Surface quotient singularities and bigness of the cotangent bundle: Part I

Abstract We investigate the components determining bigness of the cotangent bundle $$\Omega ^1_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Ω</mml:mi> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> of smooth models X in the birational class "Equation missing" of an orbifold surface of general type Y , with a focus on the contribution given by the singularities of Y . A criterion for bigness of $$\Omega _X^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Ω</mml:mi> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> is given involving only topological and singularity data on Y . We single out a special case, the Canonical Model Singularities (CMS) criterion, when Y is the canonical model of "Equation missing". We study the singularity invariants appearing in the criterion and determine them for $$A_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> singularities. Knowledge of these invariants for $$A_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> singularities allows one to evaluate the $$(c_2,c^2_1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -geographical range of the CMS criterion and compare it to other criteria. We obtain new examples of resolutions X of hypersurfaces $$Y\subset \mathbb {P}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Y</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> (with lower degrees) and of cyclic covers Y of $$\mathbb {P}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> branched along line arrangements with $$\Omega ^1_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Ω</mml:mi> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msubsup> </mml:math> big.