Lifting free divisors

Buchweitz, Ragnar-Olaf; Pike, Brian

doi:10.1112/plms/pdw013

Cited by 2 publications

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“…Moreover, in the curve case, the singularities are isolated, since we consider only reduced curves. For the next result, in the local analytic context, see also [5], section 2, especially Prop. 2.13.…”

Section: Free Surfaces In Pmentioning

confidence: 92%

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Free and nearly free surfaces in $\mathbb{P}^3$

Dimca

Sticlaru

2018

Asian Journal of Mathematics

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1 2 ALEXANDRU DIMCA AND GABRIEL STICLARU the Hilbert polynomial of a projective hypersurface V = V (f ) : f = 0 in P n for n ≥ 3 whose singular locus Σ satisfies dim Σ = 1. To illustrate the results stated there, we consider in the third section two simple cases: the case when V is a surface obtained as the union D ∪D ′ of two smooth surfaces D and D ′ in P 3 meeting transversally, and the case when V is a cone (in two natural ways) over a hypersurface W in P n−1 having only isolated singularities. The latter case can be used to show by examples that the Hilbert polynomial P (M(f )) may depend on the position of the singularities, e.g. when the singular locus Σ consists of three concurrent lines, then the polynomial P (M(f )) may depend on whether or not these lines are coplanar, see Examples 3.8 and 3.15. It also allows the construction of free (resp. nearly free) surfaces in P 3 as cones over free (resp. nearly free) curves in P 2 , see Corollaries 4.1 and 4.2, and Definition 5.3.The last two sections contain the main results of this note. Theorem 4.7 (resp. Theorem 5.12) express the Hilbert polynomial P (M(f )) and other invariants of the free (resp. nearly free) surface D : f = 0 in terms of the exponents d 1 ≤ d 2 ≤ d 3 of D. In fact, for a free surface, the exponents and the Hilbert polynomial P (M(f )) determine each other, see Corollary 4.8, (i).In Theorem 5.4 we prove an analog of Saito's criterion of freeness in the case of nearly free surfaces, which expresses the (unique) second order syzygy in terms of determinants constructed using the first order syzygies. A similar result holds for the nearly free curves in P 2 with the same proof as for surfaces, but was not stated in [21].Exactly as in the case of free curves in P 2 discussed in [33], [20], the irreducible free (resp. nearly free) surfaces are not easy to find. We give both isolated cases and countable families of examples of such surfaces in Examples 4.10 and 5.6. More involved examples, related to the discriminants of binary forms, are given in Propositions 4.14, 4.15 and 5.11. All our examples are rational surfaces (either one of the variables x, y, z, w occurs only with exponent 1 in the defining equations, or as in Propositions 4.14 and 5.11 this follows from the description of the discriminants).Theorem 5.16 and Example 5.17 discuss the point whether for a nearly free surface, the first local cohomology group H 1 Q (M(f )) of the Milnor algebra M(f ) with respect to the maximal homogeneous ideal Q in S is a finite dimensional C-vector space. By contrast, note that H 0 Q (M(f )) is finite dimensional for any hypersurface V : f = 0 in P n , see Remark 2.2. Examples provided by Aldo Conca are listed in Example 5.17 (ii) and allow one to construct rank 3 vector bundles on P 3 which are not direct sum of line bundles, see Remark 5.18.Note that a rational cuspidal curve can be characterized either as a rational curve which is simply-connected, or as an irreducible curve which is homeomorphic to P 1 . By analogy to the case of rational cuspidal c...

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Section: Free Surfaces In Pmentioning

confidence: 92%

“…For the fact that a free divisor D has a 1-dimensional singular locus, see [5], section 2, especially Prop. 2.13 or Corollary 4.8 (ii) below.…”

Section: Free Surfaces In Pmentioning

confidence: 99%

Free and nearly free surfaces in $\mathbb{P}^3$

Dimca

Sticlaru

2018

Asian Journal of Mathematics

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Homological properties of determinantal arrangements

Yim

2017

Journal of Algebra

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We explore a natural extension of braid arrangements in the context of determinantal arrangements. We show that these determinantal arrangements are free divisors. Additionally, we prove that free determinantal arrangements defined by the minors of 2×n matrices satisfy nice combinatorial properties.We also study the topology of the complements of these determinantal arrangements, and prove that their higher homotopy groups are isomorphic to those of S 3 . Furthermore, we find that the complements of arrangements satisfying those same combinatorial properties above have Poincaré polynomials that factor nicely.1991 Mathematics Subject Classification. 13N15,32S22.

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Lifting free divisors

Cited by 2 publications

References 32 publications

Free and nearly free surfaces in $\mathbb{P}^3$

Free and nearly free surfaces in $\mathbb{P}^3$

Homological properties of determinantal arrangements

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