Marked nodal curves with partitioned auxiliary marks #
Let the auxiliary marks of a marked nodal curve (labelled by R, see
TauCeti.AlgebraicGeometry.Curves.NodalGluing.AuxiliaryMarks) be coloured by a function
κ : R → Cl, so that the auxiliary labels are partitioned into the colour classes
Λ_c = κ ⁻¹' {c}, empty classes being allowed. An arrow (π, f) with unordered auxiliary marks
preserves the colours when κ (π j) = κ j for every auxiliary label j: its permutation lies
in ∏_c S(Λ_c), and it may permute the marks within each colour class but never moves a mark to
another class. For instance, two distinguished unordered auxiliary sets R₁ and R₂ use the
permutations S(R₁) × S(R₂) of R₁ ⊕ R₂, not all of S(R₁ ⊕ R₂), which could interchange their
roles (AuxiliaryIso.preservesColors_isLeft_iff).
The colour-preserving arrows form a wide subgroupoid of the groupoid of curves with unordered
auxiliary marks: it has all the objects but in general not all the arrows between them. Its
inclusion is faithful on actual arrows, as for every subgroupoid (Subgroupoid.hom.faithful).
The automorphisms preserving the colours form a subgroup of the automorphism group, between the
ordered automorphisms and all of it, and it acts on each colour class.
A single colour class recovers the groupoid with unordered auxiliary marks, and singleton colour classes recover the ordered isomorphisms, those fixing every mark. Relabelling the colours bijectively does not change the subgroupoid, and merging colour classes only enlarges it.
Main definitions #
NodalGluing.AuxiliaryIso.PreservesColors κ e: the arrowepreserves the colouringκ.NodalGluing.AuxiliaryMarkedCurve.partitionedAuxiliaryIsomorphismGroupoid κ: the wide subgroupoid of the colour-preserving arrows (isWide_partitionedAuxiliaryIsomorphismGroupoid).NodalGluing.AuxiliaryAut.partitioned I n m κ: the subgroup of colour-preserving automorphisms, with its actionAuxiliaryAut.blockPermon the colour classes, whose kernel is the subgroup of ordered automorphisms (AuxiliaryAut.ker_blockPerm).
Main results #
AuxiliaryMarkedCurve.partitionedAuxiliaryIsomorphismGroupoid_eq_top,mem_partitionedAuxiliaryIsomorphismGroupoid_iff_perm_eq_one: one colour class gives every arrow, and singleton colour classes give the arrows fixing every mark.AuxiliaryMarkedCurve.partitionedAuxiliaryIsomorphismGroupoid_comp_of_injective,partitionedAuxiliaryIsomorphismGroupoid_le_comp: relabelling and merging colour classes.AuxiliaryIso.PreservesColors.perm_image_preimage,AuxiliaryIso.PreservesColors.image_image_point_inr_preimage: a colour-preserving arrow maps each union of colour classes onto itself, and so carries the auxiliary marks of those colours onto the auxiliary marks of the same colours. For an arbitrary arrow, the marks labelled by a setSgo to those labelled by its image under the permutation (AuxiliaryIso.image_image_point_inr).AuxiliaryIso.preservesColors_isLeft_iff: the two-colour caseS(R₁) × S(R₂).
References #
- Joel W. Robbin and Dietmar A. Salamon, A construction of the Deligne–Mumford orbifold, J. Eur. Math. Soc. 8 (2006), 611–699, for marked nodal Riemann surfaces and their isomorphisms (Definition 3.2).
Colour-preserving arrows #
A colour-preserving arrow. For a colouring κ : R → Cl of the auxiliary labels, the
arrow e preserves the colours when its permutation carries each auxiliary label to a label of
the same colour, that is, when it permutes each colour class κ ⁻¹' {c}.
Instances For
An arrow preserves the colours exactly when the colouring is invariant under its
permutation, κ ∘ π = κ.
An arrow preserves the colours exactly when its permutation preserves the fibers of κ,
that is, lies in TauCeti.fiberSubgroup κ.
The identity arrow preserves the colours.
The inverse of a colour-preserving arrow preserves the colours.
The inverse of an arrow preserves the colours exactly when the arrow does.
A composite of colour-preserving arrows preserves the colours.
Merging colour classes. An arrow preserving a colouring κ preserves every coarser
colouring σ ∘ κ.
Relabelling the colours. For an injective relabelling σ of the colours, an arrow
preserves σ ∘ κ exactly when it preserves κ.
One colour class. If there is at most one colour, every arrow preserves the colours.
Singleton colour classes. For an injective colouring, an arrow preserves the colours exactly when its permutation of the auxiliary labels is trivial.
A colour-preserving arrow maps each union of colour classes onto itself.
Colour classes are transported. A colour-preserving arrow carries the auxiliary marks of any union of colour classes exactly onto the auxiliary marks of the same colours.
Two colour classes. For auxiliary labels R₁ ⊕ R₂ coloured by their summand, an arrow
preserves the colours exactly when its permutation lies in S(R₁) × S(R₂), embedded in
S(R₁ ⊕ R₂) by Equiv.Perm.sumCongrHom.
The partitioned groupoid #
The groupoid of curves with partitioned auxiliary marks. For a colouring κ : R → Cl of
the auxiliary labels, this is the subgroupoid of the groupoid of curves with unordered auxiliary
marks consisting of the colour-preserving arrows (π, f), those whose permutation π lies in
∏_c S(κ ⁻¹' {c}). It is wide (isWide_partitionedAuxiliaryIsomorphismGroupoid): it has the
same objects, but in general fewer arrows between them.
Instances For
An arrow lies in the partitioned groupoid exactly when it preserves the colours.
The partitioned groupoid is a wide subgroupoid: it contains the identity of every curve.
Merging colour classes enlarges the partitioned groupoid.
Relabelling the colours injectively does not change the partitioned groupoid.
One colour class recovers the groupoid of curves with unordered auxiliary marks: every arrow preserves the colours.
Singleton colour classes recover the ordered isomorphisms: for an injective colouring, the arrows of the partitioned groupoid are those with trivial permutation, which fix every distinguished and every auxiliary label.
Colour-preserving automorphisms #
The colour-preserving automorphisms of a curve with auxiliary marks coloured by
κ : R → Cl: the automorphisms whose permutation of the auxiliary labels preserves every colour
class, the preimage of TauCeti.fiberSubgroup κ under the permutation action auxiliaryPerm.
Instances For
An automorphism is colour-preserving exactly when its permutation preserves the colours.
Every ordered automorphism preserves the colours.
Merging colour classes enlarges the colour-preserving automorphisms.
Relabelling the colours injectively does not change the colour-preserving automorphisms.
One colour class: every automorphism preserves the colours.
Singleton colour classes: for an injective colouring, the colour-preserving automorphisms are the ordered automorphisms.
The action on the colour classes. A colour-preserving automorphism permutes each colour
class κ ⁻¹' {c}; this is its image in ∏_c S(κ ⁻¹' {c}), obtained from its permutation by the
restriction isomorphism TauCeti.fiberSubgroupMulEquivPiPerm κ.
Instances For
The action of a colour-preserving automorphism on a colour class is its permutation.
A colour-preserving automorphism acts trivially on every colour class exactly when it is ordered: the kernel of the action on the colour classes is the subgroup of ordered automorphisms.