Deligne Mumford Stacks Research Articles

Mori dream stacks

We propose a generalisation of Mori dream spaces to stacks. We show that this notion is preserved under root constructions and taking abelian gerbes. Unlike the case of Mori dream spaces, such a stack is not always given as a quotient of the spectrum of its Cox ring by the Picard group. We give a criterion when this is true in terms of Mori dream spaces and root constructions. Finally, we compare this notion with the one of smooth toric Deligne-Mumford stacks.

A relative stack of principal G-bundles over $\mathcal{M}_{g, n}^{tw}$

In this paper, we record a proof that a relative moduli stack of principal G-bundles over a moduli stack of genus g, n-pointed twisted curves is a smooth Artin stack locally of finite type with the universal family. As an application, we reprove that the stack 𝒦g, n(𝒳, β) of Abramovich, Graber and Vistoli is a Deligne–Mumford stack of finite type when 𝒳 is a quotient stack.

Mass Formulas for Local Galois Representations and Quotient Singularities. I: A Comparison of Counting Functions

We study a relation between the Artin conductor and the weight coming from the motivic integration over wild Deligne-Mumford stacks. As an application, we prove some version of the McKay correspondence, which relates Bhargava's mass formula for extensions of a local field and the Hilbert scheme of points.

F-zips with additional structure

An $F$-zip over a scheme $S$ over a finite field is a certain object of semi-linear algebra consisting of a locally free module with a descending filtration and an ascending filtration and a $\Frob_q$-twisted isomorphism between the respective graded sheaves. In this article we define and systematically investigate what might be called "$F$-zips with a $G$-structure", for an arbitrary reductive linear algebraic group $G$. These objects come in two incarnations. One incarnation is an exact linear tensor functor from the category of finite dimensional representations of $G$ to the category of $F$-zips over $S$. Locally any such functor has a type $\chi$, which is a cocharacter of $G$. The other incarnation is a certain $G$-torsor analogue of the notion of $F$-zips. We prove that both incarnations define stacks that are naturally equivalent to a quotient stack of the form $[E_{G,\chi}\backslash G_k]$ that was studied in an earlier paper. By the results obtained there they are therefore smooth algebraic stacks of dimension 0 over $k$. Using our earlier results we can also classify the isomorphism classes of such objects over an algebraically closed field, describe their automorphism groups, and determine which isomorphism classes can degenerate into which others. For classical groups we can deduce the corresponding results for twisted or untwisted symplectic, orthogonal, or unitary $F$-zips. The results can be applied to the algebraic de Rham cohomology of smooth projective varieties (or generalizations thereof such as smooth proper Deligne-Mumford stacks) and to truncated Barsotti-Tate groups of level 1. In addition, we hope that our systematic group theoretical approach will help to understand the analogue of the Ekedahl-Oort stratification of the special fibers of arbitrary Shimura varieties.

The orbitwise volume form and geometry of a Deligne–Mumford stack

We discuss a way to deal with the Hodge star-operator and integration on a Deligne–Mumford (DM) quotient stack via equivariant geometry. After reviewing quickly the geometry of a DM quotient stack, we introduce the orbitwise volume form. Then we discuss how we can deal with the star-operator and integration without looking explicitly at the stack.

Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks

Laszlo and Olsson constructed Grothendieck’s six operations for constructible complexes on Artin stacks in étale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. We give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed by Zheng (2014). As an application, we prove a Lefschetz-Verdier formula for Deligne-Mumford stacks. We include both torsion and ℓ-adic coefficients.

$K$-theoretic and categorical properties of toric Deligne–Mumford stacks

We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.

The tropicalization of the moduli space of curves

We show that the skeleton of the Deligne-MumfordKnudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the “naive” set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.

On pliability of del Pezzo fibrations and Cox rings

Abstract We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over ℙ 1 ${\mathbb{P}^{1}}$ , into other Mori fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fibre spaces is at least three and they are not rational.

The geometry of stable quotients in genus one

Stable quotients provide an alternative to stable maps for compactifying spaces of maps. When \(n \ge 2\), the space \(\overline{Q}_{g}({\mathbb {P}}^{n-1},d) = \overline{Q}_{g}(G(1,n),d)\) compactifies the space of degree \(d\) maps of smooth genus \(g\) curves to \({\mathbb {P}}^{n-1}\), while \(\overline{Q}_{g}(G(1,1),d) \simeq \overline{M}_{1, d \cdot \epsilon }/S_d\) is a quotient of a Hassett weighted pointed space. In this paper we study the coarse moduli schemes associated to the smooth proper Deligne–Mumford stacks \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\), for all \(n \ge 1\). We show these schemes are projective, unirational, and have Picard number 2. Then we give generators for the Picard group, compute the canonical divisor, the cones of ample divisors, and in the case \(n=1\) the cones of effective divisors. We conclude that \(\overline{Q}_{1}({\mathbb {P}}^{n-1},d)\) is Fano if and only if \(n(d-1)(d+2) < 20\). Moreover, we show that \({\overline{Q}}_{1}({\mathbb {P}}^{n-1},d)\) is a Mori Fiber space for all \(n,d\), hence always minimal in the sense of the minimal model program. In the case \(n=1\), we write in addition a closed formula for the Poincaré polynomial.

