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Small changes in varieties

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aisejohan committed Dec 6, 2017
1 parent 5b2f263 commit 831b0e3f8264ebebad2dbfa85ced7488d1a6b2cf
Showing with 6 additions and 4 deletions.
  1. +6 −4 varieties.tex
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@@ -9740,7 +9740,7 @@ \section{Numerical intersections}
Let $Z_i \subset X$ be the irreducible components
of $\text{Supp}(\mathcal{F})$ of dimension $d$. Let $\xi_i \in Z_i$
be the generic point and set
$m_i = \text{length}_{\mathcal{O}_{X, \xi_i}}(\mathcal{F}_{\xi_i})$
$m_i = \text{length}_{\mathcal{O}_{X, \xi_i}}(\mathcal{F}_{\xi_i})$.
Then
$$
\chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes
@@ -9847,7 +9847,7 @@ \section{Numerical intersections}
\end{definition}
\noindent
The displayed equality in the lemma follows from
The displayed equality in the definition follows from
the projection formula
(Cohomology, Section \ref{cohomology-section-projection-formula}) and
Cohomology of Schemes, Lemma
@@ -10139,8 +10139,10 @@ \section{Numerical intersections}
\end{proposition}
\begin{proof}
This follows immediately from the definitions and
Lemma \ref{lemma-ample-positive}.
This follows from the definitions,
Lemma \ref{lemma-ample-positive}, and the vanishing
of higher cohomology in
Cohomology of Schemes, Lemma \ref{coherent-lemma-vanshing-gives-ample}.
\end{proof}

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