Logarithmic differentials

Question

I have a general and a more special question. I begin with the general one: If $X/k$ is an algebraic scheme over a field $k$ and $D$ is a divisor with normal crossings on it, then there is the so-called sheaf of log-differentials $\Omega_{X/k}(\log(D))$ on it. Is there a good reference for the very basic properties of this kind of differetials? In particular I am interested in behavior under pullback and analogues to the two exact sequences one has for usual differentials.

Now I come to the special question. Let $k=\mathbb F_q$ and $B/k$ a smooth projective curve. Let $F: B\to B$ be the Frobenius morphism. ($F$ is the identity on the underlying topological space of $B$ and the map ${\cal O}_B(U)\to {\cal O}_B(U)$ induced by $F$ is $x\mapsto x^q$ for every open $U\subset B$.) Let $S=\sum a_P P$ be an effective divisor on $B$. Is it true that $F^* \Omega_{X/k}(\log(S))=\Omega_{X/k}(\log(F^* S))$ (rather then $F^* \Omega_{X/k}(\log(S))=\Omega_{X/k}(\log(S))$)? (Here $F^* S=\sum a_P^q P$.)

My interest in these matters arise from an attempt to understand the paper "Purely inseparable points on curves of higher genus" http://www.mathjournals.org/mrl/1997-004-005/1997-004-005-004.pdf I find the results in this paper very interesting. For example they have been applied towards full Mordell-Lang in positive characteristic, and I have other applications in mind.

But I am very puzzled. In the proof of the Theorem in this paper, I do not understand why $\Omega_B((F^n)^{-1} S)=\Omega_B(S)$ and why $\deg(P^*\omega)$ should be bounded. Also I do not see a reason why the separable map $g$ occurring later in the proof should be non-constant.

Remark: In this paper the ground field $k$ seems to be an issue that should be discussed a bit. The paper starts with "Let $k$ be a field of characteristic $p>0$". I think the Corollary is false as it stands, at least in the case where $k$ is algebraically closed with $trdeg(k/{\mathbb F}_p)\ge 1$. (In that case, if $K=k(X)$ and $C/K$ is a smooth projective curve of genus $\ge 2$ which is defined over $k$ will have $|C(k)|=\infty$, $|C(K)|=\infty$ and $|C(K^{\frac{1}{p^\infty}})|=\infty$, but of course $C$ needs not be birational to a curve defined over a finite field.) Also there seem to be counterexamples to the Theorem itself, if $k$ is a general field of positive characteristic. These counterexamples disappear, if we assume $k$ finitely generated over its finite prime field, and I think the whole paper was meant to address that case. So at the moment I try to understand everything in the case where $k$ is finitely generated of positive characteristic. But until now I failed even in the easiest case where $k$ is finite ...

EDIT: Maybe I should state here the Claim, whose proof I finally want to understand.

Claim: Let $k$ be a finitely generated field of positive characteristic. Let $K/k$ be a finitely generated field extension with $trdeg(K/k)=1$. Let $C/K$ be a smooth projective curve over $k$ of genus $\ge 2$. Assume that $C$ is not birational over $\overline{K}$ to a curve which is defined over a finite field. Then $C(K^{\frac{1}{p^\infty}})$ is finite.

Answer 1

For a reference on logaritmic sheaves, see Esnault-Viehweg "Lectures on vanishing theorems", in particular the first six Chapters.

The Frobenius morphism is also considered in the same book, in Chapters 8 and 9.

But in order to answer your question, no deep results are needed. In fact, over a curve $B$ the sheaf $\Omega_B(\log S)$ is invertible and isomorphic to $\Omega_B(S)$. This because, in general, there is a short exact sequence

$0 \to T_B(-S) \to T_B(- \log S) \to T_S \to 0$,

and $T_S$ is zero for a zero-dimensional scheme.

On the other hand, by [Hartshorne, II prop. 6.9] for any divisor $D$ on $B$ we have

$\deg(F^*D) = q \cdot \deg(D)$.

Then

$\deg (F^*\Omega_B(\log S))=q \cdot \deg(K_B)+q \cdot \deg(S)$,

$\deg (\Omega_B(F^*\log S))=\deg(K_B)+q \cdot \deg(S)$,

$\deg (\Omega_B(\log S))=\deg(K_B)+\deg(S)$,

hence the invertible sheaf $F^*\Omega_B(\log S)$ is in general different from both $\Omega_B(F^*\log S)$ and $\Omega_B(\log S)$.

The fact that $\Omega_B((F^n)^{-1}S))=\Omega_B(S)$ comes simply from the fact that $F$ is ramified everywhere with ramification index $q$, so the set-theoretic preimage of $S$ is nothing but $S$.