Projective varieties defined by small number of equations are complete intersections

Netsvetaev, N. Yu.

doi:10.1007/bfb0082787

Cited by 4 publications

(5 citation statements)

References 11 publications

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“…The hypothesis that the Chern class of the normal bundle of Z is a restriction from the ambient space holds automatically if Z is nonsingular of codimension 2, by [Har74, Theorem 2.2, Proposition 6.1]. Analogous results for smooth varieties in higher codimension, subject to delicate inequalities, may be found in [Net88].…”

Section: Challenges and Examplesmentioning

confidence: 99%

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The Segre zeta function of an ideal

Aluffi

2017

Advances in Mathematics

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Abstract. We define a power series associated with a homogeneous ideal in a polynomial ring, encoding information on the Segre classes defined by extensions of the ideal in projective spaces of arbitrarily high dimension. We prove that this power series is rational, with poles corresponding to generators of the ideal, and with numerator of bounded degree and with nonnegative coefficients. We also prove that this 'Segre zeta function' only depends on the integral closure of the ideal.The results follow from good functoriality properties of the 'shadows' of rational equivalence classes of projective bundles. More precise results can be given if all homogeneous generators have the same degree, and for monomial ideals.In certain cases, the general description of the Segre zeta function given here leads to substantial improvements in the speed of algorithms for the computation of Segre classes. We also compute the projective ranks of a nonsingular variety in terms of the corresponding zeta function, and we discuss the Segre zeta function of a local complete intersection of low codimension in projective space.

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Section: Challenges and Examplesmentioning

confidence: 99%

“…Requirements such as condition (*) may be bypassed, again subject to certain inequalities involving r, m, n (see [Fal81], [Net88]).…”

Section: Challenges and Examplesmentioning

confidence: 99%

The Segre zeta function of an ideal

Aluffi

2017

Advances in Mathematics

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“…Thus, the relation (7) gives a sufficient condition for a subcanonical surface in P 4 to be scheme-theoretically defined by 4 equations.…”

Section: Remarkmentioning

confidence: 99%

“…Following this approach, Faltings proved in [3] that if p ≤ n − 2 and n ≥ 8 and X is a (possibly singular) subcanonical local complete intersection, then it is a complete intersection (in any characteristic). Some years later, this result was improved in [7], proving in characteristic zero that if p ≤ n − 1, n ≥ 8, then X is a complete intersection (but assuming X smooth).…”

Section: Introductionmentioning

confidence: 99%

“…The aim of the present work is twofold: on one hand we would like to give a different (in that we use Serre's correspondence) and simpler proof of the result announced in [7], hoping to give a crystal clear version of some obscure (to our opinion) arguments. Moreover, assuming only that X is a (possibly singular) subcanonical l.c.i., we prove in any characteristic that if n ≥ 3 and p ≤ n − 1, then X is a complete intersection and we give some more results working in characteristic zero, assuming that the normal bundle of X extends to a numerically split bundle E on P n (i.e.…”

Section: Introductionmentioning

confidence: 99%

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On Two Simple Criteria for Recognizing Complete Intersections in Codimension 2

Arsie¹

2001

Communications in Algebra

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Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P n , n ≥ 3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p ≤ n − 1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P n , p ≤ n and the characteristic of the base field is zero. Finally, using our characterization, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu ([4]) on self-linking and complete intersections in positive characteristic. MSC (1991): Primary: 14M10 , Secondary: 14M06

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Choi

Kwak

2003

Geometriae Dedicata

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Projective varieties defined by small number of equations are complete intersections

Cited by 4 publications

References 11 publications

The Segre zeta function of an ideal

The Segre zeta function of an ideal

On Two Simple Criteria for Recognizing Complete Intersections in Codimension 2

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