Hello,
I noticed something in the calculation of the sheaf cohomology of a
toric divisor in sage/schemes/toric/divisor.
The method _sheaf_complex() is supposed to return an abstract complex,
that coincides with the definition of V_D,m from Cox, Little, Schenck:
"Toric Varieties" [CLS2011] (page 402):
V_D,m = U_{σ∈Σ} Conv(u_ρ|ρ∈σ(1), ⟨m,u_ρ⟩ < -a_ρ)
But the current implementation does this
fan=self.parent().scheme().fan()
ray_is_negative=[m *ray+self.coefficient(i) <0
fori, rayinenumerate(fan.rays())]
defcone_is_negative(cone): # and non-trivial
ifcone.is_trivial():
returnFalse
returnall(ray_is_negative[i] foriincone.ambient_ray_indices())
negative_cones=[coneforconeinflatten(fan.cones()) ifcone_is_negative(cone)]
returnSimplicialComplex([c.ambient_ray_indices() forcinnegative_cones])
Here the array negative_cones only consists of cones, where all
rays fulfil ⟨m,u_ρ⟩ < -a_ρ and thus the abstract complex is the union of
those cones. However we are then missing all the faces/convex hulls
coming from a cone, where not all the rays fulfil ⟨m,u_ρ⟩ < -a_ρ.
Thus, in none smooth cases we can build counterexamples like this cone:
cone=Cone([[1,1,1],[1,-1,1], [-1,1,1], [-1,-1,1]])
TorVar=ToricVariety(Fan([cone]))
N=TorVar.fan().lattice()
D=-TorVar.divisor(N(1,1,1)) -TorVar.divisor(N(-1,-1,1))
M=TorVar.fan().dual_lattice()
clpx=D._sheaf_complex(M(0,0,0))
print(D._sheaf_cohomology(clpx))
Output: (0,1,0,0)
Which should not happen, because our toric variety was affine, ie
doesn't has higher cohomology. Similarly, we can extend this problem to
a complete toric variety.
r0, r1, r2, r3, r4=[[0,0,-1],[1,1,1],[1,-1,1], [-1,1,1], [-1,-1,1]]
c0=Cone([r1,r2,r3,r4])
c1=Cone([r0, r1, r2])
c2=Cone([r0, r1, r3])
c3=Cone([r0, r4, r2])
c4=Cone([r0, r4, r3])
TorVar=ToricVariety(Fan([c0, c1, c2, c3, c4]))
N=TorVar.fan().lattice()
D=-TorVar.divisor(N(r1)) -TorVar.divisor(N(r4))
D.cohomology(dim=True)
Output: (0,1,0,0)
I think that one can fix this problem by maybe using something like this:
fan=self.parent().scheme().fan()
#once calculate which rays appear as vertices of the abstract complex
ray_is_negative=[m *ray+self.coefficient(i) <0
fori, rayinenumerate(fan.rays())]
#the set of faces of the abstract complex
simplicial_faces=[]
forconeinflatten(fan.cones()):
#by definition, we take the convex hull of the negative rays for each cone
simplicial_faces.append([ray_indexforray_indexincone.ambient_ray_indices()
ifray_is_negative[ray_index]])
returnSimplicialComplex(simplicial_faces)
Best regards,
Leif Jacob
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