ag.algebraic geometry - How to compute the dimension of a linear system on $mathbb{P}^n$

Your question is equivalent to the computation of $H^0(mathcal{I}_S(2))$.

In the example you give, $S$ is a complete intersection of $4$ quadrics and so the resolution of its ideal sheaf $mathcal{I}_S$ is given by the Koszul complex (I write $mathcal{O}$ instead of $mathcal{O}_{mathbb{P}^9}$):

$0 to mathcal{O}(-8) to mathcal{O}(-6)^{oplus 4} to mathcal{O}(-4)^{oplus 6} to mathcal{O}(-2)^{oplus 4} to mathcal{I}_S to 0$.

Tensoring with $mathcal{O}(2)$ we obtain:

$0 to mathcal{O}(-6) to mathcal{O}(-4)^{oplus 4} to mathcal{O}(-2)^{oplus 6} to mathcal{O}^{oplus 4} to mathcal{I}_S(2) to 0$.

Splitting this exact sequence into short exact ones it is immediate to check that

$H^0(mathcal{I}_S(2))=H^0(mathcal{O}^{oplus 4})=4$,

as Algori states in his comment.

Therefore the linear system of quadrics passing through $S$ has dimension $4-1=3$.

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