I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2.22.


Let $\mathcal{L}\_{2g}$ be the linear system of degree $2g+1$ hypersurfaces in $\mathbb{P}^{2g}$ passing through $2g+2$ general points say $p\_1,\cdots,p\_{2g+2}$ with multiplicity $2g-1$. Let $\mu\_g:\mathbb{P}^{2g}\dashrightarrow\Sigma\_{2g}\subset\mathbb{P}^N$ be the rational map induced by $\mathcal{L}\_{2g}$. Let $H\_{I}=H\_{i\_1,\cdots,i\_{g+1}}$ be the $g$-plane generated by $p\_{i\_1},\cdots,p\_{i\_{g+1}}\in \mathbb{P}^{2g}$ and let $J\subset I$ such that $|J|=g$. Then I have following questions.


1. Why the general element $D$ of $\mathcal{L}\_{2g}|\_{H\_I}$ must contain $H\_J$ with multiplicity $g(2g-1)-(g-1)(2g+1)=1$?
2. Why $\mu\_{g}|\_{H\_I}$ is the standard Cremona transformation on $\mathbb{P}^{g}$?


Any suggestions or reference related to questions is highly appreciated. Thanks in advance.



Question regarding linear system of projective space
The formula for the multiplicity of a linear system along a linear subspace is classical. You can find it for instance in Lemma 2.1 here <https://arxiv.org/pdf/1210.5175.pdf>


This will tell you that a general $D$ contains $H\_J$ with multiplicity one. Now, $D\_{|H\_I}$ has degree $2g+1$ and multiplicity $2g-1$ at $g+1$ general points. But you have a fixed component given by the $H\_J$, there are $g+1$ of them, and through each one of the $p\_i$ there are $g-1$ of these. So their union forms a hypersurface in $H\_I$ of degree $g+1$ having multiplicity $g$ at the $p\_i$. Once you remove this hypersurface from $D\_{|H\_I}$ you are left with a hypersurface of degree $2g+1 - (g+1) = g$ having multiplicity $2g-1-g = g-1$.


Hence, your linear system restricted to $H\_I\cong\mathbb{P}^g$ is (once you remove the part of codimension $1$ of the base locus) the linear system of hypersurfaces of degree $g$ having multiplicity $g-1$ at $g+1$ general points. This is exactly the linear system inducing the standard Cremona of $\mathbb{P}^g$. For instace, for $g = 2$ you get the conics through three points, for $g = 3$ the cubics of $\mathbb{P}^3$ with four double points, ecc...


