diff --git a/src/moduli.jl b/src/moduli.jl index 4b43c1c..f1449db 100644 --- a/src/moduli.jl +++ b/src/moduli.jl @@ -53,7 +53,7 @@ end is_coprime(M::QuiverModuli) Checks if the stability parameter is coprime with the dimension vector, -i.e., if for all subdimension vectors `e` of `d`, \$\\theta\\cdot e \\neq 0\$. +i.e., if for all subdimension vectors ``e`` of ``d``, ``\\theta\\cdot e \\neq 0``. INPUT: - `M::QuiverModuli`: a moduli space or stack of representations of a quiver. @@ -588,7 +588,7 @@ function solve(A, b) end """ -Cardinality of general linear group \$\\mathrm{GL}_n(\\mathbb{F}_v)\$. +Cardinality of general linear group ``\\mathrm{GL}_n(\\mathbb{F}_v)``. """ @memoize Dict function CardinalGl(n::Int, q) if n == 0 @@ -599,7 +599,7 @@ Cardinality of general linear group \$\\mathrm{GL}_n(\\mathbb{F}_v)\$. end """ -Cardinality of representation space \$\\mathrm{R}(Q,d)\$, over \$\\mathbb{F}_q\$. +Cardinality of representation space ``\\mathrm{R}(Q,d), over \\mathbb{F}_q``. """ function CardinalRd(Q::Quiver, d::AbstractVector{Int}, q) return q^sum( @@ -609,14 +609,17 @@ function CardinalRd(Q::Quiver, d::AbstractVector{Int}, q) end """ -Cardinality of product of general linear groups \$\\mathrm{GL}_{d}(\\mathbb{F}_q)\$. +Cardinality of product of general linear groups ``\\mathrm{GL}_{d}(\\mathbb{F}_q)``. """ @memoize Dict function CardinalGd(d::AbstractVector{Int}, q) return prod(CardinalGl(di, q) for di in d) end -"""Entry of the transfer matrix, as per Corollary 6.9""" +""" +Entry of the transfer matrix, as per Corollary 6.9 of +[MR1974891](https://doi.org/10.1007/s00222-002-0273-4) +""" function TransferMatrixEntry(Q, e, f, q) fe = f - e @@ -853,8 +856,8 @@ end Returns the index of the moduli space ``M``. -The index of a variety \$X\$ is the largest which divides the canonical divisor \$K_X\$ -in \$Pic(X)\$. +The index of a variety ``X`` is the largest which divides +the canonical divisor ``K_X`` in ``Pic(X)``. This implementation currently only works for the canonical stability. @@ -1014,7 +1017,6 @@ function Poincare_polynomial(M::QuiverModuliSpace) end -# oh my god function power(x, n::Int) if n >= 0 return x^n @@ -1043,7 +1045,7 @@ INPUT: - ``denom``: a function. Default is the sum. OUTPUT: -- The motive as an element in the function field \$\\mathbb{Q}(L)\$. +- The motive as an element in the function field \\mathbb{Q}(L). EXAMPLES: @@ -1151,7 +1153,7 @@ end Computes the Chow ring of the moduli space of ``\\theta``-semistable representations of ``Q`` with dimension vector ``d``, for a choice of linearization ``a``. -This method of the function Chow_ring also returns the ambient ring \$R\$ +This method of the function Chow_ring also returns the ambient ring ``R`` and the inclusion morphism. INPUT: @@ -1164,7 +1166,7 @@ OUTPUT: A tuple containing: - the Chow ring of the moduli space, - the polynomial ring above it, -- the inclusion map \$\\iota : A \\to R\$. +- the inclusion map ``\\iota : A \\to R``. EXAMPLES: @@ -1366,7 +1368,7 @@ end Returns the first Chern class of the line bundle L(eta). -This is given by \$L(eta) = \\bigoplus_{i \\in Q_0} \\det(U_i)^{-eta_i}\$. +This is given by ``L(eta) = \\bigoplus_{i \\in Q_0} \\det(U_i)^{-eta_i}``. INPUT: - ``M``: a moduli space of representations of a quiver. @@ -1377,7 +1379,7 @@ OUTPUT: EXAMPLES: -The line bundles \$\\mathcal{O}(i)\$ on the projective line: +The line bundles ``\\mathcal{O}(i)`` on the projective line: ```jldoctest julia> Q = mKronecker_quiver(2); M = QuiverModuliSpace(Q, [1, 1]); @@ -1447,15 +1449,15 @@ end """ total_Chern_class_universal(M::QuiverModuliSpace, i, chi) -Returns the total Chern class of the universal bundle \$U_i(\\chi)\$. +Returns the total Chern class of the universal bundle ``U_i(\\chi)``. INPUT: - ``M``: a moduli space of representations of a quiver. - ``i``: the universal bundle we want the Chern class of. -- ``chi``: a choice of linearization to construct \$U_i(\\chi)\$. +- ``chi``: a choice of linearization to construct ``U_i(\\chi)``. OUTPUT: -- the total Chern class of the universal bundle \$U_i(\\chi)\$. +- the total Chern class of the universal bundle ``U_i(\\chi)``. EXAMPLES: @@ -1561,7 +1563,7 @@ julia> Todd_class(M) ) """ - We call the series \$Q(t) = t/(1-e^{-t})\$ the Todd generating series. + We call the series ``Q(t) = t/(1-e^{-t})`` the Todd generating series. The function computes the terms of this series up to degree n. We use this instead of the more conventional notation `Q` to avoid a clash with the notation for the quiver. @@ -1620,7 +1622,8 @@ julia> Todd_class(M) num = gens(preimage(inclusion, Ideal(R, num)))[1] den = gens(preimage(inclusion, Ideal(R, den)))[1] - # renormalizing the constant term because Singular is silly like that + # renormalizing the constant term because it should be 1, but Singular does not keep + # it fixed. num /= constant_coefficient(num) den /= constant_coefficient(den) @@ -1648,7 +1651,7 @@ the integral of `f`. EXAMPLES: -The integral of \$\\mathcal{O}(i)\$ on the projective line for some `i`s. +The integral of ``\\mathcal{O}(i)`` on the projective line for some `i`s. ```jldoctest julia> Q = mKronecker_quiver(2); M = QuiverModuliSpace(Q, [1, 1]); @@ -1736,8 +1739,8 @@ Returns the "pseudodegree" of the monomial `f` in the Chow ring of the moduli space `M` passed. This method is unsafe, as it does not consider the actual degree of the MPolyRingElem -objects passed. Instead, it assumes that the Chow ring passed has variables \$x_{i, j}\$ -as in the Chow ring paper. +objects passed. Instead, it assumes that the Chow ring passed has variables +``x_{i, j}`` as in the Chow ring paper. """ function __Chow_ring_monomial_grading(M::QuiverModuliSpace, f) return __Chow_degrees(M.d)' * collect(Singular.exponent_vectors(f))[1] @@ -1761,7 +1764,7 @@ end Returns the dimension of the moduli stack. This differs from the dimension of the moduli space by 1, as we do not quotient out -the stabilizer \$ \\mathbb{G}\$. +the stabilizer `` \\mathbb{G}``. INPUT: - ``M``: a moduli stack of representations of a quiver.