Projective manifolds with small pluridegrees

Abstract:

Let $\mathcal {L}$ be a very ample line bundle on a connected complex projective manifold $\mathcal {M}$ of dimension $n\ge 3$. Except for a short list of degenerate pairs $(\mathcal {M},\mathcal {L})$, $\kappa (K_\mathcal {M}+(n-2)\mathcal {L})=n$ and there exists a morphism $\pi : \mathcal {M} \to M$ expressing $\mathcal {M}$ as the blowup of a projective manifold $M$ at a finite set $B$, with $\mathcal {K}_M:=K_M+(n-2)L$ nef and big for the ample line bundle $L:= (\pi _*\mathcal {L})^{**}$. The projective geometry of $(\mathcal {M},\mathcal {L})$ is largely controlled by the pluridegrees $d_j:=L^{n-j}\cdot (K_M+(n-2)L)^j$ for $j=0,\ldots ,n$, of $(\mathcal {M},\mathcal {L})$. For example, $d_0+d_1=2g-2$, where $g$ is the genus of a curve section of $(\mathcal {M},\mathcal {L})$, and $d_2$ is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of $(\mathcal {M},\mathcal {L})$. In this article, a detailed analysis is made of the pluridegrees of $(\mathcal {M},\mathcal {L})$. The restrictions found are used to give a new lower bound for the dimension of the space of sections of $\mathcal {K}_M$. The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to $|2(K_\mathcal {M}+ (n-2)\mathcal {L})|$.