uneven changes in the global urban hierarchy have given way to new forms of relationships between urban and rural areas based on complementarities, cooperative and specialized exchange of services and goods, abandoning the additive processes of growth guided by industrialization and urbanization. Representing a distant notion from traditional concepts in regional studies such as 'compact cities' or 'suburbs', 'gravitation' or 'hierarchy', the 'city-region' paradigm has stimulated different visions to be recomposed within the 'sustainability' framework. With global changes, the 'mega-city region' model has starting to take the lead in the development of contemporary urban agglomeration. In this study, considerations over the emergence of this urban model in the Mediterranean region will be presented to investigate the relationship between dispersed urbanization and consolidating southern European city-regions. While Mediterranean cities have been considered for long time as ‘ordinary’ cities, rather distant from the 'globalized' northern urban models, most of these cities are characterized by distinctive socioeconomic traits possibly open to competition and globalization. The present contribution describes the emergence of a Mediterranean urban area, Athens, as a new 'city-region' in the context of urbanization processes in Greece and in the Mediterranean basin as a whole. One of the clearest indications of urban competitiveness amongst emerging and established large city-regions is the fight for hosting mega-events. The final objective of the study is to understand how the efforts for increasing urban competitiveness are impacting new forms of cityregions, mainly based on low-density settlements reflecting discontinuous urbanization