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?



Is there a bound on arithmetic genus of a variety in projective n-space in terms of dimension and degree?
I don't know about an explicit bound, but a bound exists in theory for arbitrary varieties. This follows from the
fact that the set of cycles, and in particular subvarieties, in $\mathbb{P}^n$ of fixed degree $d$ and dimension $N$ are parameterized by a Chow variety $Chow\_{d,N}$ (it needn't be irreducible, but it is certainly of finite type). More formally, consider the preimage
of $Chow\_{d,N}$ in the Hilbert scheme. This preimage has finitely many components,
and therefore there are finite number of possible Hilbert polynomials for a fixed $d,N$.


This is a bit sketchy, but perhaps someone else can supply more details or precise references.


