TITLE: Is there a bound on arithmetic genus of a variety in projective n-space in terms of dimension and degree?
QUESTION [10 upvotes]: 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?

REPLY [3 votes]: 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).