Not all empty schemes are created equal

[kedlaya]

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)

Depends on #21297

Component: algebraic geometry

Keywords: schemes, empty

Author: Kiran Kedlaya

Branch/Commit: u/kedlaya/not_all_empty_schemes_are_created_equal @ 8b00ed9

Issue created by migration from https://trac.sagemath.org/ticket/20511

[nbruin]

comment:1

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

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_ideal

decomposing in irreducible components is a more expensive operation.

[kedlaya]

comment:2

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_ideal

decomposing in irreducible components is a more expensive operation.

Agreed again.

[nbruin]

comment:3

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.

[kedlaya]

Commit: 433a479

[kedlaya]

Branch: u/kedlaya/not_all_empty_schemes_are_created_equal

[kedlaya]

Author: Kiran Kedlaya

[kedlaya]

comment:4

Speaking of shaking out bugs, here is a first attempt (minus the hash), which already runs into something further afield:

sage -t --warn-long 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/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 495, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/projects/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/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/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/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/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/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/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/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/b8cc019c-1204-44b1-bea9-eb81c119388e/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/b8cc019c-1204-44b1-bea9-eb81c119388e/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/b8cc019c-1204-44b1-bea9-eb81c119388e/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/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/src/build/cythonized/sage/misc/weak_dict.c:1259)
        ep = mp.ma_lookup(mp, <PyObject*><void*>key, PyObject_Hash(key))
      File "/projects/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/sage/schemes/generic/algebraic_scheme.py", line 2299, in __eq__
        return(self.saturated_defining_ideal() == other.saturated_defining_ideal())
      File "/projects/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/sage/schemes/generic/algebraic_scheme.py", line 2280, in saturated_defining_ideal
        I3 = I1.saturation(I2)[0]
      File "/projects/b8cc019c-1204-44b1-bea9-eb81c119388e/sage/local/lib/python2.7/site-packages/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/b8cc019c-1204-44b1-bea9-eb81c119388e/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

[kedlaya]

comment:5

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).

[kedlaya]

Dependencies: #21297

[sagetrac-git]

Changed commit from 433a479 to 8b00ed9

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8b00ed9

Extra commit, not sure why