scholarly journals The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces

1999 ◽  
Vol 75 (7) ◽  
pp. 129-133 ◽  
Author(s):  
Takeshi Sasaki ◽  
Masaaki Yoshida
Keyword(s):  
Differential Equation ◽  
Moduli Space ◽  
Hyperbolic Structure ◽  
Cubic Surfaces

scholarly journals Lamé polynomials, hyperelliptic reductions and Lamé band structure

2007 ◽  
Vol 366 (1867) ◽  
pp. 1115-1153 ◽  
Author(s):  
Robert S Maier
Keyword(s):  
Differential Equation ◽  
Dispersion Relation ◽  
Band Structure ◽  
Elliptic Curve ◽  
General Formula ◽  
Moduli Space ◽  
Periodic Potential ◽  
Hyperelliptic Curve ◽  
One Dimensional ◽  
Lamé Equation

The band structure of the Lamé equation, viewed as a one-dimensional Schrödinger equation with a periodic potential, is studied. At integer values of the degree parameter ℓ , the dispersion relation is reduced to the ℓ =1 dispersion relation, and a previously published ℓ =2 dispersion relation is shown to be partly incorrect. The Hermite–Krichever Ansatz, which expresses Lamé equation solutions in terms of ℓ =1 solutions, is the chief tool. It is based on a projection from a genus- ℓ hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lamé equation parameters take complex values, are investigated. If the Lamé equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lamé polynomials, i.e. band-edge solutions, are given. A table in the earlier literature is corrected.

scholarly journals The Chow group of the moduli space of marked cubic surfaces

2004 ◽  
Vol 183 (3) ◽  
pp. 291-316 ◽  
Author(s):  
Elisabetta Colombo ◽  
Bert van Geemen
Keyword(s):  
Moduli Space ◽  
Chow Group ◽  
Cubic Surfaces

A complex hyperbolic structure for moduli of cubic surfaces

1998 ◽  
Vol 326 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Daniel Allcock ◽  
James A. Carlson ◽  
Domingo Toledo
Keyword(s):  
Hyperbolic Structure ◽  
Cubic Surfaces

scholarly journals A LINEAR SYSTEM ON NARUKI'S MODULI SPACE OF MARKED CUBIC SURFACES

2002 ◽  
Vol 13 (02) ◽  
pp. 183-208 ◽  
Author(s):  
BERT VAN GEEMEN
Keyword(s):  
Linear System ◽  
Projective Space ◽  
Weyl Group ◽  
Root System ◽  
Moduli Space ◽  
Automorphic Forms ◽  
Cubic Surfaces ◽  
Ball Quotients ◽  
Dimensional Projective Space

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to obtain these results. Here we describe the same embedding using Naruki's toric model of the moduli space. We also give an explicit parametrization of the tritangent divisors, we discuss another way to find equations for the image and we show that the moduli space maps, with degree at least ten, onto the unique quintic hypersurface in a five dimensional projective space which is invariant under the action of the Weyl group of the root system E6.

scholarly journals The complex hyperbolic geometry of the moduli space of cubic surfaces

2002 ◽  
Vol 11 (4) ◽  
pp. 659-724 ◽  
Author(s):  
Daniel Allcock ◽  
James A. Carlson ◽  
Domingo Toledo
Keyword(s):  
Moduli Space ◽  
Hyperbolic Geometry ◽  
Complex Hyperbolic Geometry ◽  
Cubic Surfaces
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