Opened 4 years ago
Last modified 4 years ago
#20511 new defect
Not all empty schemes are created equal
Reported by:  kedlaya  Owned by:  

Priority:  major  Milestone:  sage7.2 
Component:  algebraic geometry  Keywords:  schemes, empty 
Cc:  Merged in:  
Authors:  Kiran Kedlaya  Reviewers:  
Report Upstream:  N/A  Work issues:  
Branch:  u/kedlaya/not_all_empty_schemes_are_created_equal (Commits)  Commit:  8b00ed9ce22c6ca4641a09cfc0bc8ece2bdb66f0 
Dependencies:  #21297  Stopgaps: 
Description
In this example, both schemes are empty, so they should be equal as subschemes of P^{2. However... }
sage: P2.<x,y,z> = ProjectiveSpace(2, QQ) sage: P2.subscheme([x,y^2,z]) == P2.subscheme([x,y,z]) False
While I'm at it, an is_empty
method would be nice. It could be defined as follows:
def self.is_empty(): return (len(self.irreducible_components() == 0)
Change History (6)
comment:1 followup: ↓ 2 Changed 4 years ago by
comment:2 in reply to: ↑ 1 Changed 4 years ago by
Replying to nbruin:
Projective scheme comparison should compare ideals saturated with respect to the irrelevant ideal:
sage: U=P2.subscheme([x,y,z]) sage: V=P2.subscheme([x,y^2,z]) sage: J=U.defining_ideal() sage: U.defining_ideal().saturation(J)[0] == V.defining_ideal().saturation(J)[0] True
Agreed. I guess we don't want to transform the generating set at creation, so we can recover the generators as specified, but maybe we want to cache the saturation?
Similarly, the
is_empty
should do:sage: one_ideal= J^0 #just get the ideal generated by 1 sage: U.defining_ideal().saturation(J)[0] == one_idealdecomposing in irreducible components is a more expensive operation.
Agreed again.
comment:3 Changed 4 years ago by
Also note that if we change equality we also have to change the hash, i.e., the hash should be the hash of the generators of the groebner basis of the saturation of the defining ideal. That'll probably shake out another few bugs out of the doctests.
comment:4 Changed 4 years ago by
 Branch set to u/kedlaya/not_all_empty_schemes_are_created_equal
 Commit set to 433a479a3d98d1fe0916d9ab3bcb06053004551e
Speaking of shaking out bugs, here is a first attempt (minus the hash), which already runs into something further afield:
sage t warnlong 237.1 src/sage/schemes/hyperelliptic_curves/hyperelliptic_padic_field.py ********************************************************************** File "src/sage/schemes/hyperelliptic_curves/hyperelliptic_padic_field.py", line 809, in sage.schemes.hyperelliptic_curves.hyperelliptic_padic_field.HyperellipticCurve_padic_field.coleman_integral Failed example: HK.coleman_integral(w,S,P) Exception raised: Traceback (most recent call last): File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/doctest/forker.py", line 495, in _run self.compile_and_execute(example, compiler, test.globs) File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/doctest/forker.py", line 858, in compile_and_execute exec(compiled, globs) File "<doctest sage.schemes.hyperelliptic_curves.hyperelliptic_padic_field.HyperellipticCurve_padic_field.coleman_integral[70]>", line 1, in <module> HK.coleman_integral(w,S,P) File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/schemes/hyperelliptic_curves/hyperelliptic_padic_field.py", line 837, in coleman_integral basis_values = self.coleman_integrals_on_basis(P, Q, algorithm) File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/schemes/hyperelliptic_curves/hyperelliptic_padic_field.py", line 621, in coleman_integrals_on_basis M_frob, forms = self._frob_calc = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(self) File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/schemes/hyperelliptic_curves/monsky_washnitzer.py", line 1799, in matrix_of_frobenius_hyperelliptic S = SpecialHyperellipticQuotientRing(Q, extra_prec_ring, True) File "sage/misc/classcall_metaclass.pyx", line 330, in sage.misc.classcall_metaclass.ClasscallMetaclass.__call__ (/projects/b8cc019c120444b1bea9eb81c119388e/sage/src/build/cythonized/sage/misc/classcall_m etaclass.c:1251) return cls.classcall(cls, *args, **kwds) File "sage/misc/cachefunc.pyx", line 1057, in sage.misc.cachefunc.CachedFunction.__call__ (/projects/b8cc019c120444b1bea9eb81c119388e/sage/src/build/cythonized/sage/misc/cachefunc.c:5558) return self.cache[k] File "sage/misc/weak_dict.pyx", line 874, in sage.misc.weak_dict.WeakValueDictionary.__getitem__ (/projects/b8cc019c120444b1bea9eb81c119388e/sage/src/build/cythonized/sage/misc/weak_dict.c:3905) cdef PyObject* wr = PyDict_GetItemWithError(self, k) File "sage/misc/weak_dict.pyx", line 150, in sage.misc.weak_dict.PyDict_GetItemWithError (/projects/b8cc019c120444b1bea9eb81c119388e/sage/src/build/cythonized/sage/misc/weak_dict.c:1259) ep = mp.ma_lookup(mp, <PyObject*><void*>key, PyObject_Hash(key)) File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/schemes/generic/algebraic_scheme.py", line 2299, in __eq__ return(self.saturated_defining_ideal() == other.saturated_defining_ideal()) File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/schemes/generic/algebraic_scheme.py", line 2280, in saturated_defining_ideal I3 = I1.saturation(I2)[0] File "/projects/b8cc019c120444b1bea9eb81c119388e/sage/local/lib/python2.7/sitepackages/sage/rings/polynomial/multi_polynomial_ideal.py", line 2083, in saturation ideal, expo = sat(self, other) File "sage/libs/singular/function.pyx", line 1319, in sage.libs.singular.function.SingularFunction.__call__ (/projects/b8cc019c120444b1bea9eb81c119388e/sage/src/build/cythonized/sage/libs/singular/functio n.cpp:14767) raise TypeError("Cannot call Singular function '%s' with ring parameter of type '%s'"%(self._name,type(ring))) TypeError: Cannot call Singular function 'sat' with ring parameter of type '<class 'sage.rings.polynomial.multi_polynomial_ring.MPolynomialRing_polydict_domain_with_category'>' **********************************************************************
So it seems that there is something wrong with the definition of saturation
in sage/rings/polynomial/multi_polynomial_ideal.py
.
New commits:
433a479  Correct equality testing for projective schemes

comment:5 Changed 4 years ago by
 Dependencies set to #21297
It might be that dealing with hashing will help with this. But hashing ideals itself need to be fixed; see #21297 (which itself has dependencies on hashing for polynomials).
comment:6 Changed 4 years ago by
 Commit changed from 433a479a3d98d1fe0916d9ab3bcb06053004551e to 8b00ed9ce22c6ca4641a09cfc0bc8ece2bdb66f0
Branch pushed to git repo; I updated commit sha1. New commits:
8b00ed9  Extra commit, not sure why

Projective scheme comparison should compare ideals saturated with respect to the irrelevant ideal:
Similarly, the
is_empty
should do:decomposing in irreducible components is a more expensive operation.