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INPUT:
- l -- a list
EXAMPLES::
sage: from admcycles.stratarecursion import classes
sage: classes([])
[]
sage: classes([4,4,3,3,3,1,1])
[[0, 1], [2, 3, 4], [5, 6]]
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Return a list of partitions of the set S into the disjoint union of s sets.
A partition is a list of sets.
EXAMPLES::
sage: from admcycles.stratarecursion import SetPart
sage: S = set(['b','o','f'])
sage: SetPart(S,2)
[[set(), {'b', 'f', 'o'}],
...
[{'b', 'f', 'o'}, set()]]
sage: SetPart(S,1)
[[{'b', 'f', 'o'}]]
sage: SetPart(S,-1)
[]
sage: SetPart(S,0)
[]
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Returns the fundamental class of the closure of the stratum of k-differentials in genus g in Mbar_{g,n}
with vanishing and pole orders mu.
The class is computed using a formula from papers by Farkas-Pandharipande and Schmitt,
proven by [HoSc]_ and [BHPSS]_.
The formula for differentials with residue conditions is based on unpublished work
relying on [BaChGeGrMo]_.
If the mu is of spin type, we can compute the spin stratum class H^+ - H^-. In [CSS21]_, Costantini,
Sauvaget and Schmitt made a similar conjecture for the Pixton formula.
INPUT:
- ``g`` -- integer ; genus of the curves
- ``k`` -- integer ; power of the canonical line bundle in definition of stratum
- ``mu`` -- tuple ; tuple of integers of length n giving zero and pole multiplicities
of k-differential, required to sum up to k*(2g-2)
- ``virt`` -- bool (default: `False`); if True, k=1 and all entries of mu nonnegative, this
computes the virtual codimension g class supported on the codimension g-1 stratum of
holomorphic 1-differentials.
- ``res_cond`` -- tuple (default: `()`); tuple of residue conditions. Each entry of
res_cond can be of one of the following two types:
- an integer i from 1, ..., n indicating a marking pi with mu[i]<0, such that the
differential eta is required to have a pole with vanishing residue at the marking.
- a tuple (c1, ..., cn) of rational numbers indicating that a condition
c1 * Res_{p1}(eta) + ... + cn * Res_{pn}(eta) = 0
is imposed. Currently only implemented for ci in {0,1}.
The function then computes the class of the closure of the locus of smooth curves
having such a differential eta. Currently only implemented for k=1.
- ``xi_power`` -- integer (default: `0`); if positive, returns the pushforward of
the corresponding power of the first Chern class xi = c_1(O(-1)) of the tautological
bundle on the moduli space of multi-scale differentials from [BCGGM3].
Currently only implemented for k=1 and with method = 'diffstrata'.
- ``method`` -- string (default: `'pull'`); when computing a stratum of 1-differentials
with residue conditions, there are two choices here: 'pull' will compute it via boundary
pullbacks of higher genus strata, 'diffstrata' will use the package `diffstrata`, which
iteratively replaces residue conditions with equivalent divisorial conditions. The two
results should be equal.
- ``spin`` -- bool (default: `False`); if true, we will compute the spin stratum class
- ``spin_conj`` -- bool (default: `False`); if we assume the conjecture in [CSS21]_ to be true,
we will compute the spin stratum class by first compute the holomorphic strata of the same genus,
then use the conjecture to recursively compute the input stratum class
WARNING::
Imposing residue conditions at poles of high order leads to very long computations,
since the method works by computing strata of differentials on high-genus moduli
spaces.
TESTS::
sage: from admcycles import Hyperell, Biell
sage: from admcycles.stratarecursion import Strataclass
sage: L=Strataclass(2,1,(3,-1)); L.is_zero()
True
sage: L=Strataclass(3,1,(5,-1)); L.is_zero() # doctest: +SKIP
True
sage: L=Strataclass(2,1,(2,)); (L-Hyperell(2,1)).is_zero()
True
In g=2, the locus Hbar_2(2,2) decomposes into the codimension 1 set of Hbar_2(1,1) and the
codimension 2 set of curves (C,p,q) with p,q Weierstrass points. The latter is equal to the cycle
Hyperell(2,2). We can obtain it by subtracting the virtual cycle for the partition (1,1) from the
virtual cycle for the partition (2,2)::
sage: H1 = Strataclass(2, 2, (2, 2), virt = True)
sage: H2 = Strataclass(2, 1, (1, 1), virt = True)
sage: T = H1 - H2 - Hyperell(2, 2)
sage: T.is_zero()
True
In g=1, the locus Hbar_1(2,-2) is the locus of curves (E,p,q) with p-q being 2-torsion in E.
