Surfaces of general type with 𝑝_𝑔=𝑞=1,𝐾²=8 and bicanonical map of degree 2

2003/11/30 by Francesco Polizzi · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Degree (music) #Geology #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Mathematics #Pure mathematics #Type (biology) #math.AG #msc:14H37 #msc:14J10 #msc:14J29

paper · pdf · doi:10.1090/s0002-9947-05-03673-1

published as Trans. Amer. Math. Soc. 358 (2006), no. 2, 759--798 · 36 pages. To appear in Transactions of the American Mathematical Society

arxiv created 2005/03/14 · openalex publication_date 2005/03/25 · arxiv updated 2014/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We classify the minimal algebraic surfaces of general type with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Subscript g Baseline equals q equals 1 comma upper K squared equals 8"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:msup> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p_g=q=1, K²=8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and bicanonical map of degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It will turn out that they are isogenous to a product of curves, i.e. if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is such a surface, then there exist two smooth curves <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C comma upper F"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C, F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a finite group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting freely on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C times upper F"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo> × </mml:mo> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C × F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals left-parenthesis upper C times upper F right-parenthesis slash upper G"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>C</mml:mi> <mml:mo> × </mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S = (C × F)/G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We describe the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C comma upper F"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi>F</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">C, F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that occur. In particular the curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a hyperelliptic-bielliptic curve of genus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the bicanonical map <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi"> <mml:semantics> <mml:mi> ϕ </mml:mi> <mml:annotation encoding="application/x-tex">φ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is composed with the involution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">σ</mml:annotation> </mml:semantics> </mml:math> </inline-formula> induced on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation>

Citations

• Complex Algebraic Surfaces
• Geometry of Algebraic Curves
• On the classification of surfaces of general type with non-birational bicanonical map and Du Val Double planes
• Canonical models of surfaces of general type
• Algebraic Geometry

Cited by

• Standard isotrivial fibrations with pg=q=1