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Register a new userGeometry of the Wiman Pencil, I: Algebro-Geometric Aspects
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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.
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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.
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 $mathcal B$.
We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.
Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling geometric constraints that are physically meaningful, real algebraic geometry is a core mathematical input for geometric constraint systems.
This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.
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