Equivalently, this is the locus of bielliptic curves with a pair of bielliptic conjugate points::
sage: (Strataclass(1,1,(2,-2)) - Biell(1,0,1)).is_zero()
True
Some tests of computations involving residue conditions::
sage: from admcycles import Strataclass
sage: OmegaR = Strataclass(1,1,(6,-4,-2),res_cond=(3,))
sage: OmegaRalt = Strataclass(1,1,(6,-4,-2),res_cond=(2,)) # long time
sage: (OmegaR - OmegaRalt).is_zero() # long time
True
sage: (OmegaR.forgetful_pushforward([2,3])).fund_evaluate()
42
sage: a=4; (a+2)**2 + a**2 - 10 # formula from [Castorena-Gendron, Cor. 5.5]
42
sage: OmegaR2 = Strataclass(1,1,(4,-2,-2),res_cond=(3,))
sage: (OmegaR2.forgetful_pushforward([2,3])).fund_evaluate()
10
sage: OmegaR3 = Strataclass(1,1,(5,-3,-2),res_cond=(2,)) # not tested
sage: (OmegaR3.forgetful_pushforward([2,3])).fund_evaluate() # not tested
24
sage: a=3; (a+2)**2 + a**2 - 10 # formula from [Castorena-Gendron, Cor. 5.5] # not tested
24
sage: OmegaR5 = Strataclass(2,1,(5,-1,-2),res_cond=(3,)) # not tested
sage: OmegaR5.is_zero() # vanishes by residue theorem # not tested
True
We can also check that the two ways of computing residue conditions (via pullbacks and
via the package diffstrata) coincide::
sage: a = Strataclass(1,1,(4,-2,-2), res_cond=(2,))
sage: b = Strataclass(1,1,(4,-2,-2), res_cond=(2,), method='diffstrata')
sage: (a-b).is_zero()
True
The following computes the locus of genus 1 curves admitting a differential with multiplicity
vector (8,-2,-2,-2,-2) at the markings such that the sum of residues at p2 and p3 equals zero::
sage: c = Strataclass(1,1,(8,-2,-2,-2,-2), res_cond=((0,1,1,0,0),))
Using the parameter xi_power, we can observe an interesting relationship between strata with
xi-insertions and higher terms in Pixton's formula of the double ramification cycle::
sage: from admcycles import DR_cycle
sage: g=0; mu = [11,4,-8,-2,-5,-2]; A = [a+1 for a in mu]
sage: n = len(mu); adddeg = 1
sage: D = DR_cycle(g,A,rpoly=True,d=g+adddeg,chiodo_coeff=True)
sage: v = vector(QQ,[a[adddeg] for a in D.basis_vector()])
sage: S = Strataclass(g,1,mu,xi_power=adddeg)
sage: print(v); print(-S.basis_vector())
(0, 0, -4, 7, 2, 4, 2, 0, -1, 0, 0, 0, 0, 2, 0, 2)
(0, 0, -4, 7, 2, 4, 2, 0, -1, 0, 0, 0, 0, 2, 0, 2)
We can use the function to compute spin stratum class, with or without the assumption of the
conjecture of spin Pixton formula::
sage: cl1=Strataclass(2,1,(8,-4,-2),spin=True)
sage: cl1.basis_vector()
(-1107/2, 369/2, -246, -552, -899/2, 288, 590, 807/2, 1254, 1167/2, 1947/2, 613, -793, -1354, 351, -615, -666, 939/2, -563, -2077/2, 1302, 1107/2, -2583/2, 369/2, 21, -165, -181, -369/2, -201/2, 201/2, 551/2, -31/2, -155/2, -203/2, 18, -18, 114, 119/2, -58, 102, -102, -109/2, -76, 142)
sage: cl2=Strataclass(2,1,(8,-4,-2),spin=True,spin_conj=True)
sage: cl1==cl2
True
If we assume the conjecture above, we can also compute spin strata of meromorphic k-differentials
for k > 1. Below we check that they restrict under a boundary gluing map as expected::
sage: from admcycles import StableGraph
sage: from admcycles.admcycles import prodtautclass
sage: H = Strataclass(2,3,[10,-4],spin=True,spin_conj=True)
sage: gamma = StableGraph([1,1],[[1,3],[2,4]],[(3, 4)])
sage: pb = gamma.boundary_pullback(H)
sage: H1 = Strataclass(1,3,[10,-10],spin=True,spin_conj=True)
sage: H2 = Strataclass(1,3,[-4,4],spin=True,spin_conj=True)
sage: prot = prodtautclass(gamma, protaut=[H1, H2])
sage: pb.totensorTautbasis(2) == prot.totensorTautbasis(2)
True
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Returns sum of boundary terms in Conjecture A for entries of bdry_list.
These entries have the format from the output of twistenum, but might be a subset of these.
INPUT:
- g (integer): genus
- k (integer): power of canonical bundle
- bdry_list (list, default=None): a sublist of all the twisted stable graphs; if not
specified, it will just use the whole list
- termsout (bool, default=False): if true, then the output will not be simplified
- spin (bool, default=False): if true, then compute the spin strata boundary sum
- spin_conj (bool, default=False): if true, then we will assume the spin Conjecture A
to be true and the summand will be computed by recursion on this formula
EXAMPLES::
sage: from admcycles.stratarecursion import Strataboundarysum
sage: from admcycles.double_ramification_cycle import DR_cycle
sage: Strataboundarysum(2,1,(6,-4),spin=True).basis_vector()
(-139, 46, -109, -99, 219, 194, 174, 136, -318, 91/2, -45/2, -15, -46, -25/2)
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Return number of automorphisms of gr leaving the twist I on gr invariant.
EXAMPLES::
sage: from admcycles import StableGraph
sage: from admcycles.stratarecursion import automorphism_number_fixing_twist
sage: gr = StableGraph([0,0],[[1,2,3],[4,5,6]],[(1,2),(3,4),(5,6)])
sage: twist = {i:0 for i in range(7)}
sage: automorphism_number_fixing_twist(gr,twist)
8
sage: twist[1] = 1
sage: automorphism_number_fixing_twist(gr,twist)
2
sage: twist[2] = 1
sage: automorphism_number_fixing_twist(gr,twist)
4
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