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Chen
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Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(X\backslash D)=0$$H^1(\mathcal{O}_{X\backslash D})=0$ (we know that $H^1(X)=0$$H^1(\mathcal{O}_X)=0$)? If not true in general are there examples of $D$ (other than hyperplane sections of $X$) for which this is true?

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(X\backslash D)=0$ (we know that $H^1(X)=0$)? If not true in general are there examples of $D$ (other than hyperplane sections of $X$) for which this is true?

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(\mathcal{O}_{X\backslash D})=0$ (we know that $H^1(\mathcal{O}_X)=0$)? If not true in general are there examples of $D$ (other than hyperplane sections of $X$) for which this is true?

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Chen
Chen
  • 1.6k
  • 9
  • 13

Projective surfaces with vanishing first cohomology

Let $X$ be a smooth, projective surface in $\mathbb{P}^3$ and $D$ an effective divisor in $X$. Is it true that $H^1(X\backslash D)=0$ (we know that $H^1(X)=0$)? If not true in general are there examples of $D$ (other than hyperplane sections of $X$) for which this is true?