Hurwitz components of groups with socle PSL(3, q)

For a finite group G, the Hurwitz space Hinr,g(G) is the space of genus g covers of the Riemann sphere P1 with r branch points and the monodromy group G. In this paper, we give a complete list of some almost simple groups of Lie rank two. That is, we assume that G is a primitive almost simple groups of Lie rank two. Under this assumption we determine the braid orbits on the suitable Nielsen classes, which is equivalent to finding connected components in Hinr,g(G).

Monodromy of projections of hypersurfaces

Let X be an irreducible, reduced complex projective hypersurface of degree d. A point P not contained in X is called uniform if the monodromy group of the projection of X from P is isomorphic to the symmetric group $$S_d$$ S d . We prove that the locus of non-uniform points is finite when X is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least 2, except possibly for a special class of hypersurfaces with singular locus linear in codimension 1. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.

On the LHC signatures of $$SU(5)\times U(1)'$$ F-theory motivated models

We study low energy implications of F-theory GUT models based on SU(5) extended by a $$U(1)'$$ U ( 1 ) ′ symmetry which couples non-universally to the three families of quarks and leptons. This gauge group arises naturally from the maximal exceptional gauge symmetry of an elliptically fibred internal space, at a single point of enhancement, $$E_8\supset SU(5)\times SU(5)'\supset SU(5)\times U(1)^4.$$ E 8 ⊃ S U ( 5 ) × S U ( 5 ) ′ ⊃ S U ( 5 ) × U ( 1 ) 4 . Rank-one fermion mass textures and a tree-level top quark coupling are guaranteed by imposing a $$Z_2$$ Z 2 monodromy group which identifies two abelian factors of the above breaking sequence. The $$U(1)'$$ U ( 1 ) ′ factor of the gauge symmetry is an anomaly free linear combination of the three remaining abelian symmetries left over by $$Z_2$$ Z 2 . Several classes of models are obtained, distinguished with respect to the $$U(1)'$$ U ( 1 ) ′ charges of the representations, and possible extra zero modes coming in vector-like representations. The predictions of these models are investigated and are compared with the LHC results and other related experiments. Particular cases interpreting the B-meson anomalies observed in LHCb and BaBar experiments are also discussed.

Sequential discontinuities of Feynman integrals and the monodromy group

We generalize the relation between discontinuities of scattering amplitudes and cut diagrams to cover sequential discontinuities (discontinuities of discontinuities) in arbitrary momentum channels. The new relations are derived using time-ordered perturbation theory, and hold at phase-space points where all cut momentum channels are simultaneously accessible. As part of this analysis, we explain how to compute sequential discontinuities as monodromies and explore the use of the monodromy group in characterizing the analytic properties of Feynman integrals. We carry out a number of cross-checks of our new formulas in polylogarithmic examples, in some cases to all loop orders.

38406501359372282063949 and All That: Monodromy of Fano Problems

A Fano problem is an enumerative problem of counting $r$-dimensional linear subspaces on a complete intersection in ${\mathbb{P}}^n$ over a field of arbitrary characteristic, whenever the corresponding Fano scheme is finite. A classical example is enumerating lines on a cubic surface. We study the monodromy of finite Fano schemes $F_{r}(X)$ as the complete intersection $X$ varies. We prove that the monodromy group is either symmetric or alternating in most cases. In the exceptional cases, the monodromy group is one of the Weyl groups $W(E_6)$ or $W(D_k)$.

Connected Components of the Hurwitz Space for the Symmetric Group of Degree 7

The Hurwitz space is the space of genus covers of the Riemann sphere with branch points and the monodromy group . Let be the symmetric group . In this paper, we enumerate the connected components of . Our approach uses computational tools, relying on the computer algebra system GAP and the MAPCLASS package, to find the connected components of . This work gives us the complete classification of primitive genus zero symmetric group of degree seven.

Arithmeticity of the monodromy of some Kodaira fibrations

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

ON THE MONODROMY AND GALOIS GROUP OF CONICS LYING ON HEISENBERG INVARIANT QUARTIC K3 SURFACES

In [5], Eklund showed that a general (ℤ/2ℤ)4 -invariant quartic K3 surface contains at least 320 conics. In this paper, we analyse the field of definition of those conics as well as their Monodromy group. As a result, we prove that the moduli space of (ℤ/2ℤ)4-invariant quartic K3 surface with a certain marked conic has 10 irreducible components.