Bounds on degrees of birational maps with arithmetically Cohen-Macaulay graphs (Q515597)

Suppose that \(\mathcal F:X\to Y\) is a birational map of closed, reduced, irreducible projective varieties over an algebraically closed field \(k\). If \(X\subseteq \mathbb P^n_k\) and \(Y\subseteq \mathbb P^m_k\), then \(\mathcal F\) may be represented by \(m+1\) homogeneous polynomials of the same degree in \(k[x_0,\dots,x_n]\). The goal of the present paper is to express bounds on the degrees of the representatives of \(\mathcal F\) (and also on the representatives of \(\mathcal F^{-1}\)) in terms of known numerical invariants of the relevant homogeneous coordinate rings. If no additional assumption is made then the bounds are calculated using [\textit{A. V. Doria} et al., Adv. Math. 230, No. 1, 390--413 (2012; Zbl 1251.14007)] and [\textit{E. W. Mayr} and \textit{S. Ritscher}, J. Symb. Comput. 49, 78--94 (2013; Zbl 1258.13032)] and these bounds are doubly exponential in the data. The present paper focuses on the case where the bi-homogeneous coordinate ring of the graph of \(\mathcal F\) is Cohen-Macaulay. This bi-homogeneous coordinate ring is the Rees Algebra \(\bigoplus_{0\leq i} I^i\), where \(A\) is the homogeneous coordinate ring of \(X\) and \(I\) is the ideal in \(A\) generated by the homogeneous forms that represent \(\mathcal F\). With the Cohen-Macaulay assumption it is shown that the homogeneous forms that represent \(\mathcal F\) have degree at most \(m(\dim X+1)\) and the homogeneous forms that represent \(\mathcal F^{-1}\) have degree at most \(n(\dim X+1)\).