feat: framed quiver moduli spaces, local structure, Proposition 6 fibres#43
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Stacked on top of #33 (
bocklandt). Adds framed quiver moduli spaces and the fibres of theframed projection (Proposition 6 of arXiv:2607.12895,
https://arxiv.org/abs/2607.12895), together with the etale-local structure of Luna strata
(Proposition 5), and makes
local_quiver_settingpublic (closes #41).Projection-to-the-wall fibres (Proposition 8) are intentionally left to #33, which already
implements them functionally (
fibre_dimension,is_flat,is_semismall).What is added
Quiver-level framing (parity with the Python/Sage QuiverTools):
framed_quiver(Q, n): prepends a framing vertex withn[i]arrowsi0 -> i.coframed_quiver(Q, n): appends a coframing vertex withn[i]arrowsi -> i0.A framed quiver moduli space, encoded as a new type
FramedQuiverModuliSpacestoring thebase
(Q, d, theta)and the framing datumn:base(X): the base moduli spaceM^theta(Q, d), the codomain of the projectionp.total_space(X):Xitself as an ordinaryQuiverModuliSpaceon the framed quiver.framing_vector(X): the framing datumn.fibre(X, tau): Proposition 6. The fibre ofpover the Luna stratumS_tau, as aNilpotentLocusinside a framed moduli of the local quiver, with local framing datumn_tau = sum_k (n . d_k) i_k(using thesummandskey that feat: Bocklandt reduction and smoothness of quiver moduli #33 adds tolocal_quiver_setting).NilpotentLocus(ambient)marks the nilpotent sublocus of a moduli space;ambient(N)recovers the ambient.
local_structure(M, tau)returns the etale-local modelM^0(Q_tau, d_tau)of the singularity alongS_tau(Proposition 5).local_quiver_settingis now exported (closes #41).
API note (deliberate deviation)
FramedQuiverModuliSpaceandNilpotentLocusintroduce an object-oriented flavour thatdeviates from the current function-based API (e.g.
local_quiver_settingreturns aDict).Consequently
fibrereturns an object rather than a data dictionary. This was a deliberatechoice: a framed quiver moduli space carries data (base plus framing) that a plain
QuiverModuliSpaceon the framed quiver would discard, and Proposition 6 states the fibre asitself a framed moduli space.
Nilpotency caveat
The fibre is the nilpotent locus inside
total_space(ambient(fibre(X, tau))); thatnilpotency is recorded, not computed. (#33's
dimension_nullcone/nullcone_motiveare thenatural way to compute it, which a follow-up can wire into
NilpotentLocus.)Testing
fourfold, and its five fibre types (Lemmas 18-22) have the local dimension vectors and
local framing data stated in the paper.
local_structure, and the publiclocal_quiver_setting.