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kostya
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In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:

For a general (smooth) surface $S$ in $\mathbb{P}^3$ over the complex numbers which is not ruled, and for a general curve $C$ on this surface which is proportional to some multiple of a plane section $H$, the restriction map $$\mathrm{Pic}\ S \rightarrow \mathrm{Pic}\ C$$ is injective.

I can only see the following: we always can assume that the surface has irregularity $0$, and we can check injectivity only for divisors $L$ such that $L.H = 0$, and hence $L^2 < 0$ by Hodge index theorem. But I cannot see how to use that the surface is non-ruled. Thanks in advance for any suggestions.

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:

For a general (smooth) surface $S$ in $\mathbb{P}^3$ over the complex numbers which is not ruled, and for a general curve $C$ on this surface which is proportional to some multiple of a plane section, the restriction map $$\mathrm{Pic}\ S \rightarrow \mathrm{Pic}\ C$$ is injective.

I can only see the following: we always can assume that the surface has irregularity $0$, and we can check injectivity only for divisors $L$ such that $L.H = 0$, and hence $L^2 < 0$ by Hodge index theorem. But I cannot see how to use that the surface is non-ruled. Thanks in advance for any suggestions.

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:

For a general (smooth) surface $S$ in $\mathbb{P}^3$ over the complex numbers which is not ruled, and for a general curve $C$ on this surface which is proportional to some multiple of a plane section $H$, the restriction map $$\mathrm{Pic}\ S \rightarrow \mathrm{Pic}\ C$$ is injective.

I can only see the following: we always can assume that the surface has irregularity $0$, and we can check injectivity only for divisors $L$ such that $L.H = 0$, and hence $L^2 < 0$ by Hodge index theorem. But I cannot see how to use that the surface is non-ruled. Thanks in advance for any suggestions.

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kostya
kostya
  • 111
  • 5

Restriction of the Picard group of a surface to a curve

In a paper by Griffiths and Harris on the Noether-Lefschetz theorem, they use the following fact which they don't comment as if it is obvious:

For a general (smooth) surface $S$ in $\mathbb{P}^3$ over the complex numbers which is not ruled, and for a general curve $C$ on this surface which is proportional to some multiple of a plane section, the restriction map $$\mathrm{Pic}\ S \rightarrow \mathrm{Pic}\ C$$ is injective.

I can only see the following: we always can assume that the surface has irregularity $0$, and we can check injectivity only for divisors $L$ such that $L.H = 0$, and hence $L^2 < 0$ by Hodge index theorem. But I cannot see how to use that the surface is non-ruled. Thanks in advance for any suggestions.