Conic bundles that are not birational to numerical Calabi--Yau pairs

Abstract

Let $X$ be a general conic bundle over the projective plane with branch curve of degree at least 19. We prove that there is no normal projective variety $Y$ that is birational to $X$ and such that some multiple of its anticanonical divisor is effective. We also give such examples for 2-dimensional conic bundles defined over a number field.

Keywords

• mathematics - algebraic geometry
• 14m22, 14j45, 14j20 (primary), 14j32, 14e05 (secondary)