arXiv subjects
Publications and source records attributed to Barbara Fantechi.
We give criteria for the existence of geometric smoothings of a proper lci scheme or a DM stack $X$ as well as for a polarized lci scheme $(X,L)$, without assuming that $X$ is reduced. As applications, we give criteria for the smoothability of polarized K3 surfaces and of stable varieties.
Let $X$ be a variety. In this survey, we study (decompositions of) the motivic class, in the Grothendieck ring of stacks, of the stack $\mathscr{C}oh^n(X)$ of $0$-dimensional coherent sheaves of length $n$ on $X$. To do so, we review the construction of the support map $\mathscr{C}oh^n(X) \to \mathrm{Sym}^n(X)$ to the symmetric product and we prove that, for any closed point $p \in X$, the motive of the punctual stack $\mathscr{C}oh^n(X)_p$ parametrising sheaves supported at $p$ only depends on a formal neighbourhood of $p$. We perform the same analysis for the Quot-to-Chow morphism $\mathrm{Quot}_X(\mathcal E,n) \to \mathrm{Sym}^n(X)$, for a fixed sheaf $\mathcal E \in \mathrm{Coh}(X)$.
Let $X$ be a quasiprojective scheme. In this expository note we collect a series of useful structural results on the stack $\mathscr{C}oh^n(X)$ parametrising $0$-dimensional coherent sheaves of length $n$ over $X$. For instance, we discuss its functoriality (in particular its behaviour along \'etale maps), the support morphism to $\mathrm{Sym}^n(X)$, and its relationship with the Quot scheme of points $\mathrm{Quot}_X(\mathcal E,n)$ for fixed $\mathcal E\in \mathrm{Coh}(X)$.
Mukai proved that the moduli space of simple sheaves on a smooth projective K3 surface is symplectic, and in \cite{FM2} we gave two constructions allowing one to construct new locally closed Lagrangian/isotropic subspaces of the moduli from old ones. In this paper, we extend both Mukai's result and our construction to reduced projective K3 surfaces; for the former we need to restrict our attention to perfect sheaves. There are two key points where we cannot get a straightforward generalization. In each, we need to prove that a certain differential form on the moduli space of simple, perfect sheaves vanishes, and we introduce a smoothability condition to complete the proof.
Let $X$ be a K3 surface and let $\text{Spl}(r;c_1,c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. Under suitable assumptions, to a pair $(F,W)$ (respectively, $(F,V)$) where $F\in \text{Spl}(r;c_1,c_2)$ and $W\subset H^0(F)$ (resp.~$V^*\subset H^1(F^*)$) is a vector subspace, we associate a simple syzygy bundle (resp.~extension bundle) on $X$. We show that both syzygy bundles and extension bundles can be constructed in families and that the induced morphism to a different component of the moduli of simple sheaves is a locally closed embedding. We show that this construction associates to every Lagrangian (resp.~isotropic) algebraic subspace of $\text{Spl}(r;c_1,c_2)$ an induced Lagrangian (resp.~isotropic) algebraic subspace of a different component of the moduli of simple sheaves.
Given a vector bundle $F$ on a variety $X$ and $W\subset H^0(F)$ such that the evaluation map $W\otimes \mathcal{O}_X\to F$ is surjective, its kernel $S_{F,W}$ is called generalized syzygy bundle. Under mild assumptions, we construct a moduli space $\mathcal{G}^0_U$ of simple generalized syzygy bundles, and show that the natural morphism $\alpha$ to the moduli of simple sheaves is a locally closed embedding. If moreover $H^1(X,\mathcal{O}_X)=0$, we find an explicit open subspace $\mathcal{G}^0_V$ of $\mathcal{G}^0_U$ where the restriction of $\alpha$ is an open embedding. In particular, if $\dim X\ge 3$ and $H^1(\mathcal{O}_X)=0$, starting from an ample line bundle (or a simple rigid vector bundle) on $X$ we construct recursively open subspaces of moduli spaces of simple sheaves on $X$ that are smooth, rational, quasiprojective varieties.
We show that all the semi-smooth stable complex Godeaux surfaces, classified in [FPR18a], are smoothable, and that the moduli stack is smooth of the expected dimension 8 at the corresponding points.
For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are \'etale locally the product of a double crossing point (uv=0) or a pinch point (u^2-v^2w=0) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf T^1_X for a semi-smooth variety X in terms of the gluing data.
For any locally free coherent sheaf on a fixed smooth projective curve, we study the class, in the Grothendieck ring of varieties, of the Quot scheme that parametrizes zero-dimensional quotients of the sheaf. We prove that this class depends only on the rank of the sheaf and on the length of the quotients. As an application, we obtain an explicit formula that expresses it in terms of the symmetric products of the curve.
