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

naturemath

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?