Concerning the isomorphism of complex varieties.
markz...@yahoo.com
n. Is it decidable whether V and V' are isomorphic as algebraic
varieties? If this is not so, what is the smallest n-for which the
answer to this question is known? It seems to me that for n=1, the
question is yes.
Thanks a bundle.
Mariano Suárez-Alvarez
A (reduced) variety in C^1 is simply a set of points,
and two of them are isomorphic iff they have the
same number of points.
If each of the two varieties is given as the common
set of zeros of a few polynomials in C[X], taking
the GCD, you can give each of the two by a single
polynomial. The varieties being reduced implies that
the polynomials are square-free. Therefore the number
of points is simply the degree of the defining
polynomials.
So, more or less independently of what you mean
by `deciding', the answer for n=1 should be yes.
In the general case, you'd need to precise that
meaning though.
-- m
markz...@yahoo.com
Yes, for n=1 the answer is yes, as I stated. However, the question
is: If two algebraic varieties V and W are given in K^n (asume both
algebraic varieties are equidimensional of dimension other than zero),
where K is the complex field or some algebraically closed field, is
there an algorithm that can tell you that V is isomorphic to W, and if
such a general algorithm does not exist, what is the smallest n, other
than one, for which it exists?
Lee Rudolph
>Yes, for n=1 the answer is yes, as I stated. However, the question
>is: If two algebraic varieties V and W are given in K^n (asume both
>algebraic varieties are equidimensional of dimension other than zero),
>where K is the complex field or some algebraically closed field, is
>there an algorithm that can tell you that V is isomorphic to W, and if
>such a general algorithm does not exist, what is the smallest n, other
>than one, for which it exists?
Every finite simplicial complex of dimension k is homotopy-equivalent
to a complex algebraic variety of complex dimension k (with its
transcendental topology) consisting of an arrangement of affine
complex k-planes whose combinatorics mimicks that of the simplicial
complex; this variety will surely embed in affine (2k+1)-space, if
not in lower dimension. The unsolvability of the isomorphism problem
for group presentations implies the unsolvability of the homotopy-equivalence
problem (and _a fortiori_ the homeomorphism problem) for finite simplicial
complexes of dimension 2 or greater, and thus the unsolvability of the
algebraic isomorphism for arrangements of affine complex 2-planes
in complex 5-space.
Do you, perhaps, want your varieties to be non-singular, or irreducible?
Lee Rudolph
Jannick Asmus
> On Mar 30, 10:30 am, Mariano Suárez-Alvarez
> <mariano.suarezalva...@gmail.com> wrote:
>> On 27 mar, 17:30, markzorr...@yahoo.com wrote:
>>
>>> Question: Let V and V' be two complex varieties in C^n for some fixed
>>> n. Is it decidable whether V and V' are isomorphic as algebraic
>>> varieties? If this is not so, what is the smallest n-for which the
>>> answer to this question is known? It seems to me that for n=1, the
>>> question is yes.
>> A (reduced) variety in C^1 is simply a set of points,
unless it is equal to C^1
>> and two of them are isomorphic iff they have the
>> same number of points.
[snip]
... since they are affine and the coordinate rings are n-fold products
of C where n is the number of points. This generalizes to the case where
C is substituted by an arbitrary field K which is not necessarily
algebraically closed. Then the local rings of V and V', resp., are
fields finite over K.
> Yes, for n=1 the answer is yes, as I stated. However, the question
> is: If two algebraic varieties V and W are given in K^n (asume both
> algebraic varieties are equidimensional of dimension other than zero),
> where K is the complex field or some algebraically closed field, is
> there an algorithm that can tell you that V is isomorphic to W, and if
> such a general algorithm does not exist, what is the smallest n, other
> than one, for which it exists?
For general n>0 we can translate the problem, since the varieties are
affine: If I(V) and I(W) denote the defining ideals of V and W, resp.,
in R=C[X1,...,Xn] when are the quotient rings R/I(V) and R/I(W)
isomorphic over C. As there is a structure theory of the quotient rings
with dimension 0, the result could be derived (see above).
The ideals I(V) and I(W) are finitely generated which gives the chance
that there is an algorithm. So a research could start from here.
What could be investigated is are the following questions: Are the
following objects assigned to V and W,resp., isomorphic and are these
isomorphisms compatible to each other:
- function field
- normalization
- projective closure
- ... many more
On this very general level this certainly generalizes to an arbitrary
field K instead of C.
--
Best wishes,
J.
Torsten Ekedahl
I don't think so. An algebraic isomorphism of such a configuration will take
irreducible components to irreducible components and hence, it seems to me,
will induce a combinatorial isomorphism between the complexes (in the
strictest sense, no subdivisions even). This is certainly decidable (only a
finite number of possibilities).
Note that to start with one should specify what one means by algorithm, the
complex field is in no sense algorithmic in the usual sense. One can
probably use Smale et al's notion of real Turing machine (or some more
algebraic version of it). It was some since I looked at that theory but I
believe there are some undecidability results which are possible relevant.
Alternatively one can use the algebraic closure of Q which can be made a
recursively computable field.
markz...@yahoo.com
> Lee Rudolph wrote:
Those interested might want to look at the following paper:
http://www.springerlink.com/content/am1xwaxajh1b145y/fulltext.pdf
which can be downloaded from here:
http://www.springerlink.com/content/am1xwaxajh1b145y/
Abstract. We show the undecidability of the question of isomorphism
of forms over a polynomial ring $ R[t_1,\ldots,t_n] $, assuming a
hypothesis about units in certain quaternion rings. Assuming this, it
follows that isomorphisms of modules, and of affine algebraic
varieties over R are undecidable.
Mathematics Subject Classification (2000): 03D40; 20F10,
14A10,14A99,11U05
Keywords: - Quaternion algebra - equivalence of forms - diophantine
decidability - regular equivalence of affine algebraic varieties