A relative GAGA principle for families of curves

We prove a relative GAGA principle for families of curves, showing: (i) analytic families of pointed curves whose fibres have finite automorphism groups are algebraizable; and (ii) analytic birational models of M g , n possessing modular interpretations with the finite automorphism property are algebraizable. This is accomplished by extending some well-known GAGA results for proper schemes to non-separated Deligne–Mumford stacks.

Twisted orbifold Gromov–Witten invariants

AbstractLet χ be a smooth proper Deligne–Mumford stack over ℂ. One can define twisted orbifold Gromov–Witten invariants of χ by considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable maps χg,n,d, cupping them with evaluation and cotangent line classes, and then integrating against the virtual fundamental class. These are more general than the twisted invariants introduced by Tseng. We express the generating series of the twisted invariants in terms of the generating series of the untwisted ones. We derive the corollaries which are used in a paper with Givental about the quantum K-theory of a complex compact manifold X.

Twisted orbifold Gromov–Witten invariants

AbstractLetχbe a smooth proper Deligne–Mumford stack over ℂ. One can define twisted orbifold Gromov–Witten invariants ofχby considering multiplicative invertible characteristic classes of various bundles on the moduli spaces of stable mapsχg,n,d, cupping them with evaluation and cotangent line classes, and then integrating against the virtual fundamental class. These are more general than the twisted invariants introduced by Tseng. We express the generating series of the twisted invariants in terms of the generating series of the untwisted ones. We derive the corollaries which are used in a paper with Givental about the quantumK-theory of a complex compact manifoldX.

On semi-continuity problems for minimal log discrepancies

Abstract We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne–Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric pairs.

Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes

Abstract A quasi-smooth derived enhancement of a Deligne–Mumford stack 𝒳 naturally endows 𝒳 with a functorial perfect obstruction theory in the sense of Behrend–Fantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, for X = S an algebraic K3-surface, g ∈ ℕ, and β ≠ 0 in H 2(S,ℤ) a curve class, we construct a derived stack ℝ 𝐌 ¯ g , n red ( S ; β ) ${\mathbb {R}\overline{\mathbf {M}}^{\text{red}}_{g,n}(S;\beta )}$ whose truncation is the usual stack 𝐌 ¯ g , n ( S ; β ) ${\overline{\mathbf {M}}_{g,n}(S;\beta )}$ of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion 𝐌 ¯ g ( S ; β ) ↪ ℝ 𝐌 ¯ g red ( S ; β ) ${\overline{\mathbf {M}}_{g}(S;\beta )\hookrightarrow \mathbb {R}\overline{\mathbf {M}}^{\text{red}}_{g}(S;\beta )}$ induces on 𝐌 ¯ g ( S ; β ) ${\overline{\mathbf {M}}_{g}(S;\beta )}$ a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov–Maulik–Pandharipande–Thomas. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory – not relying on any result on semiregularity maps – but also a new global geometric interpretation. We give two further applications to moduli of complexes. For a K3-surface S we show that the stack of simple perfect complexes on S is smooth. This result was proved with different methods by Inaba for the corresponding coarse moduli space. Finally, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X) to the derived stack of simple perfect complexes on X with vanishing negative Ext's, and show how this map induces a morphism of the corresponding obstruction theories when X is a Calabi–Yau 3-fold. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is given by Illusie's trace map for perfect complexes.

Frobenius morphisms and derived categories on two dimensional toric Deligne–Mumford stacks

For a toric Deligne–Mumford (DM) stack X, we can consider a certain generalization of the Frobenius endomorphism. For such an endomorphism F:X→X on a 2-dimensional toric DM stack X, we show that the push-forward F∗OX of the structure sheaf generates the bounded derived category of coherent sheaves on X.We also choose a full strong exceptional collection from the set of direct summands of F∗OX in several examples of two dimensional toric DM orbifolds X.

From exceptional collections to motivic decompositions via noncommutative motives

Abstract Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(𝒳)ℚ of every smooth and proper Deligne–Mumford stack 𝒳, whose bounded derived category 𝒟 b (𝒳) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(𝒳)ℚ decomposes into a direct sum of tensor powers of the Lefschetz motive and moreover 𝒟 b (𝒳) admits a semi-orthogonal decomposition, then the noncommutative motive of each one of the pieces of the semi-orthogonal decomposition is a direct sum of ⊗-units. As an application we obtain a simplification of Dubrovin's conjecture.

A compactification of the space of maps from curves

We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper Deligne-Mumford stack equipped with a natural virtual fundamental class.

Deriving Deligne–Mumford stacks with perfect obstruction theories

We give conditions for a n-connective quasicoherent obstruction theory on a Deligne-Mumford stack to come from the structure of a connective spectral Deligne-Mumford stack on the underlying topos.

Białynicki-Birula decomposition of Deligne-Mumford stacks

This short note considers the Białynicki-Birula decomposition of Deligne-Mumford stacks under one-dimensional torus actions and extends a result of Oprea.