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Projective generation of curves in polynomial extensions of an affine domain and a question of Nori. (English) Zbl 0936.13005
Let $$A$$ 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.

##### 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
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