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10 - Twenty points in ℙ3
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- By D. Eisenbud, University of California, Berkeley, R. Hartshorne, University of California, Berkeley, F.-O. Schreyer, Universität des Saarlandes
- Edited by Christopher D. Hacon, University of Utah, Mircea Mustaţă, University of Michigan, Ann Arbor, Mihnea Popa, University of Illinois, Chicago
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- Book:
- Recent Advances in Algebraic Geometry
- Published online:
- 05 January 2015
- Print publication:
- 15 January 2015, pp 180-199
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Summary
Abstract
Using the possibility of computationally determining points on a finite cover of a unirational variety over a finite field, we determine all possibilities for direct Gorenstein linkages between general sets of points in ℙ3 over an algebraically closed field of characteristic 0. As a consequence, we show that a general set of d points is glicci (that is, in the Gorenstein linkage class of a complete intersection) if d ≤ 33 or d = 37, 38. Computer algebra plays an essential role in the proof. The case of 20 points had been an outstanding problem in the area for a dozen years [8].
For Rob Lazarsfeld on the occasion of his 60th birthday
1 Introduction
The theory of liaison (linkage) is a powerful tool in the theory of curves in ℙ3 with applications, for example, to the question of the unirationality of the moduli spaces of curves (e.g., [3, 26, 29]). One says that two curves C, D ⊂ ℙ3 (say, reduced and without common components) are directly linked if their union is a complete intersection, and evenly linked if there is a chain of curves C = C0, C1, …, C2m = D such that Ci is directly linked to Ci+1 for all i. The first step in the theory is the result of Gaeta that any two arithmetically Cohen-Macaulay curves are evenly linked, and in particular are in the linkage class of a complete intersection, usually written licci. Much later Rao [23] showed that even linkage classes are in bijection with graded modules of finite length up to shift, leading to an avalanche of results (reported, e.g., in [19, 20]). However, in codimension > 2 linkage yields an equivalence relation that seems to be very fine, and thus not so useful; for example, the scheme consisting of the four coordinate points in ℙ3 is not licci.
A fundamental paper of Peskine and Szpiro [22] laid the modern foundation for the theory of linkage.