Stable pairs and DT ideal-sheaf moduli on projective threefolds

Assume the Axiom of Choice and Dependent Choice carried by the declared derived and deformation suppliers. Let $X$ be a nonsingular projective threefold over $\mathbb{C}$ and fix an ample line bundle $H$ on $X$. For any coherent sheaf $G$ with support of dimension at most one, define its associated integral one-cycle by $[G{]}_1={\sum }_C{\mathrm{length}}_{{\mathcal{O}}_{X,{\eta }_C}}(G_{{\eta }_C})[C]$, where $C$ runs over the one-dimensional irreducible support components and ${\eta }_C$ is their generic point; the sum is zero for zero-dimensional $G$. Its homology class is the sum of the same multiplicities times the fundamental classes of the complex curves, in singular homology $H_2(X(\mathbb{C}),\mathbb{Z})$. Fix $\beta$ in this group. In moduli subscripts, equality of a one-cycle with $\beta$ means equality of its homology class, rather than equality of cycles. Write $[Z{]}_1=[{\mathcal{O}}_Z{]}_1$. For such a sheaf the Hilbert polynomial with respect to $H$ is $P_G(k)=\chi (X,G\otimes H^{\otimes k})=({\int }_{[G{]}_1}{c}_1(H))k+\chi (X,G)$; the integral is the sum of the generic multiplicities times the degrees of $H$ on the component curves.

A stable pair is a morphism $s:{\mathcal{O}}_X\to F$ of coherent sheaves such that $F$ is pure of dimension one and $\mathrm{coker}(s)$ is zero-dimensional. Purity means every nonzero coherent subsheaf of $F$ has one-dimensional support. A family over a scheme $S$ consists of a coherent sheaf on $X\times S$ flat over $S$ and a section from ${\mathcal{O}}_{X\times S}$ whose geometric fibres satisfy these conditions; fixing the homology class of $[F{]}_1$ to be $\beta$ and $\chi (F)=n$ fixes the Hilbert polynomial $P_F(k)=({\int }_{\beta }{c}_1(H))k+n$ and defines the moduli problem $P_n(X,\beta )$. Its associated complex is $I^{\bullet }=[{\mathcal{O}}_X\stackrel{s}{\to }F]$ in cohomological degrees $0,1$.

The DT ideal-sheaf datum $I_n(X,\beta )$ parametrizes ideal sheaves $I_Z\subseteq {\mathcal{O}}_X$ of subschemes $Z\subset X$ with dimension at most one, one-dimensional cycle class $[Z{]}_1=\beta$, and $\chi ({\mathcal{O}}_Z)=n$. The zero-dimensional part of $Z$ is allowed. These are distinct moduli problems: a stable pair is not identified with an ideal sheaf or with a stable map.

The cited derived-scheme supplier fixes the cotangent-complex convention if a moduli functor is given a derived enhancement. The cited square-zero-extension theorem applies to deformations of a specified flat, locally finitely presented ambient scheme over a small Artin extension. It does not identify the deformation-obstruction complex of a pair $(F,s)$ on fixed $X$; that pair-specific obstruction theory is a separate assertion and is required before constructing a PT virtual class. In particular, no stable-pair obstruction claim is deduced from the imported Ext${}^2$ theorem.

Derived schemes and the cotangent complex of a morphism