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 Collections: Mathematics