Let X be a smooth projective variety over the complex numbers. Recall that a Cohen-Macaulay curve is a one-dimensional closed subscheme without embedded or isolated points (fat components are allowed).



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> I want to show that if you fix a curve class β in H2(X), then there is a bounded interval which contains the genus of every CM curve whose homology class is β.
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> Do you know of a reference for this result?
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Motivation: I am trying to prove a certain class of sheaves forms a bounded family (namely, the collection of sheaves underlying [stable pairs](http://arxiv.org/abs/0707.2348)). The above result will allow me to reduce to the case where the support is a fixed CM curve, and from there, I know how to finish the proof.


I am aware that Le Potier (who built the moduli space of stable pairs, among others things, in "System Coherents et Structures de Niveau" --- pardon my lack of accents) has proven a much stronger result. However, for my purposes, it would be very helpful to have a proof (hopefully much less technically demanding than Le Potier's) that is no more general than what I need.


EDIT: in hindsight, and in light of the negative answer below, I realize that Le Potier does not claim to prove quite what I claimed he claimed.



Are the arithmetic genera of Cohen-Macaulay curves in a fixed homology class bounded?
I don't think this is true. Take $X = \mathbb P^1 \times \mathbb P^2$. Let $C$ be $\mathbb P^1 \times 0$, and let $C\_1$ be the first infinitesimal neighborhood of $C$. The curve $C\_1$ is the relative spectrum of $\mathcal O\_{\mathbb P^1} \oplus O\_{\mathbb P^1}^{\oplus 2}$, where $\mathcal O\_{\mathbb P^1}^{\oplus 2}$ is a square-zero ideal. Any sheaf $\mathcal O\_{\mathbb P^1}(d)$, with $d \ge 0$, is a quotient of $\mathcal O\_{\mathbb P^1}^{\oplus 2}$. If $C(d)$ denotes the relative spectrum of $\mathcal O\_{\mathbb P^1} \oplus O\_{\mathbb P^1}(d)$, then $C(d)$ is contained in $C\_1$ for all $d \ge 0$; hence it is embedded in $X$ with fundamental class $2[C]$. But the arithmetic genus of $C(d)$ is $-d-1$.


[Added later] For a surface, a Cohen-Macaulay curve is a divisor, and the adjunction formula shows that the arithmetic genus is determined by the cohomology class, so the answer is positive. I believe that the answer is negative for all $X$ of dimension at least three.


