Projective Generation of Curves (III)

Shrikant Mahadeo Bhatwadekar, Mrinal Kanti Das
2013 International mathematics research notices  
Let A be an affine domain of dimension n over a field of characteristic zero. Let I ⊂ A[T ] be a local complete intersection ideal of height n such that µ(I/I 2 ) = n. This paper examines under what condition I is surjective image of a projective A[T ]module of rank n. More specifically, one is interested in knowing when is an element (I, ωI ) of the Euler class group E(A[T ]) obtained as the Euler class of a projective A[T ]module. It is proved that such a phenomenon occurs if and only if the
more » ... if and only if the naturally induced element (I(0), ω I(0) ) of the Euler class group E(A) is obtained as the Euler class of a projective A-module. Theorem 1.1. Let A be an affine domain of dimension n ≥ 3 over a field k of characteristic zero. Let I ⊂ A[T ] be a local complete intersection ideal of height n such that µ(I/I 2 ) = n. Assume that I(0) is of height n. If there exists a projective A-module Q of rank n with trivial determinant and a surjection from Q[T ] to I/(I 2 ∩ (T )), then I is projectively generated. Remark 1.2. Note that if I(0) = A then in the above theorem we can take Q to be free and then by [BRS1, 3.9] there is a surjection Q[T ] I/(I 2 T ), proving that I is projectively generated. We do not mention this case in the results below. Applying our main theorem we prove the following results (see 3.7, 3.11, 3.10). Corollary 1.3. Let A be an affine domain of dimension n ≥ 3 over a field k of characteristic zero. Let I ⊂ A[T ] be a local complete intersection ideal of height n such that I/I 2 is generated by n elements. Then I is projectively generated in the following cases : (1) n is even and I(0) is a projectively generated ideal of height n; (2) k is a C 1 field and I(0) is a projectively generated ideal of height n; (3) k is algebraically closed.
doi:10.1093/imrn/rnt230 fatcat:dva6xp2c6zdltpxyfrygr3tqka