scholarly journals Rational curves on Del Pezzo manifolds

2018 ◽  
Vol 18 (4) ◽  
pp. 451-465
Author(s):  
Adrian Zahariuc
Keyword(s):  
Simple Formula ◽  
Rational Curves ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Normal Bundles ◽  
Splitting Type ◽  
Del Pezzo

Abstract We exploit an elementary specialization technique to study rational curves on Fano varieties of index one less than their dimension, known as del Pezzo manifolds. First, we study the splitting type of the normal bundles of the rational curves. Second, we prove a simple formula relating the number of rational curves passing through a suitable number of points in the case of threefolds and the analogous invariants for del Pezzo surfaces.

scholarly journals Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

2009 ◽  
Vol 42 (4) ◽  
pp. 531-557
Author(s):  
Fabrizio Catanese ◽  
Frédéric Mangolte
Keyword(s):  
Rational Curves ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals Real singular Del Pezzo surfaces and threefolds fibred by rational curves, I

2008 ◽  
Vol 56 (2) ◽  
pp. 357-373 ◽  
Author(s):  
Fabrizio Catanese ◽  
Frédéric Mangolte
Keyword(s):  
Rational Curves ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals Maximally mutable Laurent polynomials

2021 ◽  
Vol 477 (2254) ◽  
Author(s):  
Tom Coates ◽  
Alexander M. Kasprzyk ◽  
Giuseppe Pitton ◽  
Ketil Tveiten
Keyword(s):  
Mirror Symmetry ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Laurent Polynomials ◽  
Fano Manifold ◽  
One To One ◽  
Del Pezzo

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.

scholarly journals Counting imaginary quadratic points via universal torsors

2014 ◽  
Vol 150 (10) ◽  
pp. 1631-1678 ◽  
Author(s):  
Ulrich Derenthal ◽  
Christopher Frei
Keyword(s):  
Arbitrary Number ◽  
Number Fields ◽  
Quadratic Fields ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Imaginary Quadratic Fields ◽  
Del Pezzo Surface ◽  
Singularity Type ◽  
Del Pezzo ◽  
Manin’S Conjecture

AbstractA conjecture of Manin predicts the distribution of rational points on Fano varieties. We provide a framework for proofs of Manin’s conjecture for del Pezzo surfaces over imaginary quadratic fields, using universal torsors. Some of our tools are formulated over arbitrary number fields. As an application, we prove Manin’s conjecture over imaginary quadratic fields$K$for the quartic del Pezzo surface$S$of singularity type${\boldsymbol{A}}_{3}$with five lines given in${\mathbb{P}}_{K}^{4}$by the equations${x}_{0}{x}_{1}-{x}_{2}{x}_{3}={x}_{0}{x}_{3}+{x}_{1}{x}_{3}+{x}_{2}{x}_{4}=0$.

scholarly journals Cox rings of degree one del Pezzo surfaces

2009 ◽  
Vol 3 (7) ◽  
pp. 729-761 ◽  
Author(s):  
Damiano Testa ◽  
Anthony Várilly-Alvarado ◽  
Mauricio Velasco
Keyword(s):  
Del Pezzo Surfaces ◽  
Cox Rings ◽  
Del Pezzo

scholarly journals Tropicalization of del Pezzo surfaces

2016 ◽  
Vol 300 ◽  
pp. 156-189 ◽  
Author(s):  
Qingchun Ren ◽  
Kristin Shaw ◽  
Bernd Sturmfels
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo
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