Rational surfaces having only a finite number of exceptional curves

Lahyane, Mustapha

doi:10.1007/s00209-002-0474-y

Cited by 9 publications

(6 citation statements)

References 12 publications

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“…(c, d) These follow from [Lahyane 2001a;2004], over ‫,ރ‬ but the proofs work over any algebraically closed field.…”

Section: Criteria For Finite Generation Of M(x)mentioning

confidence: 99%

“…In fact, if X is rational and anticanonical, we show in Corollary 4.2 that M(X ) is not finitely generated if and only if X contains infinitely many irreducible curves C of self-intersection −2 ≤ C 2 ≤ −1. Although a rational surface with K 2 X ≥ 0 is always anticanonical, our results on finite generation of M(X ) are of interest mainly in the case of anticanonical rational surfaces X with K 2 X < 0, since M(X ) is always finitely generated when K 2 X > 0, and [Lahyane 2001a;2001b;2004] give a complete characterization when K 2 X = 0 (see Proposition 4.3(c,d)). This characterization is in terms of the set of −2-curves, which in this situation must be a finite set.…”

Section: Introductionmentioning

confidence: 99%

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Irreducibility of −1-classes on anticanonical rational surfaces and finite generation of the effective monoid

Lahyane¹,

Harbourne²

2005

Pacific J. Math.

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We give necessary and sufficient conditions for a divisor class on smooth projective anticanonical rational surfaces to be the class of a smooth rational curve of self-intersection −1. We characterize smooth projective anticanonical rational surfaces for which the monoid of classes (modulo algebraic equivalence) of effective divisors is not finitely generated, extending results of Lahyane for the case of rational surfaces X with K 2 X = 0.

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“…(c, d) These follow from [Lahyane 2001a;2004], over ‫,ރ‬ but the proofs work over any algebraically closed field.…”

Section: Criteria For Finite Generation Of M(x)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Irreducibility of −1-classes on anticanonical rational surfaces and finite generation of the effective monoid

Lahyane¹,

Harbourne²

2005

Pacific J. Math.

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“…Such results were published after in [37]. Nowadays, such problem is still open even in the case of surfaces, some contributions in this direction are [9], [21], [4], [30], [15], [31], [17], [3], [33], [10], [11], [12], [32], [18], [13], [14], [6], [19] and [34].…”

Section: Introductionmentioning

confidence: 99%

On the effective, nef, and semi-ample monoids of blowups of Hirzebruch surfaces at collinear points

Rosa-Navarro

Frías-Medina²,

Lahyane³

2023

Can. Math. Bull.

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This paper is devoted to determine the geometry of a class of smooth projective rational surfaces whose minimal models are the Hirzebruch ones, concretely, are obtained as the blow-up of a Hirzebruch surface at collinear points. Explicit descriptions of their effective monoids are given and we present a decomposition for every effective class. Such decomposition is used to confirm the effectiveness of some divisor classes when Riemann-Roch theorem does not give information about their effectiveness. Furthermore, we study the nef divisor classes on such surfaces. We provide an explicit description for their nef monoids and moreover, we present a decomposition for every nef class. On the other hand, we prove that these surfaces satisfy the anticanonical orthogonal property. As a consequence, the surfaces are Harbourne-Hirschowitz and their Cox rings are finitely generated. Finally, we prove that the complete linear system associated to any nef divisor is base-point free, thus, the semi-ample and nef monoids coincide. The base field is assumed to be algebraically closed of arbitrary characteristic.

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“…Henceforth, an interesting (but still) open problem is to classify theoretically and/or effectively and constructively all smooth projective rational surfaces S for which the k-algebra Cox(S) is finitely generated. Masayoshi Nagata (see Theorem 4a, page 283, in [28]) showed that the surface Z obtained by blowing up of the projective plane P 2 at nine or more points in general position has an infinite number of (−1)-curves (see also [24], [21], [22], [25], [29], [27], [26] and [23] for cases when the points need not be in general position), consequently its Cox ring Cox(Z) is not finitely generated. Here a (−1)-curve on Z means a smooth projective curve on Z of self-intersection equal to −1.…”

Section: Introductionmentioning

confidence: 99%

A geometric criterion for the finite generation of the Cox rings of projective surfaces

Rosa-Navarro¹,

Medina²,

Lahyane³

et al. 2015

Rev. Mat. Iberoam.

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The aim of this paper is to give a geometric characterization of the finite generation of the Cox rings of anticanonical rational surfaces. This characterization is encoded in the finite generation of the effective monoid. Furthermore, we prove that in the case of a smooth projective rational surface having a negative multiple of its canonical divisor with only two linearly independent global sections (e.g., an elliptic rational surface), the finite generation is equivalent to the fact that there are only a finite number of smooth projective rational curves of self-intersection −1. The ground field is assumed to be algebraically closed of arbitrary characteristic.

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Rational surfaces having only a finite number of exceptional curves

Cited by 9 publications

References 12 publications

Irreducibility of −1-classes on anticanonical rational surfaces and finite generation of the effective monoid

Irreducibility of −1-classes on anticanonical rational surfaces and finite generation of the effective monoid

On the effective, nef, and semi-ample monoids of blowups of Hirzebruch surfaces at collinear points

A geometric criterion for the finite generation of the Cox rings of projective surfaces

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