CSUN Algebra, Number Theory, and Discrete Mathematics Seminar

Let $X$ be a smooth complex projective surface. A conjecture attributed to Mukai says for any ample line bundle $A$ and integer $k\ge 4$, the line bundle $L={\omega }_X\otimes A^{\otimes k}$ is normally generated. This in particular implies that $X\subseteq \mathbb{P}\left(H^0\left(L\right)\right)$ can be cut out by quadratic or cubic equations. In this talk, I will survey some results and methods concerning the conjecture for both curve and surface case. I will show that if the surface has a cyclic covering structure over an anticanonical rational surface and $A$ is a pullback of some ample line bundle, then $L$ is normally generated. This presents further evidence for the conjecture.