On linear systems of \(\mathbb P^3\) through multiple points
Fix an integer \(d>0\), at most \(8\) general points of \(\mathbb {P}^3\) and consider the linear system \(M\) of all degree \(d\) surfaces with prescribed multiplicities at these points. Here the authors compute \(\dim (M)\) in the following way. They apply a space cubic Cremona transformation to transform \(M\) into another a similar problem \(N\) in standard form. They show that \(\dim (M)=\dim (N)\) and they give an explicit formula for \(\dim (N)\) in terms of the degree and the multiplicities needed to define \(N\).
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