2 years ago

On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

G\Y. After removing the

G\Y. After removing the ramification locus of G\Y → G\X which has realcodimension two this space remains connected since we assumed that our spacesare normal, cf. [GR2, Chapter 7.4.2]. This connectivity result implies that thehomomorphismπ top1 (G\X − D, q(x 0 )) ↠ π top1 (X, G, x 0 ) (∗)is surjective. It remains to compute its kernel.However, first we want to define a surjective homomorphismψ : π top1 (G\X − D, q(x 0 )) ↠ G.For this we lift a loop γ in the group on the left to a path in X starting at x 0 . Thislift ends at a point g · x 0 where the element g ∈ G is unique. This defines thehomomorphism we are looking for. Of course if we take the pull-back via themorphism X → G\X we are in the situation of Section 4.3 where we defined ahomomorphism π top1 (X, G, x 0 )↠G in a similar way via lifting elements of thegroup on the left to the point x 0 × 1 of X × G. Chasing through the diagrams wesee that the homomorphisms onto G are compatible with the homomorphism (∗).For the divisor D i we define the following loop Γ i in Z := G\X − D: Wechoose a point w i on D i that is a smooth point of D. We let γ i be a path connectingq(x 0 ) to w i inside Z. We shorten γ i a little bit before reaching w i . Then we puta little circle around w i starting at the end of γ i . This defines a loop Γ i based atq(x 0 ). Such a loop is usually called a simple loop.If we lift this loop to a path based at x 0 ∈ X it “winds around“ a component R iof q −1 (D i ): We choose a small neighbourhood U(w i ) of the point w i ∈ D i thatwe have chosen above. We let V (w i ) be the connected component of q −1 (U(w i ))such that the lift of Γ i to x 0 meets V (w i ). The map q : X → G\X looks in localcoordinates likeV (w i ) → U(w i )e(z 1 , z 2 , ...) ↦→ (z i 1 , z 2 , ...)where e i is the ramification index of D i . The reason for this is that locally aroundw i the map q is a branched Galois cover with group ¡ e iand branch locus D i .In these coordinates R i is given by the equation z 1 = 0. The automorphism ofX induced by the lift of Γ i to x 0 clearly is the map x ↦→ ψ(Γ i ) · x. It is clear fromthis local description that R i must be fixed by ψ(Γ i ).We let ˜p : ˜X → X be a universal cover of X. We choose a point ˜x0 lyingabove x 0 ∈ X. Lifting Γ i to ˜x 0 we get a path that “winds“ around a component˜R ′ i of ˜p −1 (R i ). It corresponds to an automorphism of ˜X that fixes ˜R ′ i. Via basechange to X × G → X we get exactly an element that corresponds to the inertiaautomorphism of ˜R ′ i corresponding to ψ(Γ i) as described in Section 4.3.38

eThis automorphism has order e i and so the image of Γ i i under (∗) must beetrivial. In particular, the subgroup normally generated by the Γ i i ’s lies in thekernel of (∗). We stop for a moment to define a new object:With respect to the Γ i and e i we define the orbifold fundamental group withrespect to G\X, D i , e i to be the quotientπ orb1 (G\X, {D i , e i }, q(x 0 )) := π top1 (G\X − D, q(x 0 )) / ≪ Γ ie i≫ .If we choose different set of loops Γ ′ i around the D i’s as described above thenthey are conjugate to the original Γ i ’s and so this set generates the same normalsubgroup. Hence this quotient is well-defined.This orbifold fundamental group is the opposite automorphism group of sometopological cover ˜c ′ : Ỹ → G\X − D. By what we have said above the homomorphism(∗) factors through the orbifold fundamental group and so Ỹ dominates˜X − (q ◦ ˜p) −1 (D). For a smooth point w i on D i we let U(w i ) be an admissibleneighbourhood, i.e. a neighbourhood such that ˜c ′−1 (U(w i )) is a disjoint union ofspaces that are homeomorphic to U(w i ). We assume that D i is smooth in U(w i )so that U(w i )−D i is homeomorphic to (¨ −{0})רdim X−1 . This means that thefundamental group of U(w i ) − D i is isomorphic to ¡ . It is generated by a loop Γ ias described above. Looking at this locally we can extend Ỹ → ˜X − (q ◦ ˜p) −1 (D)to some map ¯Ỹ → ˜X − (q ◦ ˜p) −1 (S) where S is the finite set of singularities ofD. Since both spaces are locally homeomorphic this is a topological cover map.The space ˜X is normal and simply connected. Since (q ◦ ˜p) −1 (S) is a discrete setof real codimension 4 also ˜X − (q ◦ ˜p) −1 (S) is simply connected. Since ¯Ỹ is aconnected topological cover of ˜X − (q ◦ ˜p) −1 (S) they must be homeomorphic.Then there is only one way to complete this to a cover of ˜X: namely to take thetrivial cover of ˜X. So we conclude that Ỹ is homeomorphic to ˜X − (q ◦ ˜p) −1 (D)and this means that the homomorphism (∗) induces an isomorphismπ orb1 (G\X, {D i , e i }, q(x 0 )) ∼ = πtop1 (X, G, x 0 ).We already noted above that both groups possess surjective homomorphisms ontoG that are compatible under this isomorphism.4.5 The quotient in the topological setupWe let X be smooth projective surface over the complex numbers and f : X → 2be a good generic projection of degree n. From Proposition 2.7 we know that S nacts on X gal . With respect to this action and the action of the subgroup S (1)n−1 we39

Groups of integral representation type - MSP
Carbon reductions generate positive ROI - Carbon Disclosure Project
Projected Costs of Generating Electricity - OECD Nuclear Energy ...
The Path Ahead: Fundamentals of Generative ... - Right 4 Your Type
Generic process improvement approach applied to IT projects - PMI ...
Renewable Projects Portal for CLP Group - OSIsoft
Project Insight - Power Generation Construction Projects in Asia-Pacific
Building Next Generation Design Support Programmes - See Project
GALOIS THEORY - Tata Institute of Fundamental Research
On The Structure of Certain Galois Cohomology Groups
The Fundamental Theorem of Galois Theory and Normal Subgroups
Generic Galois groups for q-difference equations
periods of eisenstein series: the galois case - Project Euclid
Introduction to Differential Galois Theory
Galois module structure of local unit groups
fields and galois theory - Neil Strickland - University of Sheffield
On Galois Algebras with an Inner Galois Group - Bradley Bradley
1 The Galois Group of a Quadratic 2 The Galois Group of a Cubic
Supersingular Galois representations and a generalization of a ...
A family of polynomials with Galois group $ PSL_5 (2) $ over ...
Galois groups with restricted ramification Romyar Sharifi Harvard ...
Automorphy lifting for residually reducible l-adic Galois ...
On Galois Algebras with a Unique Galois Group - Bradley Bradley
The Frattini subgroup of the absolute Galois group - Tel Aviv University
The Grothendieck Conjecture on the Fundamental Groups of ...
Connected Linear Groups as Differential Galois Groups (with C ...