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S. Carnahan ♦
S. Carnahan ♦
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Is there a bound on arithmetic genus of a variety in projective n-space in terms of dimension and degree?

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Wanderer
Wanderer
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Suppose you are given a closed subvariety $V$ of projective space $\mathbb{P}^n_k$. Let's say we fix the degree and the dimension of $V$. Can we then bound the arithmetic genus of $V$, or does no such bound exist?

For example, in the case of (possibly singular) curves of some given degree, are there finitely many possible values for the arithmetic genus, or is it "unbounded"? I don't really know what answer to expect.

Suppose you are given a closed subvariety $V$ of projective space $\mathbb{P}^n_k$. Let's say we fix the degree and the dimension of $V$. Can we then bound the arithmetic genus of $V$, or does no such bound exist?

For example, in the case of (possibly singular) curves of some given degree, are there finitely many possible values for the arithmetic genus, or is it "unbounded"? I don't really know what answer to expect.

Suppose you are given a closed subvariety $V$ of projective space $\mathbb{P}^n_k$. Let's say we fix the degree and the dimension of $V$. Can we then bound the arithmetic genus of $V$, or does no such bound exist?

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Wanderer
Wanderer
  • 5.2k
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  • 49

arithmetic genus

Suppose you are given a closed subvariety $V$ of projective space $\mathbb{P}^n_k$. Let's say we fix the degree and the dimension of $V$. Can we then bound the arithmetic genus of $V$, or does no such bound exist?

For example, in the case of (possibly singular) curves of some given degree, are there finitely many possible values for the arithmetic genus, or is it "unbounded"? I don't really know what answer to expect.