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Arkadij
Arkadij
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From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings.

Now, if I start in the projective world I would like to be able to stay there under pushout. However, it seems that the answer to this questions is a counterexample in which one glues two copies of $\mathbb{P}^3$ along two different embedded curves. So naturally, I have a question if there is a condition under which the pushout of embeddings of projective varieties is again projective.

The main example that I would like to understand is as follows. Let $D\hookrightarrow X$$i:D\hookrightarrow X$ be a particular embedding of a normal crossing divisor ininto a smooth projective variety $X$ over $\mathbb{C}$. Is $X\cup_D X$$X\cup_{i,D,i} X$ projective again?

From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings.

Now, if I start in the projective world I would like to be able to stay there under pushout. However, it seems that the answer to this questions is a counterexample in which one glues two copies of $\mathbb{P}^3$ along two different embedded curves. So naturally, I have a question if there is a condition under which the pushout of embeddings of projective varieties is again projective.

The main example that I would like to understand is as follows. Let $D\hookrightarrow X$ be a normal crossing divisor in a smooth projective variety $X$ over $\mathbb{C}$. Is $X\cup_D X$ projective again?

From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings.

Now, if I start in the projective world I would like to be able to stay there under pushout. However, it seems that the answer to this questions is a counterexample in which one glues two copies of $\mathbb{P}^3$ along two different embedded curves. So naturally, I have a question if there is a condition under which the pushout of embeddings of projective varieties is again projective.

The main example that I would like to understand is as follows. Let $i:D\hookrightarrow X$ be a particular embedding of a normal crossing divisor into a smooth projective variety $X$ over $\mathbb{C}$. Is $X\cup_{i,D,i} X$ projective again?

Source Link
Arkadij
Arkadij
  • 988
  • 4
  • 21

When is the pushout of projective varieties along embeddings a projective variety?

From Karl Schwede's paper "Gluing schemes and a scheme without closed points'', I know that there exists a pushout of schemes for closed embeddings.

Now, if I start in the projective world I would like to be able to stay there under pushout. However, it seems that the answer to this questions is a counterexample in which one glues two copies of $\mathbb{P}^3$ along two different embedded curves. So naturally, I have a question if there is a condition under which the pushout of embeddings of projective varieties is again projective.

The main example that I would like to understand is as follows. Let $D\hookrightarrow X$ be a normal crossing divisor in a smooth projective variety $X$ over $\mathbb{C}$. Is $X\cup_D X$ projective again?