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Let $S$ be a compact Riemann surface of genus $g\ge 2$ and $\iota_k:S\longrightarrow \mathbb{P}^{g-1}$ canonical mapping of $S$, $\iota_ {k}(S)=C \subset \mathbb{P}^{g-1}$.

Let be a $H\subset\mathbb{P}^{g-1}$ hyperplane.

What does it mean: $\iota_k^{-1}(H\cap C)$ contains multiple points???

Thank you!

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$C \cap H$ is the set of points where $C$ and $H$ intersect in $\mathbb P^{g-1}$, and $\iota_k^{-1}(C \cap H)$ is the preimage of this set under the morphism $\iota_k : C \to \mathbb P^{g-1}$. The statement is that this preimage contains more than one point.

For generic choices of the hyperplane $H$, the number of points in the set $\iota_k^{-1}(C \cap H)$ is equal to the degree of the line bundle $ \iota_k^{\star}\mathcal O_{\mathbb P^{g-1}}(H)$ on $C$. Since $\iota_k^{\star}\mathcal O_{\mathbb P^{g-1}}(H)$ is isomorphic to the canonical bundle $\mathcal K_C$ on $C$, it has degree $2g - 2$. So $\iota_k^{-1}(C \cap H)$ contains $2g - 2$ points.

For certainly special choices of $H$, the intersections may fail to be transversal. In these cases, the set-theoretic count of points in $\iota_k^{-1}(C \cap H)$ will be smaller than $2g - 2$, but the degree of the line bundle $\mathcal O_{\mathbb P^{g-1}}(H)|_C$ will remain as $2g - 2$; in other words, we get $2g - 2$ intersection points only when we count with multiplicity.

Finally, note the relationship between the number of points in $\iota_k^{-1}(C \cap H)$ (counted with multiplicity) and the number of points in $C \cap H$ (counted with multiplicity):

  • If $\iota_k$ is a closed immersion, then $|C \cap H| = |\iota_k^{-1}(C \cap H)| = 2g - 2$.

  • If $C$ is hyperelliptic, then $\iota_k : C \to \iota_k(C)$ is a morphism of degree two, so $|C \cap H| = \frac 1 2 |\iota_k^{-1}(C \cap H)| = g - 1$.

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By definition $\iota^{-1}_k(H \cap C)$ is a effective divisor $D = \sum_i k_i p_i$ of degree $2g - 2$ on $C$, so the author probably means that there is some $i$ with $k_i > 1$. Geometrically it means that $H$ is tangent to $\iota_k(C)$ at $\iota_k(p_i)$.

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  • $\begingroup$ Geometrically it means that $H$ is tangent to $\iota_k(C)$ at $\iota_k(p_i)$, or if $H$ passes through any of the points in the branch locus of $\iota_k$, if $S$ is hyperelliptic $\endgroup$ – Manoel Nov 16 '17 at 20:52

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