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feat: Bocklandt reduction and smoothness of quiver moduli#33

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feat: Bocklandt reduction and smoothness of quiver moduli#33
pbelmans wants to merge 23 commits into
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bocklandt

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@pbelmans pbelmans commented Jul 9, 2026

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Implements the reduction algorithm of Bocklandt (MR1929191) and the classification of cofree quiver settings by Bocklandt and Van de Weyer (doi:10.1016/j.jalgebra.2007.08.019), and uses them for the geometry of moduli spaces:

  • strongly_connected_components for quivers,
  • bocklandt_reduction and is_coregular for quiver settings, with every reduction step documented against the paper,
  • is_smooth for moduli spaces with properly semistable representations, by checking coregularity of the local quiver setting of every Luna type (MR1972892),
  • codimension_singular_locus for moduli spaces, as the singular locus is a union of Luna strata,
  • is_cofree for quiver settings, via wedging, prime components and the classification lists,
  • nullcone_motive, dimension_nullcone and defect: the motive of the nullcone is a polynomial in the Lefschetz motive, computed by the socle recursion of Gösmann--Reineke (doi:10.3842/SIGMA.2026.020); cofree = coregular + zero defect (Popov) is verified on 100 settings in the test suite,
  • fibre_dimension, is_flat and is_semismall for projections to walls, computed through nilpotent semistable moduli of local quiver settings; the source stability parameter is required to be King-normalized.

Tests include the ten nodes of the Segre cubic, the accidental isomorphism to Gr(2, 4) for the 6-subspace quiver on a wall together with its fibre dimensions and semismallness, and P^5 for the 3-Kronecker quiver with d = (2, 2) and d = (2, 4).

@pbelmans

pbelmans commented Jul 9, 2026

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@giannipetrella This is a Claude-assisted implementation of Raf's smoothness check. Let's review this carefully, and then we have another very useful feature ready.

pbelmans added 18 commits July 9, 2026 08:46
The Groebner computation over the quasiprimitive cycle traces cuts out a
variety that can strictly contain the nullcone; the socle recursion of
Goesmann--Reineke (doi:10.3842/SIGMA.2026.020) is exact and much faster.
The quasiprimitive cycle traces do not generate the ring of invariants, so
the criterion is unreliable, and infeasible on the settings where it is not.
A classification-based check (Bocklandt for symmetric settings, Joo for thin
ones, Le Bruyn--Teranishi for matrix invariants) can replace it later.
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