Using stacks to impose tangency conditions on curves

By Charles Cadman. 2003

275 citations recorded by the source corpus.

We define a Deligne-Mumford stack XD,r which depends on a scheme X, an effective Cartier divisor D ⊂ X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into XD,r provides compactifications of the locally closed substacks of M g,n(X, β) corresponding to relative stable maps.

Subfields: Geometry and Topology, Mathematical Physics, Organic Chemistry

View Paper: https://www.semanticscholar.org/paper/9c850fef43399aecfec6b9a174fe8b8caa109da2

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In Geometry and Topology

In Mathematical Physics

In Organic Chemistry

Citations in the corpus

The category of toric stacks (2006, methodology): “his structure is obtained by a simple technique called “taking n-th roots of an invertible sheaf”. We will explain it. Recall the notion of the stack of roots of an invertible sheaf (for example, see [Cad07]). Let X be an algebraic stack and Lan invertible sheaf on X. Let P : X →BGm be the morphism ”
Local Systems on Proper Algebraic V -manifolds (2010, background): “One of the main constructions of Deligne-Mumford stacks is to look at the Cadman-Vistoli root stacks.”
Tropical geometry and correspondence theorems via toric stacks (2010, methodology): “eduction of Xtr RL → SpecRL only, and X tr RL is obtained from X tr′ RL by extracting the roots of order l(ρ)along Oρ for all ρ∈ Σ1 Γtr,η, i.e., X tr RL = X tr′ RL D,~r , where D=∪Oρand~r =(l(ρ))(see [5] for the deﬁnition and the basic properties of the root stacks). Next, we deﬁne the stack Ctr RL ”
Abelian Hurwitz-Hodge integrals (2008, methodology): “Since P1[a] is a root stack, we may use the classification results on maps to root stacks proven in [5].”
Parabolic sheaves on logarithmic schemes (2010, background): “ogarithmic structure is generated by a single eﬀective Cartier divisor D ⊆ X, so that M is the constant sheaf ND on D, and we take B to be 1 dND, then XB/M is the root stack d p (X,D) (see [AGV08] or [Cad07]). Our main result (Theorem 6.1) is that the category of quasi-coherent parabolic sheaves on ”
Grothendieck Duality for Deligne-Mumford Stacks (2008, methodology): “smooth orbifold points is just the pullback of the dualizing sheaf of its moduli space. Smooth orbifold points give a non trivial contribution that can be computed using the root construction (Cadman [Cad07], Abramovich-Graber-Vistoli [AGV06]). We compute also the dualizing sheaf of a local complete”
Smooth toric Deligne-Mumford stacks (2010, methodology): “In the articles [12] and [2], the authors define the notion of root of an invertible sheaf with a section on an algebraic stack: here, we only consider roots of e¤ective Cartier divisors on a smooth algebraic stack, since this is what we will use.”
Relative and orbifold Gromov-Witten invariants (2010, background): “stable curve might expand to prevent a marked point from colliding with a node. The deformation theory of maps from curves into expanded targets still remains quite subtle, however. A second solution [Cad07] is to change the target variety by a root construction. The variety is replaced by a stack t”
Katz's middle convolution algorithm (2006, methodology): “Katz’s original method. If the weights of a parabolic structure are rational, i.e. for every point qi the parabolic type is a divisor concentrated over roots of unity in S1, then as discussed in [19] [27] [108] [77], the parabolic bundle may also be viewed as a bundle on a Deligne-Mumford stack Z[Q ”