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Project descriptions
Group leaders can edit this wiki page to provide a description and background for their projects.
Leader: Keller VandeBogert (Notre Dame)
The general goal of the "complexes of sheaves" group is to extend current functionality for morphisms of sheaves to be compatible with the new type "Complex of Sheaves" over rings constructed as Proj of some homogeneous quotient ring. Currently morphisms of sheaves along with induced maps on cohomology are only implemented for projective varieties, so another direction for this group would be to work on implementing morphisms of sheaves to be compatible with the "NormalToricVarieties" type; this should be done in close communication with the "Toric Varieties" working group. Some concrete goals that we will be working toward are the following:
- Having the types "Complex of Sheaves" and "Morphism of Complexes" (of sheaves) implemented along with checks for well-definedness.
- Implementing all functionality that currently exists for complexes of modules for complexes of sheaves (such as tensor products, accessing differentials, mapping cones, homology, etc), and also being able to take sheaves associated to complexes of modules in a functorial way.
- Compatibility of complexes of sheaves with cohomology and hypercohomology spectral sequences, and more generally having the capability to perform concrete computations in the derived category of sheaves (such as derived global sections functors/derived global Hom).
- Implementing concrete and interesting examples of complexes of sheaves (such as Beilinson's resolution of the diagonal).
By the end of the week, it would be nice to be able to give examples of complexes of sheaves whose hypercohomology spectral sequences have interesting page maps, and to see these page maps constructed explicitly.
Leaders: Liam McAllister (Cornell) and Mike Stillman (Cornell)
Leader: Michael Brown (Auburn)
By a theorem of Orlov, given a Calabi-Yau complete intersection X embedded in projective space with affine cone Spec(R), there is an equivalence of triangulated categories between the bounded derived category of X and the singularity category of R. This follows from Theorem 2.13 in the following paper:
https://arxiv.org/pdf/math/0503632
The functor from the singularity category of R to the bounded derived category of X in this equivalence is difficult to compute by hand; the goal of this project is to implement this functor in Macaulay2.
Leader: Hunter Simper (Utah)
There is a natural action of GL_n x GL_m on a polynomial ring K[x_{i,j}]. The ideals which are invariant under this action were classified by De Concini, Eisendbud and Procesi and correspond to collections of partitions. The objective of this group is to create tools to go back and forth between ideals and partitions as well as other tools to assist in manipulating such ideals in Macaulay2.
Leader: Karl Schwede (Utah)
Leader: David Eisenbud (Berkeley)
The goal of our group will be to create a package to implement and benchmark the algorithms described in sections 3-10 of the attached short paper by Craig Huneke. It would be important for the members of the group to have read the paper, at least in first approximation, in advance of our meeting.
"Finding nilpotents with Wolmer Vasconcelos" by Craig Huneke
- Make benchmarks for computing the radical and the related ``minimalPrimes'' commands in
Macaulay2. We need several ranges of radical problems of
variable difficulty eg:
- radical of a power of a random polynomial, random ideal
- radical of a power of a determinantal ideal
- radical of the kernel of a map from a polynomial ring onto a ring of the form k{t^S, x^[p])) where S is a numerical semigroup, and x^[p] is the ideal of pth powers of some variables.
- radical of ideals as above, but in singular and/or inhomogeneous settings.
- (understand and) Document the Strategies for these
- implement the algorithm in Huneke's paper, derived from Levin's theorem (proof in Huneke's notes, in the repo); compare with the methods in M2. (It may be that this algorithm is of more theoretical than practical interest.)
Leader: Devlin Mallory (Utah)
The goal of this group is to implement some basic computations involving ruled surfaces (i.e., projectivizations of rank-2 vector bundles on curves). Ruled surfaces provide a tractable yet interesting class of varieties, which provide examples of many phenomena, some of which we hope to be able to see in Macaulay2. For example, they are the simplest varieties for which Kawamata Viehwig vanishing can fail in characteristic p; let's work this out explicitly in M2. Some specific goals or directions include:
- Provide a constructor that takes a vector bundle on a variety, and returns the projectivization of the vector bundle. We'll focus on the case of a rank-2 vector bundle on a curve, but depending on what we decide we can work in greater generality.
- Given a projectivized vector bundle, describe it as a subvariety of ambient (multi)projective spaces, e.g., by finding very ample line bundles and describing the resulting embedding.
- If E is a vector bundle on a curve C, X = P(E), and E -> L is a surjection onto a line bundle, describe the resulting section C -> X of the projective bundle.
- Implement all the operations and calculations for ruled surfaces described in Chapter 5, Section 2 of Hartshorne.
- Work out examples where Kawamata--Viehwig (or Kodaira) vanishing fails. This will also involve working with Tango bundles on curves.
If you have other suggestions for what this package could or should do, please feel free to suggest them!
Leader: Gregory G. Smith (Queen's)
The general goal of the "toric variety working group" will be to improve the functionality of the existing NormalToricVarieties package. Some specific ideas are detailed below. We hope to find tasks that appeal to everyone in the group—we will almost surely subdivide into smaller subgroups. Here are a few concrete features that should be implemented:
- Provide a general constructor for a normal toric variety that takes a list of line bundles on some normal toric variety and returns a new normal toric variety representing the projectivization of the direct sum of line bundles. This method should also construct the canonical toric morphism from the new toric variety to the underlying one; see Proposition 7.3.3. in Cox-Little-Schenck for more details.
- Provide constructors for all smooth projective toric varieties having Picard rank 3 following Batyrev's paper "On the classification of smooth projective toric varieties"; see Theorems 5.7 and 6.6.
- Provide better methods for working with linear series on projective toric varieties. One should be able to compute the homogeneous coordinate ring of the associated image in projective space (and the corresponding projective variety) easily and efficiently. For an equivariant linear series, one should also be able to obtain the corresponding toric morphism easily.
- Provide methods for constructing the toric morphisms associated to faces in the nef cone for a projective toric variety; see Subsection 15.4 in Cox-Little-Schenck.
Are their other routines that would facilitate your personal research activities? Are there features that would make using this package more convenient?