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

Author(s):  
Alice Garbagnati

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.

2007 ◽  
Vol 54 (2) ◽  
pp. 137-162 ◽  
Author(s):  
András I. Stipsicz ◽  
Zoltán Szabó ◽  
Ágnes Szilárd

2020 ◽  
Vol 2020 (762) ◽  
pp. 167-194
Author(s):  
Salim Tayou

AbstractWe prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight 2 with {h^{2,0}=1} over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of K3 surfaces.


2007 ◽  
Vol 169 (3) ◽  
pp. 519-567 ◽  
Author(s):  
V.A. Gritsenko ◽  
K. Hulek ◽  
G.K. Sankaran

1996 ◽  
Vol 11 (27) ◽  
pp. 2199-2211 ◽  
Author(s):  
RON DONAGI ◽  
ANTONELLA GRASSI ◽  
EDWARD WITTEN

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.


Author(s):  
Marian Aprodu ◽  
Gavril Farkas ◽  
Angela Ortega

AbstractThe Minimal Resolution Conjecture (MRC) for points on a projective variety


2020 ◽  
pp. 1-12
Author(s):  
John Kopper

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.


Author(s):  
Julian Lawrence Demeio

Abstract For a number field $K$, an algebraic variety $X/K$ is said to have the Hilbert Property if $X(K)$ is not thin. We are going to describe some examples of algebraic varieties, for which the Hilbert Property is a new result. The first class of examples is that of smooth cubic hypersurfaces with a $K$-rational point in ${\mathbb{P}}_n/K$, for $n \geq 3$. These fall in the class of unirational varieties, for which the Hilbert Property was conjectured by Colliot-Thélène and Sansuc. We then provide a sufficient condition for which a surface endowed with multiple elliptic fibrations has the Hilbert Property. As an application, we prove the Hilbert Property of a class of K3 surfaces, and some Kummer surfaces.


2018 ◽  
Vol 2020 (23) ◽  
pp. 9420-9439
Author(s):  
Vasily Rogov

Abstract An Iwasawa manifold is a compact complex homogeneous manifold isomorphic to a quotient $G/\Lambda $, where $G$ is the group of complex unipotent $3 \times 3$ matrices and $\Lambda \subset G$ is a cocompact lattice. In this work, we study holomorphic submanifolds in Iwasawa manifolds. We prove that any compact complex curve in an Iwasawa manifold is contained in a holomorphic subtorus. We also prove that any complex surface in an Iwasawa manifold is either an abelian surface or a Kähler non-projective isotrivial elliptic surface of Kodaira dimension one. In the Appendix, we show that any subtorus in Iwasawa manifold carries complex multiplication.


Sign in / Sign up

Export Citation Format

Share Document