Singular Fibers in Elliptic Fibrations on the Rational Elliptic Surface

2007 ◽  
Vol 54 (2) ◽  
pp. 137-162 ◽  
Author(s):  
András I. Stipsicz ◽  
Zoltán Szabó ◽  
Ágnes Szilárd
Keyword(s):  
Elliptic Surface ◽  
Elliptic Fibrations ◽  
Singular Fibers ◽  
Rational Elliptic Surface

scholarly journals Elliptic Fibrations and Kodaira Dimensions of Schreieder’s Varieties

2021 ◽  
Author(s):  
Alice Garbagnati
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Kodaira Dimension ◽  
Elliptic Surface ◽  
Minimal Resolution ◽  
Elliptic Fibrations ◽  
Smooth Model ◽  
Birational Model ◽  
Hodge Numbers ◽  
Rational Elliptic Surface

Abstract We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

scholarly journals A NONPERTURBATIVE SUPERPOTENTIAL WITH E8-SYMMETRY

1996 ◽  
Vol 11 (27) ◽  
pp. 2199-2211 ◽  
Author(s):  
RON DONAGI ◽  
ANTONELLA GRASSI ◽  
EDWARD WITTEN
Keyword(s):  
Theta Function ◽  
Elliptic Surface ◽  
Four Dimensions ◽  
Rational Elliptic Surface

We compute the nonperturbative superpotential in F-theory compactification to four dimensions on a complex threefold P1×S, where S is a rational elliptic surface. In contrast to examples considered previously, the superpotential in this case has interesting modular properties; it is essentially an E8 theta function.

scholarly journals The Nef Cone of the Hilbert Scheme of Points on Rational Elliptic Surfaces and the Cone Conjecture

2020 ◽  
pp. 1-12
Author(s):  
John Kopper
Keyword(s):  
Hilbert Scheme ◽  
Elliptic Surface ◽  
Elliptic Surfaces ◽  
Hilbert Scheme Of Points ◽  
Rational Elliptic Surfaces ◽  
Nef Cone ◽  
Rational Elliptic Surface

Abstract We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

scholarly journals UNWEIGHTED DONALDSON–THOMAS THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES

2020 ◽  
Vol 71 (3) ◽  
pp. 867-942
Author(s):  
Oliver Leigh
Keyword(s):  
Partition Function ◽  
Singular Locus ◽  
Elliptic Surface ◽  
Abelian Surfaces ◽  
Rational Elliptic Surface ◽  
Singular Fibre

Abstract We further the study of the Donaldson–Thomas theory of the banana 3-folds which were recently discovered and studied by Bryan [3]. These are smooth proper Calabi–Yau 3-folds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a ‘banana configuration’. In [3], the Donaldson–Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article, we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande–Thomas theory for a rational elliptic surface and present new Gopakumar–Vafa invariants for the banana 3-fold.

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