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Interpolation on jets. (English) Zbl 0918.14003

We work over a field \(k\) of characteristic zero. A jet in \(n\)-dimensional projective space \(\mathbb{P}^n_k\) over \(k\) will be any divisor on a one-dimensional linear subspace (i.e., a line in \(\mathbb{P}^n_k)\) with support a point. The length of a jet will be its degree as a divisor and we will say \(r\)-jet for a jet of length \(r\). We say that a closed subscheme \(Y\hookrightarrow \mathbb{P}^n_k\) has maximal rank in degree \(d\geq 0\) if the canonical map \(H^0 (\mathbb{P}^n_k, {\mathcal O}_{\mathbb{P}^n}(d))\to H^0(Y,{\mathcal O}_Y(d))\) has maximal rank as a linear map.
Theorem. Let \(L=(L_1, \dots,L_m)\) be a sequence of \(m\) lines in general position in \(\mathbb{P}^n_k\), \(r=(r_1, \dots,r_m)\) be a sequence of positive integers in nonincreasing order, and let \(d\) be an integer such that the sum \(r_1+ \cdots+r_m\) is at most \({n+d\choose d}\). Then the union \(J_1\cup\cdots\cup J_m\), where \(J_i\) is the generic \(r_i\)-jet on \(L_i\) has maximal rank in degree \(d\) if and only if the following (necessary) numerical condition \(C(n,d)\) holds: \(r_1\leq d+1\) and, if \(n=2\), then for any \(1\leq s\leq d+1\), \(r_1+ \cdots+ r_s\leq d.s+1-{s-1\choose 2}\).
This theorem slightly refines a result proved by A. Eastwood [Manuscr. Math. 67, No. 3, 227-249 (1990; Zbl 0722.41010) and J. Algebra 139, No. 2, 273-310 (1991; Zbl 0742.41003)] characterizing those generic unions of jets having maximal rank. Our proof is independent of this earlier one and much simpler, using, as was done in the papers cited above an old theorem by R. Hartshorne and A. Hirschowitz [in: Algebraic geometry, Proc. int. Conf., La Rabida 1981, Lect. Notes Math. 961, 169-188 (1982; Zbl 0555.14011)]. This theorem says that generic unions of lines have maximal rank in any degree.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14N05 Projective techniques in algebraic geometry
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[1] Eastwood, A., Collision de biais et interpolation, Manuscripta Math., 67, 227-249 (1990) · Zbl 0722.41010
[2] Eastwood, A., Interpolation à N variables, J. Algebra, 139, 273-310 (1991) · Zbl 0742.41003
[3] R. Hartshorne, A. Hirschowitz, Droites en position générale, Proceedings La Rabida 1981, Lecture Notes in Mathematics, 961, 169, 189, Springer-Verlag, Berlin/New York; R. Hartshorne, A. Hirschowitz, Droites en position générale, Proceedings La Rabida 1981, Lecture Notes in Mathematics, 961, 169, 189, Springer-Verlag, Berlin/New York · Zbl 0555.14011
[4] A. Hirschowitz, 1986, Problémes de Brill-Noether en rang supérieur; A. Hirschowitz, 1986, Problémes de Brill-Noether en rang supérieur
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