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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?

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Yes, this is called the castelnuovo bound. For curves you'll find it in Hartshorne. In this case, g is bounded by a simple quadratic polynomial in the degree, and I think there are higher-dimensonal analogues as well. – J.C. Ottem Feb 17 '11 at 16:25
Is this true also for singular varieties? I know about the bounds in Hartshorne, but the result there (Theorem 6.6.4) is for non-singular curves in P^3. – Wanderer Feb 17 '11 at 16:31
References are most welcome! – Wanderer Feb 17 '11 at 16:32
There is a lot of literature on this out there. Try for example Zak's article… – J.C. Ottem Feb 17 '11 at 16:39
You don't even need to fix the dimension as it is bounded by $n$. – Sándor Kovács Feb 17 '11 at 19:16
up vote 8 down vote accepted

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.

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I believe it was first proved by Kleiman (you only need to assume fixed dimension and degree, no need to assume a subvariety of some fixed $\mathbb P^n$), see Corollary 6.11

S. Kleiman, Exp XIII in A. Grothendieck et al., Theorie des Intersections et Theoreme de Riemann-Roch (SGA 6), Lecture Notes in Math No. 225, Springer-Verlag, Heidelberg (1971).

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