Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The intersection of two varieties, each of which is a complete intersection

  1. May 5, 2012 #1
    I posted this problem before here it is with a bit more assumption on the hypothesis.

    Consider f_1,..., f_k, g_1,..., g_l in a polynomial ring C[x_1,...,x_M], where f_i's are homogeneous of degree 2 while g_j's are linear polynomials.

    Suppose the codim of the variety cut out by S_1 = {f_1,..., f_k} is k while the codim of the variety cut out by S_2 = {g_1,..., g_l} is l.

    Assume any or all of the following:
    * {f_i, g_j} is a complete intersection for some i and for some j;
    * {f_i, g_j} is a complete intersection for any i and for any j;
    * S_1 union {g_j} is a complete intersection for any g_j in S_2;
    * {f_i} union S_2 is a complete intersection for any f_i in S_1;

    Isn't there a result somewhere in a commutative algebra book or in some paper which says that the variety cut out by all of S_1 union S_2 is a complete intersection?
     
    Last edited: May 5, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: The intersection of two varieties, each of which is a complete intersection
Loading...