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

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)

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

An Assessment of miRNA Manipulation on Senescence and Ageing Phenotypes in vitro and in vivo

Ageing has been widely described as a progressive functional deterioration of tissues that causes diminished organ function and increased mortality risk. It has been established that the proportion of senescent cells in tissues rises with age in many organs and in age-related illnesses, suggesting that cellular senescence plays a significant role in the functional decline related to ageing. Correspondingly, it has previously been shown in animal models that eliminating senescent cells might mitigate the deleterious consequences of ageing. As a key regulator of several cellular mechanisms, there are microRNAs (miRNAs) known to be associated with senescence. However, miRNAs that may directly trigger or reverse senescence remain to be elucidated. Here, the first goal of thesis was to identify the miRNA profile of proliferating, senescent, and rescued senescent endothelial cells to determine miRNAs that may be causal or influential of cellular senescence. I found that miR-361-5p not only associated with senescence but also reduced the load of senescent cells in vitro in human endothelial cells upon induction in late passage cells. Secondly, C. elegans was used to examine the role of miR-361-5p targeted genes on ageing in vivo. I found that 56% of genes which were dysregulated in vitro adversely affected healthspan and/or lifespan in vivo. Finally, a previous finding from our lab (Holly et al., 2015) identified three miRNAs-associated with human ageing and senescence in human primary fibroblasts of which miR-15b-5p may reduce senescence markers and secretory phenotypes (SASP) in the human dermal fibroblast cells. This thesis presents new miRNAs (miR-361-5p and miR-15b-5p) which may be involved in the aetiology of senescence and may be used in future in ageing intervention

Exploring the Heidelberg Retinal Tomograph 3 diagnostic accuracy across disc sizes and glaucoma stages: a multicenter study

To investigate and compare the diagnostic accuracy of the Heidelberg Retinal Tomograph 3 (HRT3) diagnostic algorithms and establish whether they are affected by optic disc size and glaucoma severity

Superstring scattering amplitudes in higher genus

In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ans\"atze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.Comment: Minor changes; final version to appear in Comm. Math. Phy

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