A characteristic-free criterion of birationality. (English) Zbl 1251.14007
The authors develop a purely algebraic theory at the level of graded rings of rational maps between reduced, but not necessarily irreducible, projective varieties in arbitrary characteristic.
Let \(R\) be a graded \(k\)-algebra, where \(k\) is any algebraically closed field. A rational \((m+1)\)-datum of degree d on \(R\) is defined to be an ordered set of \(m+1\) forms \(\mathbf{f}=\{f_0, \dots, f_m\}\) of degree \(d\) such that \(R\) is torsion free over its subring \(k[f_0,\dots,f_m]\) and the ideal \((f_0, \dots, f_m)\) is not contained in any minimal prime ideal of \(R\). This means that the induced map \(\mathrm{Proj} R \dasharrow \mathbb{P}^m\) is defined on a dense open set. Then a numerical invariant of a rational map is introduced, called the Jacobian dual rank, with data in the Rees algebra \(\mathcal{R}(I)\), where \(I\) is the ideal of \(R\) generated by \(f_0,\dots,f_m\). It is proved that a rational map is birational if and only if the Jacobian dual rank is well defined and attains its maximal possible value. As an application of this, certain results about birational maps between projective spaces in characteristic zero are extended to any characteristic.
Let \(R\) be a graded \(k\)-algebra, where \(k\) is any algebraically closed field. A rational \((m+1)\)-datum of degree d on \(R\) is defined to be an ordered set of \(m+1\) forms \(\mathbf{f}=\{f_0, \dots, f_m\}\) of degree \(d\) such that \(R\) is torsion free over its subring \(k[f_0,\dots,f_m]\) and the ideal \((f_0, \dots, f_m)\) is not contained in any minimal prime ideal of \(R\). This means that the induced map \(\mathrm{Proj} R \dasharrow \mathbb{P}^m\) is defined on a dense open set. Then a numerical invariant of a rational map is introduced, called the Jacobian dual rank, with data in the Rees algebra \(\mathcal{R}(I)\), where \(I\) is the ideal of \(R\) generated by \(f_0,\dots,f_m\). It is proved that a rational map is birational if and only if the Jacobian dual rank is well defined and attains its maximal possible value. As an application of this, certain results about birational maps between projective spaces in characteristic zero are extended to any characteristic.
Reviewer: Nikolaos Tziolas (Nicosia)
MSC:
| 14E05 | Rational and birational maps |
| 14E07 | Birational automorphisms, Cremona group and generalizations |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
Keywords:
rational map; base ideal; Jacobian dual rank; embedding dimension; birationality criterion; Cremona mapReferences:
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