The Wiman-Edge pencil is a pencil of genus $6$ curves for which the generic member has automorphism group the alternating group $\mathfrak{A}_5$. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group $\mathfrak{S}_5$. Farb and Looijenga proved that the monodromy of the Wiman-Edge pencil is commensurable with the Hilbert modular group $\mathrm{SL}_2(\mathbb{Z}[\sqrt{5}])$. In this note, we give a complete description of the monodromy by congruence conditions modulo $4$ and $5$. The congruence condition modulo $4$ is new, and this answers a question of Farb-Looijenga. We also show that the smooth resolution of the Baily-Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.
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 $\mathcal B$.
In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's 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.
The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ 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 $Γ\backslash \Hf$ of the hyperbolic plane $\Hf$, where $Γ<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpret…
We prove that the bimeromorphic class of a hyperkähler manifold deformation equivalent to O'Grady's six dimensional one is determined by the Hodge structure of its Beauville-Bogomolov lattice by showing that the monodromy group is maximal. As applications, we give the structure for the Kähler and the birational Kähler cones in this deformation class and we prove that the existence of a square zero divisor implies the existence a rational lagrangian fibration with fixed fibre types.
A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the $d^2$ differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer etale topology. In the case of curves, these data a…
We describe a 3-step filtration on all logarithmic abelian varieties with constant degeneration. The obstruction to descending this filtration, as a variegated extension, from logarithmic geometry to algebraic geometry is encoded in a bilinear pairing valued in the characteristic monoid of the base. This pairing is realized as the monodromy pairing in p-adic, l-adic, and Betti cohomolgies, and recovers the Picard-Lefschetz transformation in the case of Jacobians. The Hodge realization of the filtration is the monodromy weight filtration on the limit mixed Hodge structure.
We study the topology of the punctured disc defined over a non-archimedean field of characteristic zero. Chapter two includes a new proof of the so-called p-adic Riemann existence theorem. This release completes the study of breaks and break decompositions of the monodromy representation of a sheaf around the origin of the punctured disc.
By exploring the consequences of the triviality of the monodromy group for a class of surfaces of which the mixed Hodge structure is pure, we extend results of Miyanishi and Sugie, Dimca, Zaidenberg and Kaliman.
We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $ω$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $ω$ as the monodromy operator. We prov…
A p-typical cover of a connected scheme on which p=0 is a finite etale cover whose monodromy group (i.e., the Galois group of its normal closure) is a p-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the p-typical quotients of the etale fundamental groups, and a decomposition theorem for p-typical covers of polynomial rings over an algebraically closed field.
We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy taking values in the Zariski resp. Artin braid group. In both cases we give the generators of the image. These results are compared with the corresponding geometric monodromy. It takes values in the mapping class group of braided surfaces. Our final result gives a precise statement ab…
We establish a combinatorial connection between the real geometry and the $K$-theory of complex Schubert curves $S(λ_\bullet)$, which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. In a previous paper, the second author showed that the real geometry of these curves is described by the orbits of a map $ω$ on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of $\mathbb{RP}^1$, with $ω$ as the monodromy operator. We prov…
Let X be a smooth, geometrically connected variety over a p-adic local field. We show that the pro-unipotent fundamental group of X (in both the etale and crystalline settings) satisfies the weight-monodromy conjecture, following Vologodsky. We deduce (in the etale setting) that Frobenii act semisimply on the Lie algebra of the pro-unipotent fundamental group of X, and (in the crystalline setting) that the same is true for a K-linear power of the crystalline Frobenius. We give applications to the representability and geometry of the Selmer varieties appearing in the Chabauty-Kim program, even …
Computation of the fundamental group of the complement in the complex plane of the branch curve S , of a generic projection of the Veronese surface to the plane is presented. This paper is a continuation of our previous papers: Braid Group Technique I - IV. In I and II we developed algorithms to compute braid monodromy of a brunch curve, provided there exist a degeneration of the surface to union of planes in a configuration where the associated branch curve is partial to a line arrangement dual to generic. In III we constructed a degeneration of the Veronese surface of order 3 to union of pla…
In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2…
We the study of the monodromy of local systems with bounded ramification on a punctured disc defined over a non-archimedean valued field of characteristic zero. First, we construct the local Fourier transforms and we establish their main properties. Also, we show that the Fourier transform of a "perverse sheaf" with bounded ramification on the affine line, is again a "perverse sheaf" with bounded ramification (we put that in quotes, because actually the language of perversity is not used). These foundations are then used to exhibit a natural break decomposition for local systems with bounded r…
Let X be a normal complex algebraic variety, and p a prime. We show that there exists an integer N=N(X, p) such that: any non-trivial, irreducible representation of the fundamental group of X, which arises from geometry, must be non-trivial mod p^N. The proof involves an analysis of the action of the Galois group of a finitely generated field on the etale fundamental group of X. We also prove many arithmetic statements about fundamental groups which are of independent interest, and give several applications.
We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of G-bundles Bun_G for an arbitrary reductive group G, and their relationship to the Langlands dual group of G. In the first part of the article we study the monodromy action on the nearby cycles sheaf along the principal degeneration of Bun_G. We describe the weight-monodromy filtration in terms of the combinatorics of the Langlands dual group of G and generalizations of the Picard-Lefschetz oscillators found in [Sch1]. Our proofs use certain local models f…
Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the compl…
Georgia Institute of Technology（佐治亚理工学院）；Czech Institute of Informatics, Robotics and Cybernetics, Czech Technical University in Prague（布拉格捷克理工大学信息学、机器人学与控制论研究所）；Duke University（杜克大学）
We consider Galois/monodromy groups arising in computer vision applications, with a view towards building more efficient polynomial solvers. The Galois/monodromy group allows us to decide when a given problem decomposes into algebraic subproblems, and whether or not it has any symmetries. Tools from numerical algebraic geometry and computational group theory allow us to apply this framework to classical and novel reconstruction problems. We consider three classical cases--3-point absolute pose, 5-point relative pose, and 4-point homography estimation for calibrated cameras--where the decomposi…
We consider the moduli spaces of flat $SL(n, C)$-bundles on Riemann surfaces with one puncture when we fix the conjugacy class ${\cal C}$ of the monodromy transformation around the puncture. We show that under a certain condition on the class ${\cal C}$ (namely the product of $k<n$ eigenvalues is not equal to $1$) that we call property P the moduli space in question is smooth and its natural closure is a normal algebraic variety with rational singularities. The set of conjugacy classes having property P constitutes a Zariski open subset of $SL(n, C)$ and it is also possible to define property …
In this paper we give an exposition of Dennis Johnson's work on the first homology of the Torelli groups and show how it can be applied, alone and in concert with Saito's theory of Hodge modules, to study the geometry of moduli spaces of curves. For example, we show that the picard groups of moduli spaces of curves with a fixed level structure are finitely generated, classify all "natural" normal functions defined over moduli spaces of curves with a fixed level, and also "compute" the height paring between cycles over moduli spaces of curves which are homologically trivial and disjoint over th…
