On the Space of Holomorphic Maps From the Riemann Sphere to the Quadric Cone

Guest, Martin A.

doi:10.1093/qmath/45.1.57

Cited by 10 publications

(12 citation statements)

References 18 publications

(33 reference statements)

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“…This is entirely analogous to the proof in §3 of [Se] for the case CP n . Another treatment of the same argument was given in [Gu1] (in the case of the quadric cone) and in [Gu2] (in the case of CP n ).…”

Section: This Extends To a Continuous Map Smentioning

confidence: 91%

“…By the method of [Gu1], Proposition 3.2, it may be deduced from Proposition 4.2 that this is a homology equivalence up to dimension min{e 1 −c 1 (d 0 +(j −1)l)−c 1 l, . .…”

Section: Consider Next the Diagram Belowmentioning

confidence: 99%

“…This can be reduced to the corresponding fact for the symmetric product. The method we use here is based on [GKY], [Gu1] as the method used in [Se] for the case X = CP n does not seem to extend to the case of general X. Second, we show (using the homology-like properties of π i Q X D (C)) that Q X ∞ (C) is actually homotopy equivalent to a component of Map(S 2 , X).…”

mentioning

confidence: 99%

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The topology of the space of rational curves on a toric variety

Guest

1995

Acta Math.

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A toric variety is, roughly speaking, a complex algebraic variety which is the (partial) compactification of an algebraic torus T C = (C * ) r . It admits (by definition) an action of T C such that, for some point * ∈ X, the orbit of * is an embedded copy of T C . The most significant property of a toric variety is the fact that it is characterized entirely by a combinatorial object, namely its fan, which is a collection of convex cones in R r . As a general reference for the theory of toric varieties we use [Od1], together with the recent lecture notes [Fu].In this paper we shall study the space of rational curves on a compact toric variety X. We shall obtain a configuration space description of the space Hol(S 2 , X) of all holomorphic (equivalently, algebraic) maps from the Riemann sphere S 2 = C ∪ ∞ to X. Our main application of this concerns fixed components Hol * D (S 2 , X) of Hol * (S 2 , X), where the symbol D will be explained later, and where the asterisk indicates that the maps are required to satisfy the condition f (∞) = * . If Map * D (S 2 , X) denotes the corresponding space of continuous maps, we shall show that the inclusioninduces isomorphisms of homotopy groups up to some dimension n(D), and we shall give a procedure for computing n(D).A theorem of this type was proved in the case X = CP n by Segal ([Se]), and indeed that theorem provided the motivation for the present work. Our main idea is that the result of Segal may be interpreted as a result about configurations of distinct points in C which have labels in a certain partial monoid. We shall show that Hol * D (S 2 , X) may be identified with a space Q X D (C) of configurations of distinct points in C which have labels in a partial monoid M X , where M X is derived from the fan of X. Then we shall extend Segal's method so that it applies to this situation.A feature of the method is the idea that the functor U → π i Q X D (U ) resembles a homology theory. This functor has some similarities with the Lawson homology functor introduced in 1991 Mathematics Subject Classification. 55P99, 14M25.

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Section: This Extends To a Continuous Map Smentioning

confidence: 91%

“…By the method of [Gu1], Proposition 3.2, it may be deduced from Proposition 4.2 that this is a homology equivalence up to dimension min{e 1 −c 1 (d 0 +(j −1)l)−c 1 l, . .…”

Section: Consider Next the Diagram Belowmentioning

confidence: 99%

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confidence: 99%

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The topology of the space of rational curves on a toric variety

Guest

1995

Acta Math.

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“…The space A n,m is an example of (the complement of) a "subspace arrangement"; the topology of such spaces has been studied intensively (see, for example, part III of [GM], as well as [Va1]). Segal's method extends to the case of Hol(S 2 , X), where X is a toric variety (see [GKY1], [Gu1], [Gu2]). By imposing conditions of bounded multiplicity on these spaces, we obtain many further results of the above type.…”

Section: This Theorem Implies Thatmentioning

confidence: 99%

Spaces of polynomials with roots of bounded multiplicity

Guest¹,

Kozłowski²,

Yamaguchi³

1999

Fund. Math.

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“…In fact, these pose no problem. For, by a result of Guest (1994) quoted in Johnsen and Kleiman (1996, proof of lemma 3.2), Hom 10 1 Q is at most 23-dimensional. We now conclude by applying the same projection-based argument used in the preceding case.…”

Section: Special Subloci Of the Mapping Spacementioning

confidence: 97%

Rational Curves of Degree 10 on a General Quintic Threefold

Cotterill

2005

Communications in Algebra

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We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in 4 , there are only, finitely, many smooth rational curves of degree 10, and each curve C is embedded in F with normal bundle −1 ⊕ −1 . Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on F.

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On the Space of Holomorphic Maps From the Riemann Sphere to the Quadric Cone

Cited by 10 publications

References 18 publications

The topology of the space of rational curves on a toric variety

The topology of the space of rational curves on a toric variety

Spaces of polynomials with roots of bounded multiplicity

Rational Curves of Degree 10 on a General Quintic Threefold

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