Proposition 33.45.13 (0BJ8): Asymptotic Riemann-Roch

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Part 2: Schemes
Chapter 33: Varieties
Section 33.45: Numerical intersections
Proposition 33.45.13: Asymptotic Riemann-Roch (cite)

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Proposition 33.45.13 (Asymptotic Riemann-Roch). Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $d$ . Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$ -module. Then

$\dim _ k \Gamma (X, \mathcal{L}^{\otimes n}) \sim c n^ d + l.o.t.$

where $c = \deg _\mathcal {L}(X)/d!$ is a positive constant.

Proof. This follows from the definitions, Lemma 33.45.9, and the vanishing of higher cohomology in Cohomology of Schemes, Lemma 30.17.1. $\square$

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