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It is known that in characteristic zero, in a linear system (not necessarily complete) in $\mathbb{P}^3$ a general element is irreducible. Now my question is the following: Suppsoe we have a subspace $V \subset H^0(\mathbb{P}^3, \mathcal{O}(3))$ of dimension $\ge 2$$\ge 3$. Then is it possible always to find an element in $V$ which is reducible ?

It is known that in characteristic zero, in a linear system (not necessarily complete) in $\mathbb{P}^3$ a general element is irreducible. Now my question is the following: Suppsoe we have a subspace $V \subset H^0(\mathbb{P}^3, \mathcal{O}(3))$ of dimension $\ge 2$. Then is it possible always to find an element in $V$ which is reducible ?

It is known that in characteristic zero, in a linear system (not necessarily complete) in $\mathbb{P}^3$ a general element is irreducible. Now my question is the following: Suppsoe we have a subspace $V \subset H^0(\mathbb{P}^3, \mathcal{O}(3))$ of dimension $\ge 3$. Then is it possible always to find an element in $V$ which is reducible ?

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LAPRAS
LAPRAS
  • 549
  • 2
  • 8

reducibility of an element in a linear system in $\mathbb{P}^3$

It is known that in characteristic zero, in a linear system (not necessarily complete) in $\mathbb{P}^3$ a general element is irreducible. Now my question is the following: Suppsoe we have a subspace $V \subset H^0(\mathbb{P}^3, \mathcal{O}(3))$ of dimension $\ge 2$. Then is it possible always to find an element in $V$ which is reducible ?