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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 $\mathbb{C}(x_1, ..., x_n)$.

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    $\begingroup$ 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. $\endgroup$ – Enrico Apr 21 at 19:39
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    $\begingroup$ If you have no control over the rank of $L$, the codimension can vary wildly. $\endgroup$ – Mohan Apr 21 at 19:48
  • $\begingroup$ @Mohan I added extra assumption to deal with this. Thank you. $\endgroup$ – Johnny T. Apr 21 at 20:54
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    $\begingroup$ 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, $\endgroup$ – Mohan Apr 21 at 23:55

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