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 $72$

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

Spectral decomposition of starbursts and AGNs in 5-8 micron Spitzer IRS spectra of local ULIRGs

We present an analysis of the 5-8 micron Spitzer-IRS spectra of a sample of 68 local Ultraluminous Infrared Galaxies (ULIRGs). Our diagnostic technique allows a clear separation of the active galactic nucleus (AGN) and starburst (SB) components in the observed mid-IR emission, and a simple analytic model provides a quantitative estimate of the AGN/starburst contribution to the bolometric luminosity. We show that AGNs are ~30 times brighter at 6 micron than starbursts with the same bolometric luminosity, so that even faint AGNs can be detected. Star formation events are confirmed as the dominant power source for extreme infrared activity, since ~85% of ULIRG luminosity arises from the SB component. Nonetheless an AGN is present in the majority (46/68) of our sources.Comment: 5 Pages, 3 figures. MNRAS Letters, Accepte

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

Sediment-Induced Amplification in the Northeastern United States: a Case Study in Providence, Rhode Island

We employed ambient-noise measurements to assess the potential for seismic site response in sediment-filled valleys that intersect beneath downtown Providence, Rhode Island. At eight valley stations and at two sites on an adjacent bedrock highland, we recorded ground motion from two types of sources: pile drivers at a local construction site and ambient microtremors. At all valley sites where sediment thicknesses exceed 10 m, spectral ratios contain amplitude peaks at frequencies of 1.5 to 3.0 Hz. In contrast, spectral ratios from the two sites on the bedrock highland where sediment cover is less than 4-m thick are relatively flat within this frequency range. A variety of borehole logs identified two fundamental sediment types (soft sediment and a consolidated glacial till) and were used to map layer thicknesses over the entire study region. Refraction data constrained P-wave velocity in the bedrock to be 3680 ± 160 m/sec and indicated two soft-sediment layers with P-wave velocities of 300 ± 50 and 1580 ± 120 m/sec. Using a one-dimensional reflection matrix technique, we matched the spectral-ratio peak observed at each valley site with the frequency of fundamental resonance predicted for local layer thicknesses and velocities. A positive correlation between the best-fitting soft-sediment velocities and bedrock depth may reflect greater compaction in the deepest sediments or a locally two-dimensional sediment resonance at the deepest sediment sites. We conclude that unconsolidated sediment layers under downtown Providence have the potential to amplify earthquake ground motion at frequencies damaging to engineered structures

Extending the Belavin-Knizhnik "wonderful formula" by the characterization of the Jacobian

A long-standing question in string theory is to find the explicit expression of the bosonic measure, a crucial issue also in determining the superstring measure. Such a measure was known up to genus three. Belavin and Knizhnik conjectured an expression for genus four which has been proved in the framework of the recently introduced vector-valued Teichmueller modular forms. It turns out that for g>3 the bosonic measure is expressed in terms of such forms. In particular, the genus four Belavin-Knizhnik "wonderful formula" has a remarkable extension to arbitrary genus whose structure is deeply related to the characterization of the Jacobian locus. Furthermore, it turns out that the bosonic string measure has an elegant geometrical interpretation as generating the quadrics in P^{g-1} characterizing the Riemann surface. All this leads to identify forms on the Siegel upper half-space that, if certain conditions related to the characterization of the Jacobian are satisfied, express the bosonic measure as a multiresidue in the Siegel upper half-space. We also suggest that it may exist a super analog on the super Siegel half-space.Comment: 15 pages. Typos corrected, refs. and comments adde