Projective normality and basepoint-freeness thresholds of general polarized abelian varieties

For a polarized abelian variety $(X,L)$, Z. Jiang and G. Pareschi introduce an invariant $\beta (X,L)$, called the basepoint-freeness threshold. Using this invariant, we show that a general polarized abelian variety $(X,L)$ of dimension $g$ is projectively normal if $\chi (L)\ge 2^{2g-1}$ and the type of $L$ is not $(2,4,\dots ,4)$. This bound is sharp since it is known that any polarized abelian variety of type $(2,4,\dots ,4)$ is not projectively normal. We also give an application of $\beta (X,L)$ to the Infinitesimal Torelli Theorem for $Y\in \mathrm{\mid }L\mathrm{\mid }$.