Special subschemes of the scheme of singularities of a plane foliation

Campillo, A.; Olivares, Jorge Uroz

doi:10.1016/j.crma.2007.03.018

Cited by 6 publications

(9 citation statements)

References 7 publications

(6 reference statements)

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“…Given such an [s], in a series of papers, the authors have deal with the problem of existence of subschemes Y such that [s] is determined by Y , starting from Campillo and Olivares (2001), where it is shown that [s] is determined by S = [s] 0 . For non-degenerate [s], we have shown in Campillo and Olivares (2007) that there exist subschemes Z ⊂ S which still determine [s]. These were called special therein and renamed as Type C subschemes in Olivares (2018).…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Cayley–Bacharach and Singularities of Foliations

Campillo

Olivares

2020

Bull Braz Math Soc, New Series

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This paper deals with foliations by curves [s] of degree r ≥ 2 on P 2 with isolated singularities S, called non-degenerate if S is reduced and otherwise degenerate. Say that [s] is uniquely determined by a zero-dimensional Y ⊂ S if [s] is the unique foliation that vanishes on Y and say that Y is minimal for [s] if, moreover, the degree of Y is the minimal possible to do so. Previous work of the authors show that every nondegenerate foliation in degrees 2 ≤ r ≤ 5 does have a minimal subscheme and that the set of non-degenerate foliations of degree r ≥ 6 that have a minimal subscheme contains a Zariski-open subset of the space of foliations of degree r. For non-degenerate foliations [s] we present both characterizations and sufficient conditions for [s] to have a minimal subscheme. We also give examples of degenerate foliations of degree r ≥ 7 that do not have a minimal subscheme at all.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Cayley–Bacharach and Singularities of Foliations

Campillo

Olivares

2020

Bull Braz Math Soc, New Series

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“…This contradiction shows that (2.13) is dominant and the proof of Theorem 1.3 (b) has been completed. [7], the attentive reader will notice that the results therein hold for an algebraically closed ground field K).…”

Section: Letmentioning

confidence: 97%

Foliations by curves uniquely determined by minimal subschemes of its singularities

Campillo¹,

Olivares²

2018

jsing

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It is well-known that a foliation by curves of degree greater than or equal to two, with isolated singularities, in the complex projective space of dimension greater than or equal to two, is uniquely determined by the scheme of its singular points. The main result in this paper is that the set of foliations which are uniquely determined by a subscheme (of the minimal possible degree) of its singular points, contains a nonempty Zariski-open subset. Our results hold in the projective space defined over any algebraically closed ground field.

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“…In this setting, Bs = p 2 • p −1 1 . There is no inverse map to Bs in (8), because the map Bs is not injective. The definition of Bir • d is such that two different bases of the same homaloidal net gives different elements in Bir • d whilst have the same base locus.…”

Section: Cremona Transformations Of Fixed Degreementioning

confidence: 99%

“…To conclude, observe that [f 1 : f 2 : f 3 ] and [λf 1 : λf 2 : λf 3 ], λ ∈ C * , represent the same element in Bir • d . The previous lemma suggests how to change the target space in (8) in order to get a birational map. For this purpose, we construct suitable morphisms.…”

Section: Cremona Transformations Of Fixed Degreementioning

confidence: 99%

“…For this reason we think that it would be interesting to find closed formulas for the numbers N • (d) and N(d) of irreducible components of Bir • d and Bir d and to study the connectedness of Bir • d for each d. This study of Bir • d , as pointed out by Cerveau and Déserti in [10], also allows to build a bridge between the classical algebraic geometry of Cr(2) and foliations on P k , k 2. The reconstruction of a foliation from its singular set, [7], [8], the study of irreducible components of foliations on P k , k 3, [12], [13], and the description of the orbits by the action of PGL(3, C) on the foliations of fixed degree 2 on P 2 , [11], have been extensively studied and suggest the possibility to use our methods also in this context. We will not dwell on this here.…”

Section: Introductionmentioning

confidence: 99%

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On Plane Cremona Transformations of Fixed Degree

2013

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IntroductionThe birational geometry of algebraic varieties is governed by the group of birational self-maps. It is in general very difficult to determine this group for an arbitrary variety and as a matter of facts only few examples are completely understood. The special case of the projective plane attracted lots of attention since the XIX th -century. The pioneering work of Cremona and then the classical geometers of the Italian and German school were able to give partial descriptions of it but it was only after Noether and Castelnuovo that generators of the group were described. The Noether-Castelnuovo Theorem, see for instance [1], states that the group of birational self-maps of P 2 , usually called the plane Cremona group and denoted by Cr (2), is generated by linear automorphisms of P 2 and a single birational non biregular map, the so-called elementary quadratic transformation σ : P 2 P 2 defined by σ([x : y : z]) = [yz : xz : xy]. Even if the generators of Cr(2) have been known for a century now, many other properties of this group are still mysterious. Only after decades, see [26], a complete set of relations has been described, and more recently the non simplicity of Cr(2) has been showed, [9], and a good understanding of its finite subgroups has been achieved, see [17] and [5]. This brief and fairly incomplete list is only meant to stress the difficulties and the large unknown parts in the study of Cr(2), for a more complete picture the interested reader should refer to [15] and [14]. Amid all its subgroups the one associated to polynomial automorphisms of the plane, Aut(C 2 ), attracted even more attention than Cr(2) itself, [3]. The generators of Aut(C 2 ) are known since 1942, [29], and later on, [32], has been proved that Aut(C 2 ) is the amalgamated product of two of its subgroups, more precisely of the affine and elementary ones. Nevertheless this group is not less mysterious and challenging than the entire Cremona group. Jung's description yields a natural decompositioninto sets of polynomial automorphisms of multidegree (d 1 , · · · , d m ) and the affine subgroup A. In [22], Friedland and Milnor proved that G[d 1 , · · · , d m ] is a smooth analytic manifold of dimension (d 1 +d 2 +· · · d m +6). Later on, Furter, [23], computed the number of irreducible components of polynomial automorphisms of C 2 with fixed degree less or equal to 9 and proved that the variety of polynomial automorphisms of the plane with degree bounded by the positive integer n, is reducible when n 4. Later on Edo and Furter,[18], studied some degenerations of the multidegrees: for example they were able to show that G[3] ∩ G[2, 2] = ∅ using the lower semicontinuity of the length of a plane polynomial automorphism as a word of the amalgamated product, [24]. Contrarily to what happens in the Cremona group Cr(2), see [6], the group of polynomial automorphisms G of the plane can be endowed with a structure of an infinite-dimensional algebraic group. Denoted by G d the set of polynomial automorphisms of fixed degree d, Furte...

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Special subschemes of the scheme of singularities of a plane foliation

Cited by 6 publications

References 7 publications

Cayley–Bacharach and Singularities of Foliations

Cayley–Bacharach and Singularities of Foliations

Foliations by curves uniquely determined by minimal subschemes of its singularities

On Plane Cremona Transformations of Fixed Degree

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