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Register a new userArithmeticity of the monodromy of the Wiman-Edge pencil
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The {em Wiman-Edge pencil} is the universal family $Cs/mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $Af_5$. The goal of this paper is to prove that the monodromy of $Cs/mathcal B$ is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of $mathcal B$.
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The {em Wiman-Edge pencil} is the universal family $C_t, tinmathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $Af_5$. The curve $C_0$, discovered by Wiman in 1895 cite{Wiman} and called the {em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $Sf_5$. In this paper we give an explicit uniformization of $mathcal B$ as a non-congruence quotient $Gammabackslash Hf$ of the hyperbolic plane $Hf$, where $Gamma<PSL_2(Z)$ is a subgroup of index $18$. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of $C_t$ into $10$ lines (resp. $5$ conics) whose intersection graph is the Petersen graph (resp. $K_5$). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve $C_0$ itself as the quotient $Lambdabackslash Hf$, where $Lambda$ is a principal level $5$ subgroup of a certain unit spinor norm group of M{o}bius transformations. We then prove that $C_0$ is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.
In 1981 W.L. Edge discovered and studied a pencil $mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edges work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.
We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and get a 3-cuspidal affine quartic curve whose braid monodromy we compute. This calculation of the local braid monodromy is a step towards the determination of global braid monodromies, e.g. for the (a,b,c) surfaces previously considered by the authors. In the revision we fill a gap (noticed by the referee) in the proof of the classical theorem that any quartic surface which has the twisted cubic as cuspidal curve is its tangential developable, and we changed a base point in order to make a picture correct.
There have been several constructions of family of varieties with exceptional monodromy group. In most cases, these constructions give Hodge structures with high weight(Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with low weight(Hodge numbers equal to (2,3,2)) and geometric monodromy group G2. In this article I will give an alternative description of Katzs construction and give an extension of his result.
Let $Y$ be an $(m+1)$-dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$, and $delta$ be a positive integer such that $mathcal I_{Z,Y}(delta)$ is generated by global sections. Fix an integer $dgeq delta +1$, and assume the general divisor $X in |H^0(Y,ic_{Z,Y}(d))|$ is smooth. Denote by $H^m(X;mathbb Q)_{perp Z}^{text{van}}$ the quotient of $H^m(X;mathbb Q)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$. In the present paper we prove that the monodromy representation on $H^m(X;mathbb Q)_{perp Z}^{text{van}}$ for the family of smooth divisors $X in |H^0(Y,ic_{Z,Y}(d))|$ is irreducible.
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