Parametrization of rational maps on a variety of general type, and the finiteness theorem (Q2862173)

Let \(X\) be a variety of general type over the complex field. It is known that the set of the dominant rational maps \(X\dashrightarrow Y\) of finite degree with \(Y\) of general type, up to birational isomorphism, is a finite set. In the article under review the authors study the problem of giving an effective estimate for this number (see also [\textit{L. Guerra} and \textit{G. P. Pirola}, Collect. Math. 60, No. 3, 261--276 (2009; Zbl 1184.14020)]).NEWLINENEWLINEFor a subvariety \(X\subset \mathbb P^m=\mathbb P(V)\) of dimension \(n\), they consider the set \(R\) of linear rational maps \(\mathbb P^m\dashrightarrow \mathbb P^m\) whose restriction to \(X\) is of finite degree, and \(R_k\subset R\) the subset of maps of degree \(k\). Then \(R\) is an open subset of \(\mathbb P(\mathrm{End}(V))\) and \(R_k\) is a constructible subset. Moreover \(k\) is bounded from above in terms of \(r_n\), the minimum integer number such that the multicanonical divisor \(r_nK_V\) of any \(n\)-dimensional variety of general type \(V\) defines a birational embedding of \(V\).NEWLINENEWLINETheir main result is the following: if \(X\) is smooth projective of general type, any birational equivalence class of rational maps of degree \(k\) from \(X\) to smooth projective varieties of general type is in bijection with a union of connected components of \(R_k\).