We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker's specialization inequality to arbitrary dimension.

A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf ℱ on X, there exists an integer m0 such that m≥m0 implies Hi (X,ℱ⊗𝒪(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.

Let $\cal X$ be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field Fq2 of order q2. If the number of Fq2-rational points of $\cal X$ satisfies the Hasse–Weil upper bound, then $\cal X$ is said to be Fq2-maximal. For a point P0 ∈ $\cal X$(Fq2), let π be the morphism arising from the linear series $\cal D$: = |(q + 1)P0|, and let N: = dim($\cal D$). It is known that N [ges ] 2 and that π is independent of P0 whenever $\cal X$ is Fq2-maximal.

The number N of rational points on an algebraic curve of genus g over a finite field ${\mathbb F}_q$ satisfies the Hasse–Weil bound $N \leq q + 1 +2g\sqrt {q}$. A curve that attains this bound is called maximal. With $g_0 =\textstyle {1\over 2}(q -\sqrt {q})$ and $g_1= \textstyle {1\over 4}(\sqrt {q}- 1)^2$, it is known that maximalcurves have $g= g_0$ or $g \leq g_1$. Maximal curves with $g=g_0$ or $g_1$ have been characterized up to isomorphism. A natural genus to be studied is $g_2 =\textstyle {1 \over 8}(\sqrt {q}- 1)(\sqrt {q}- 3),$ and for this genus there are two non-isomorphic maximal curves known when $\sqrt {q}\equiv 3\;({\rm mod}\; {4})$. Here, a maximal curve with genus $g_2$ and a non-singular plane model is characterized as a Fermat curve of degree $ {1\over 2}(\sqrt {q} + 1)$.