Computing the codimension of the variety defined by a system of quadratic forms

Suppose I have an $$m \times n$$ matrix $$L$$, where $$m \leq n$$ and each entry is $$L_{i,j}(x_1, ..., x_s)$$ which is a linear form over $$\mathbb{C}$$. Let $$\mathbf{y} = (y_1, \ldots, y_n)$$. Let us consider the system of quadratic equations $$L. \mathbf{y} = \mathbf{0};$$ this is a system of $$m$$ quadratic equations and let $$V \subseteq \mathbb{C}^{s+n}$$ be the affine variety defined by these equations. I am interested in computing the codimension of $$V$$. Since the equations have a special form (that it can be expressed like above) I thought maybe there would be an easier way to compute the codimension... Since I'm not really sure, I would appreciate any comments on how one can go about on computing the codimension of $$V$$. Thank you.

PS I would like to also assume that $$L$$ has rank $$m$$ (full rank) as a matrix over the field $$\mathbb{C}(x_1, ..., x_n)$$.

• Without knowing something more specific, it is hard to give a general answer. However, if you are interested in a particular example, you might try Macaulay2, Magma or equivalent computer algebra softwares. Working over finite field with very large characteristic can help you a lot, if $s+n$ is quite large. – Enrico Apr 21 at 19:39
• If you have no control over the rank of $L$, the codimension can vary wildly. – Mohan Apr 21 at 19:48
• @Mohan I added extra assumption to deal with this. Thank you. – Johnny T. Apr 21 at 20:54
• The assumption you make is insufficient. Consider $L$ to be the $2\times 2$ diagonal matrix with $x_1$s in the diagonal, then you get a codimension one variety. Instead if you take the matrix to be diagonal with $x_1,x_2$ in the diagonal, you get a codimension 2 subvariety, – Mohan Apr 21 at 23:55