This is a somewhat naive question about the expected non-vanishing behavior of sheaf cohomology groups $H^i(X, \mathcal{L})$, where $X$ is a smooth projective variety of dimension $d$ over an algebraically closed field of characteristic $p>0$ and $\mathcal{L}$ is a line bundle on $X$. Here one starts with a few general principles: cohomology is finite-dimensional and vanishes in degrees outside the interval $0 \leq i \leq d$, while Serre duality holds.

Though my knowledge of algebraic geometry is sketchy at best, I am motivated by the case when $X$ is a flag variety for a connected reductive algebraic group $G$ and the line bundle comes from a character of some maximal torus of $G$ (lifted to a Borel subgroup). Unlike the nice classical theory in characteristic 0 where cohomology can be nonzero in at most one degree and then affords an irreducible representation of $G$, the situation in prime characteristic becomes intricate. Results here, due mainly to Kempf and Andersen, provide a few important general theorems as well as some low rank examples. But the associated representation theory is poorly understood and seems to be determined somehow in terms of Kazhdan-Lusztig theory for the affine Weyl group of Langlands dual type relative to $p$ in place of the usual Weyl group.

Two features occur frequently in low ranks and may hold in general, so I wonder if they are seen elsewhere in the world of smooth projective varieties. Here it's just a question of nonvanishing cohomology rather than more subtle module structure:

(A) Connectedness of the interval of degrees between 0 and $d$ in which non-vanishing occurs.

(B) Occurrence of non-vanishing in no more than $\frac{d}{2} +1$ degrees.

Is either (A) or (B) usual/unusual in other cases of non-vanishing cohomology in prime characteristic? (And are there relevant examples, references?)

This is a somewhat naive question about the expected non-vanishing behavior of sheaf cohomology groups $H^i(X, \mathcal{L})$, where $X$ is a smooth projective variety of dimension $d$ over an algebraically closed field of characteristic $p>0$ and $\mathcal{L}$ is a line bundle on $X$. Here one starts with a few general principles: cohomology is finite-dimensional and vanishes in degrees outside the interval $0 \leq i \leq d$, while Serre duality holds.

Though my knowledge of algebraic geometry is sketchy at best, I am motivated by the case when $X$ is a flag variety for a connected reductive algebraic group $G$ and the line bundle comes from a character of some maximal torus of $G$ (lifted to a Borel subgroup). Unlike the nice classical theory in characteristic 0 where cohomology can be nonzero in at most one degree and then affords an irreducible representation of $G$, the situation in prime characteristic becomes intricate. Results here, due mainly to Kempf and Andersen, provide a few important general theorems as well as some low rank examples. But the associated representation theory is poorly understood and seems to be determined somehow in terms of Kazhdan-Lusztig theory for the affine Weyl group of Langlands dual type relative to $p$ in place of the usual Weyl group.

Two features occur frequently in low ranks and may hold in general, so I wonder if they are seen elsewhere in the world of smooth projective varieties. Here it's just a question of nonvanishing cohomology rather than more subtle module structure:

(A) Connectedness of the interval of degrees between 0 and $d$ in which non-vanishing occurs.

(B) Occurrence of non-vanishing in no more than $\frac{d}{2} +1$ degrees.

Is either (A) or (B) usual/unusual in other cases of non-vanishing cohomology in prime characteristic? (And are there relevant examples, references?)

ADDED: The responses show that in some respects my question is too broad and therefore naive, since projective varieties come in so many shapes and sizes. Flag varieties are homogeneous spaces for a nice algebraic group (in any characteristic), though the cohomology of line bundles mysteriously gets far more complicated when the characteristic is prime. It's hard to trace the breakdown of most of the classical Borel-Weil-Bott theory, which motivates my question. At the same time, it's hard to see the role of geometry here apart from the related group theory.

By the way, (A) above is more speculative than (B) due to possible over-reliance on low rank examples. But neither property of non-vanishing has an obvious source in the geometry of the situation. I'm still interested in seeing similar examples elsewhere involving constraints on degrees where non-vanishing can occur.