Projective generation of curves in polynomial extensions of an affine domain and a question of Nori.

*(English)*Zbl 0936.13005Let \(A\) be a smooth affine domain of dimension \(n\) over a field \(k\). Let \(X =\text{Spec } A\) and \(Z = V(I)\) be a smooth subvariety of \(X \times {\mathbb A}^1 =\text{Spec } A[T]\) such that \(Z\) intersects \(X \times 0\) transversally in \(Y \times 0\), where \(Y\) is a smooth subvariety of \(X\) of dimension \(n-r\). Let \(P\) be a projective module of rank \(r \geq (n+3)/2\). Let \({\overline \phi}: P[T] \to I/(I^2T)\) be a surjection. M. V. Nori in the appendix (pp. 645-646) to the paper of S. Mandal cited below asked whether one can lift \(\overline \phi\) to a surjection from \(P[T]\) to \(I\).

S. Mandal [J. Algebr. Geom. 1, No. 4, 639-646 (1992; Zbl 0796.14011)] showed that this question has an affirmative answer when \(I\) contains a monic polynomial even without the smoothness assumptions.

The paper under review gives an affirmative answer when \(V(I)\) is a smooth curve and \(k\) is an infinite field. This result has several interesting consequences on projective generation of curves in polynomial extensions of \(A\). For instance, it is shown that if \(A\) is a regular affine domain over an algebraically closed field \(k\) (respectively \(\mathbb R\)) with \(\dim A = n \geq 3\) (respectively \(n \geq 2\)) such that every maximal ideal of \(A\) is a complete intersection and if \(I \subset A[T]\) is a local complete intersection ideal of height \(n\) such that \(I/I^2\) is generated by \(n\) elements, then \(I\) is generated by \(n\) elements. Moreover, the paper also gives examples showing that Nori’s question has negative answer if \(r \leq (n+2)/2\) or if \(A\) is not smooth.

S. Mandal [J. Algebr. Geom. 1, No. 4, 639-646 (1992; Zbl 0796.14011)] showed that this question has an affirmative answer when \(I\) contains a monic polynomial even without the smoothness assumptions.

The paper under review gives an affirmative answer when \(V(I)\) is a smooth curve and \(k\) is an infinite field. This result has several interesting consequences on projective generation of curves in polynomial extensions of \(A\). For instance, it is shown that if \(A\) is a regular affine domain over an algebraically closed field \(k\) (respectively \(\mathbb R\)) with \(\dim A = n \geq 3\) (respectively \(n \geq 2\)) such that every maximal ideal of \(A\) is a complete intersection and if \(I \subset A[T]\) is a local complete intersection ideal of height \(n\) such that \(I/I^2\) is generated by \(n\) elements, then \(I\) is generated by \(n\) elements. Moreover, the paper also gives examples showing that Nori’s question has negative answer if \(r \leq (n+2)/2\) or if \(A\) is not smooth.

Reviewer: Ngo Viet Trung (Hanoi)

##### MSC:

13C10 | Projective and free modules and ideals in commutative rings |

14H05 | Algebraic functions and function fields in algebraic geometry |

14M10 | Complete intersections |

14A05 | Relevant commutative algebra |