if a subvariety of codimension n is given by an ideal of polynomials with n generators, then the homology class of the variety is given by the intersection product of the classes of the individual generators' zero set. In this case, all the (zero set of the) individual generators intersect transversally.
The example I have in mind is something like the zero locus of $ \langle z_0, z_1 \rangle$ inside $\mathbb{P}^3$, where the z_i are the usual projective coordinates. Then the class of this zero locus should be just $[z_0] \cdot [z_1] = H^2$, where H is the hyperplane class.
However, what is the situation if the ideal has more than n generators? An example would be something like $\langle a \, d - b \, c , \, a\,x + b\,y , \, c\,x + d\,y \rangle$ in a space where $a,b,c,d,x,y$ are sections of some line bundles. How can I express the class of this variety in terms of the classes of the sections?