feat: Bocklandt reduction and smoothness of quiver moduli

docs/src/methods/quiver-moduli.md

@@ -26,6 +26,10 @@ Black-box methods are provided to study some of their properties.
 is_nonempty
 dimension
 is_smooth
+codimension_singular_locus
+fibre_dimension
+is_flat
+is_semismall
 is_projective
 index
 motive

docs/src/methods/quivers.md

@@ -22,6 +22,7 @@ indegree
 outdegree
 is_acyclic
 is_connected
+strongly_connected_components
 is_sink
 is_source
 underlying_graph

docs/src/methods/representation-theory.md

@@ -18,3 +18,44 @@ canonical_decomposition
 in_fundamental_domain
 first_hochschild_cohomology
 ```
+
+## Invariant theory of quiver representations
+
+The affine quotient of the representation variety by the base change group
+parametrizes semisimple representations of the quiver, and its ring of functions is
+the ring of invariants, generated by traces along oriented cycles.
+
+A quiver setting is *coregular* if this ring of invariants is a polynomial ring, or
+equivalently if the affine quotient is smooth (in which case it is an affine space).
+This is decided by the reduction algorithm of
+[[Bocklandt](https://mathscinet.ams.org/mathscinet/relay-station?mr=1929191)],
+which simplifies a quiver setting without changing its invariant theory and then
+compares the result against a short list.
+
+A stronger property is *cofreeness*: the coordinate ring of the representation
+variety is a graded free module over the ring of invariants, which by a criterion of
+Popov amounts to coregularity together with equidimensionality of the nullcone.
+Cofree quiver settings are classified by
+[[Bocklandt--Van de Weyer](https://doi.org/10.1016/j.jalgebra.2007.08.019)].
+
+Beyond deciding smoothness of the affine quotient itself, these notions drive the
+study of moduli spaces of quiver representations: étale-locally around a polystable
+representation, a moduli space is the affine quotient of a *local quiver setting*, so
+coregularity of local quiver settings decides smoothness of moduli spaces; see
+[`is_smooth`](@ref) and [`codimension_singular_locus`](@ref).
+
+The *nullcone* is the locus of nilpotent representations, i.e., the fibre of the
+quotient map over the image of the zero representation. Its class in the Grothendieck
+ring of varieties is a polynomial in the Lefschetz motive, by
+[[Gösmann--Reineke](https://doi.org/10.3842/SIGMA.2026.020)], and its dimension
+controls the failure of equidimensionality of the quotient map, measured by the
+*defect*.
+
+```@docs
+bocklandt_reduction
+is_coregular
+is_cofree
+nullcone_motive
+dimension_nullcone
+defect
+```