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Strength and Hartshorne’s conjecture in high degree. (English) Zbl 1464.14048

Fix positive integers \(c\) and \(e\). The authors prove the existence of an integer \(N(c,e)\) such that for all fields \(k\) and all equidimensional projective varieties \(X\subset \mathbb {P}_k^n\) with codimension \(c\), degree \(e\) and singular locus of codimension \(N(c,e)\) (so \(n\ge c+N(c,e)\)) the variety \(X\) is a complete intersection. The novelty is the independence of \(N(c,e)\) on the characteristic of the field. The reader may get the classical works on Hartshorne’s conjecture and Babylonian towers from [I. Coandă, Commun. Algebra 40, No. 12, 4668–4672 (2012; Zbl 1268.14013)]. The proof uses a very powerful new approach (going to infinitely many variables and handling the Noetherianity problems arising at the limit; see [D. Erman et al., Invent. Math. 218, No. 2, 413–439 (2019; Zbl 1427.13018)].

MSC:

14M10 Complete intersections
13C40 Linkage, complete intersections and determinantal ideals
13L05 Applications of logic to commutative algebra
13D05 Homological dimension and commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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