Let $\overline{\mathcal{M}}_{g,A[n]}$ be the Hassett moduli stack of weighted stable curves, and let $\overline{M}_{g,A[n]}$ be its coarse moduli space. These are compactifications of $\mathcal{M}_{g,n}$ and $M_{g,n}$ respectively, obtained by assigning rational weights $A = (a_{1},...,a_{n})$, $0< a_{i} \leq 1$ to the markings; they are defined over $\mathbb{Z}$, and therefore over any field. We study the first order infinitesimal deformations of $\overline{\mathcal{M}}_{g,A[n]}$ and $\overline{M}_{g,A[n]}$. In particular, we show that $\overline{M}_{0,A[n]}$ is rigid over any field, if $g\geq 1$ then $\overline{\mathcal{M}}_{g,A[n]}$ is rigid over any field of characteristic zero, and if $g+n > 4$ then the coarse moduli space $\overline{M}_{g,A[n]}$ is rigid over an algebraically closed field of characteristic zero. Finally, we take into account a degeneration of Hassett spaces parametrizing rational curves obtained by allowing the weights to have sum equal to two. In particular, we consider such a Hassett $3$-fold which is isomorphic to the Segre cubic hypersurface in $\mathbb{P}^4$, and we prove that its family of first order infinitesimal deformations is non-singular of dimension ten, and the general deformation is smooth.
The stack $\overline{\mathcal{M}}_{g,n}$ of stable curves and its coarse moduli space $\overline{M}_{g,n}$ are defined over $\mathbb{Z}$, and therefore over any field. Over an algebraically closed field of characteristic zero, Hacking showed that $\overline{\mathcal{M}}_{g,n}$ is rigid (a conjecture of Kapranov). Bruno and Mella for $g=0$, and the second author for $g\geq 1$ showed that its automorphism group is the symmetric group $S_n$, permuting marked points unless $(g,n)\in\{(0,4),(1,1),(1,2)\}$. The methods used in the papers above do not extend to positive characteristic. We show that in characteristic $p>0$, the rigidity of $\overline{\mathcal{M}}_{g,n}$, with the same exceptions as over $\mathbb{C}$, implies that its automorphism group is $S_n$. We prove that, over any perfect field, $\overline{M}_{0,n}$ is rigid and deduce that, over any field, $Aut(\overline{M}_{0,n})\cong S_{n}$ for $n\geq 5$. Going back to characteristic zero, we prove that for $g+n>4$, the coarse moduli space $\overline M_{g,n}$ is rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that $\overline{M}_{1,2}$ is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family.
We give an approach for relative and degenerate Gromov--Witten invariants, inspired by that of Jun Li but replacing predeformable maps by transversal maps to a twisted target. The main advantage is a significant simplification in the definition of the obstruction theory. We reprove in our language the degeneration formula, extending it to the orbifold case.
Consider a smooth variety $X$ and a smooth divisor $D\subset X$. Kim and Sato (arXiv:0806.3819) define a natural compactification of $(X\setminus D)^n$, denoted $X_D^{[n]}$, which is a moduli space of stable configurations of $n$ points lying on expansions of $(X,D)$ in the sense of Jun Li (arXiv:math/0009097, arXiv:math/0110113). The purpose of this note is to generalize Kim and Sato's construction to the case where $X$ is an algebraic stack; and to construct an analogous projective moduli space $W_π^{[n]}$ for a degeneration $π:W \to B$. We construct $X^n_D$ and $W_π^{[n]}$ and prove their properness using a universal construction introduced in our paper arXiv:1110.2976 with Cadman and Wise. We then use these spaces for a concrete application, as explained in the next paragraph. In arXiv:1103.5132, a degeneration formula for Gromov--Witten invariants of schemes and stacks is developed, generalizing the approach of Jun Li. This in particular requires proving properness of Li's stack of pre-deformable stable maps in the case where the target $(X,D)$ or $W\to B$ is a Deligne--Mumford stack. One could simply adapt Li's proof, or follow the age-old tradition of imposing such endeavor as an exercise on "the interested reader". Instead, we prefer to provide a different proof here, which uses the properness of $X_D^{[n]}$ and $W_π^{[n]}$. Similar ideas are used by Kim, Kresch and Oh (arXiv:1105.6143) to prove the properness of their space of ramified maps. This note is identical to the text available on our web pages since March 2013. It is posted now as it has become an essential ingredient in others' work.
Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas.
We give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable sheaves.
We introduce the notion of symmetric obstruction theory and study symmetric obstruction theories which are compatible with C*-actions. We prove that the contribution of an isolated fixed point under a C*-action to equivariant Donaldson-Thomas type invariants is +/- 1. As an application, we compute weighted Euler characteristics of all Hilbert schemes of points on any 3-fold. Moreover, we calculate the zero-dimensional Donaldson-Thomas invariants of any projective Calabi-Yau 3-fold. This proves a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande.
For an orbifold X which is the quotient of a manifold Y by a finite group G we construct a noncommutative ring with an action of G such that the orbifold cohomology of X as defined in math.AG/0004129 by Chen and Ruan is the G invariant part. In the case thar Y is S^n for a surface S with trivial canonical class we prove that (a small modification of) the orbifold cohomology of X is naturally isomorphic to the cohomology ring of the Hilbert scheme of n points on S computed in math.AG/0012166 by Lehn and Sorger.