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Problem Set 14 A quick note on degree. Let V be an ane variety of dimension p. If you
 

Summary: Problem Set 14
A quick note on degree. Let V be an ane variety of dimension p. If you
intersect V with a "random" hyperplane then you will get a variety of dimension
p-1. If you intersect V with p random hyperplanes then you should get something
which is zero dimensional. On the level of ideals, if you take the union of I(V )
with the ideal of p random linear forms then you will get an ideal J. The degree
of V is dened to be dimk(R/J). We can compute the dimension of V by
determining how many hyperplane sections are needed to reduce V to a set of
points.
Suppose V is a smooth irreducible curve in A3
. To each point P V we
can glue the tangent line to V at P. In this manner, we get a surface in A3
called the tangent variety of V . Here is one strategy for computing the equation
of such a surface from I(V ). Let I(V ) k[x, y, z]. Construct an ideal J in
k[A, B, C, x, y, z] that contains all of the information of all the tangent lines at
each choice of (A, B, C) V . This can be done by the following steps:
1) Construct the Jacobian matrix, Jac(I), and multiply it by the transpose of
the matrix [x - A, y - B, z - C]. Take all of the entries of the resulting product
and form an ideal L.
2) Form the ideal I by taking the generators of I and replacing x with A, y

  

Source: Abo, Hirotachi - Department of Mathematics, University of Idaho

 

Collections: Mathematics