By Paolo Cascini, James McKernan, Jorge Vitório Pereira
Featuring a mix of unique examine papers and complete surveys from a world crew of prime researchers within the thriving fields of foliation concept, holomorphic foliations, and birational geometry, this ebook offers the court cases of the convention "Foliation conception in Algebraic Geometry," hosted by way of the Simons origin in ny urban in September 2013.
Topics coated contain: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the constitution of symmetric differentials on a gentle complicated floor and an area constitution theorem for closed symmetric differentials of rank ; an summary of lifting symmetric differentials from types with canonical singularities and the purposes to the class of AT bundles on singular types; an summary of the robust thought of the range of minimum rational tangents brought through Hwang and Mok; contemporary examples of types that are hyperbolic and but the Green-Griffiths locus is the entire of X; and a category of psuedoeffective codimension one distributions.
Foliations play a primary function in algebraic geometry, for instance within the evidence of abundance for threefolds and to an answer of the Green-Griffiths conjecture for surfaces of normal style with confident Segre classification. the aim of this quantity is to foster communique and let interactions among specialists who paintings on holomorphic foliations and birational geometry, and to compile best researchers to illustrate the strong connection of rules, tools, and pursuits shared through those components of study.
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That includes a mix of unique learn papers and entire surveys from a global group of major researchers within the thriving fields of foliation idea, holomorphic foliations, and birational geometry, this booklet offers the complaints of the convention "Foliation concept in Algebraic Geometry," hosted by means of the Simons starting place in ny urban in September 2013.
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Additional info for Foliation Theory in Algebraic Geometry
X 0 such that . 0 / TX 0 =k is ample. 10], but our needs are a little different so we spell it out. 9] a smoothing W Y=T ! Xk0 over some smooth k-curve T with special fibre exists. 1]—applied to TX 0 =P1 . t / TX 0 =P1 to the generic k k fibre Yt is ample, while 0 jYt is finite, thus . 1. There is a map S0 W P1S ! X 0 to the smooth locus of X 0 =S such that . S0 / TX 0 =S is ample relative to S. Proof. 8] to 0 to obtain O 0 W P1O ! XO 0 , specialising to 0 , in order to conclude from the existence S of the Hilbert-scheme/S and the fact that being ample is open.
On the other hand, we have the usual exact sequence, _ 0 ! ; Symm NjF ˝ L˝n / ! FmC1 ; L˝n / ! ; Symk NjF ˝ L˝n / Ä CnrC1 kD0 where the last inequality may involve a slightly different constant, but nevertheless only depends on L as required. 2. Evidently the role of the analytic topology is only for convenience of exposition, since the above is really a proposition about formal schemes. 2 Cleaning Up We will continue to concentrate on the example of the previous section. 1, we have obtained an algebraic variety W of dimension r C 1, fibred over B by , together with a section s of , such that every fibre of W over B projects to a F -invariant sub-variety of X through the corresponding point of C.
X/ ! C/ the generic rank. We may now state, Main Theorem. C/. Here we use reduction modulo p, but only to resolve the problem for a foliation such that X=F exists as a scheme quotient over C, so in fact, (b) Notations and hypothesis as above then the minimal degree of the rational curve connecting any two points in Vx is effectively computable. In particular there is a rational curve Lx 3 x tangent to F such that for any nef. C/ C The bound, cf. 1, however, on the degree of rational curves connecting any two points in item (a) may be much worse.
Foliation Theory in Algebraic Geometry by Paolo Cascini, James McKernan, Jorge Vitório Pereira
Categories: Algebraic Geometry