2018/08/03 by Vladimir Lazić, Thomas Peternell
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Dimension (graph theory) #Elliptic curve #Geometry and complex manifolds #Modular elliptic curve #Simply connected space #Social connectedness #Symplectic geometry #math.AG #msc:14H10 #msc:14J32
paper · pdf · doi:10.5802/ahl.38
published as Annales Henri Lebesgue 3 (2020), 473-500 · 27 pages
arxiv created 2018/08/03 · openalex created_date 2018/08/22 · openalex publication_date 2020/06/29 · arxiv updated 2020/07/14 · openalex updated_date 2026/08/05
In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi–Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi–Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> contain a rational curve, and so do <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -trivial varieties with finite fundamental group which are covered by elliptic curves.