Platonic Harbourne-Hirschowitz Rational Surfaces

Rosa-Navarro, Brenda Leticia De La; Frías-Medina, Juan Bosco; Lahyane, Mustapha

doi:10.1007/s00009-020-01593-5

Cited by 1 publication

(3 citation statements)

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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 [3, 4, 8–15, 17–19, 21, 30–34].…”

Section: Introductionmentioning

confidence: 99%

“…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%

“…This notion was introduced firstly in [13] in 2018. Then, it was used in the context of anticanonical rational surfaces in [6] and [14], and more generally, studied in the context of regular surfaces in [5].…”

Section: Introductionmentioning

confidence: 99%

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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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