Projective manifolds with small pluridegrees

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 $|2(K_\mathcal{M}+ (n-2)\mathcal{L})|$.