About Noether's relations for three-dimensional Cremona transformations (Q1311495)

Let \(T: \mathbb{P}^ 3\to \mathbb{P}^ 3\) be a Cremona transformation of three- dimensional projective space. It is given by a linear system \(L\) of hypersurfaces of some degree \(m\) passing through the fundamental locus \(F\) of \(T\). If \(F\) consists of some curves \(B_ i\) and points \(p_ j\), then members of \(L\) pass through each \(B_ i\) and \(p_ j\) with some multiplicity \(d_ i\) and \(m_ j\), respectively. The condition that \(L\) gives a birational transformation translates into the conditions that \(\dim L=3\) and for each \(D\in L\), \(\sigma^*(D)^ 3=1\) where \(\sigma: X\to \mathbb{P}^ 3\) is a resolution of indeterminacy of \(T\). By using standard computations in the Chow ring of \(X\) and the Riemann-Roch formula, the author deduces the relations between the numbers \(d_ i\) and \(m_ j\), known classically as the Noether relations. The assumptions made on \(F\) are the following: the curves \(B_ i\) are nonsingular and intersect each other transversally, all higher cohomology of the sheaf \({\mathcal O}_ X(\sigma^* (D)- \sum_ i d_ i \sigma^{-1} (B_ i)- \sum_ j m_ j \sigma^{-1} (p_ j))\) vanish.