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Improve section on alternative Weil cohomology

Added some lemmas on zero dimensional schemes.
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aisejohan committed Dec 2, 2019
1 parent 8a49bd8 commit 194d6c9acd9c7dc222d191a81765050ea885ae1a
Showing with 285 additions and 39 deletions.
  1. +18 −0 chow.tex
  2. +55 −0 morphisms.tex
  3. +16 −0 varieties.tex
  4. +196 −39 weil.tex
@@ -7843,6 +7843,24 @@ \section{The splitting principle}
$\pi_i \circ \pi : P \to X$ does the job. Some details omitted.
\end{proof}

\begin{remark}
\label{remark-the-proof-shows-more}
The proof of Lemma \ref{lemma-splitting-principle}
shows that the morphism $\pi : P \to X$ has the following additional
properties:
\begin{enumerate}
\item $\pi$ is a finite composition of projective space bundles
associated to locally free modules of finite constant rank, and
\item for every $\alpha \in \CH_k(X)$ we have
$\alpha = \pi_*(\xi_1 \cap \ldots \cap \xi_d \cap \pi^*\alpha)$
where $\xi_i$ is the first chern class of some invertible
$\mathcal{O}_P$-module.
\end{enumerate}
The second observation follows from the first and
Lemma \ref{lemma-cap-projective-bundle}.
We will add more observations here as needed.
\end{remark}

\noindent
Let $(S, \delta)$, $X$, and $\mathcal{E}_i$ be as in
Lemma \ref{lemma-splitting-principle}.
@@ -9030,6 +9030,61 @@ \section{Ample and very ample sheaves relative to finite type morphisms}
Lemma \ref{lemma-characterize-relatively-ample}.
\end{proof}

\begin{lemma}
\label{lemma-invertible-add-enough-ample-very-ample}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{N}$, $\mathcal{L}$ be invertible $\mathcal{O}_X$-modules.
Assume $S$ is quasi-compact, $f$ is of finite type, and $\mathcal{L}$
is $f$-ample. Then
$\mathcal{N} \otimes_{\mathcal{O}_X} \mathcal{L}^{\otimes d}$
is $f$-very ample for all $d \gg 1$.
\end{lemma}

\begin{proof}
By Lemma \ref{lemma-characterize-very-ample-on-finite-type}
we reduce to the case $S$ is affine. Combining
Lemma \ref{lemma-finite-type-over-affine-ample-very-ample} and
Properties, Proposition \ref{properties-proposition-characterize-ample}
we can find an integer $d_0$ such that
$\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$
is globally generated. Choose global sections
$s_0, \ldots, s_n$ of $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$
which generate it. This determines a morphism
$j : X \to \mathbf{P}^n_S$ over $S$. By
Lemma \ref{lemma-finite-type-over-affine-ample-very-ample}
we can also pick an integer $d_1$ such that for all $d \geq d_1$
there exist sections $t_{d, 0}, \ldots, t_{d, n(d)}$
of $\mathcal{L}^{\otimes d}$ which generate it and define
an immersion
$$
j_d = \varphi_{\mathcal{L}^{\otimes d}, t_{d, 0}, \ldots, t_{d, n(d)}} :
X
\longrightarrow
\mathbf{P}^{n(d)}_S
$$
over $S$. Then for $d \geq d_0 + d_1$ we can consider the
morphism
$$
\varphi_{\mathcal{N} \otimes \mathcal{L}^{\otimes d},
s_j \otimes t_{d - d_0, i}} :
X
\longrightarrow
\mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S
$$
This morphism is an immersion as it is the composition
$$
S \to \mathbf{P}^n_S \times_S \mathbf{P}^{n(d - d_0)}_S
\to \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_S
$$
where the first morphism is $(j, j_{d - d_0})$ and
the second is the Segre embedding
(Constructions, Lemma \ref{constructions-lemma-segre-embedding}).
Since $j$ is an immersion, so is $(j, j_{d - d_0})$
(apply Lemma \ref{lemma-immersion-permanence}). We have a composition of
immersions and hence an immersion
(Schemes, Lemma \ref{schemes-lemma-composition-immersion}).
\end{proof}




@@ -10644,6 +10644,22 @@ \section{Bertini theorems}
$r - \dim \mathfrak m_x/\mathfrak m_x^2 - 1$ as desired.
\end{proof}







\section{Enriques-Severi-Zariski}
\label{section-vanishing-negative}

\noindent
In this section we prove some results of the form: twisting by
a ``very negative'' invertible module kills low
degree cohomology. We also deduce the connectedness
of a hypersurface section of a normal proper scheme of
dimension $\geq 2$.

\begin{lemma}
\label{lemma-vanishin-h0-negative}
Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$

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