An effective restriction theorem via wall-crossing and Mercat’s conjecture

• DOI: 10.1007/s00209-022-03036-1
• Publication Date: 2022-06-02T15:05:58Z
• Authors: Soheyla Feyzbakhsh
• Keywords: Mathematics - Algebraic Geometry; FOS: Mathematics; 0101 mathematics; Algebraic Geometry (math.AG); 01 natural sciences; 14H60, 14J28, 14H51

Abstract We prove an effective restriction theorem for stable vector bundles E on a smooth projective variety: $$E|_D$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>E</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mi>D</mml:mi></mml:msub></mml:math> is (semi)stable all irreducible divisors $$D \in |kH|$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>D</mml:mi><mml:mo>∈</mml:mo><mml:mo>|</mml:mo><mml:mi>k</mml:mi><mml:mi>H</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math> k greater than explicit constant. As application, we show that Mercat’s conjecture in any rank 2 fails curves lying K3 surfaces. Our technique to use wall-crossing with respect (weak) Bridgeland stability conditions which also reprove Camere’s result slope of the tangent bundle $${\mathbb {P}}^n$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> restricted surface.