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Computing $A$-resultants via direct images

Friedemann Groh, Matthias Zach

TL;DR

A new algorithm is introduced by introducing a new algorithm for the computation of direct images of complexes of coherent sheaves on toric varieties by improving a previously known theoretic method to compute A-resultants for suitable monomial support sets due to Weyman.

Abstract

We improve a previously known theoretic method to compute A-resultants for suitable monomial support sets due to Weyman to the extent that it becomes computationally feasible and effective. This is achieved by introducing a new algorithm for the computation of direct images of complexes of coherent sheaves on toric varieties. The procedure does not rely on Gröbner basis computations at any stage.

Computing $A$-resultants via direct images

TL;DR

A new algorithm is introduced by introducing a new algorithm for the computation of direct images of complexes of coherent sheaves on toric varieties by improving a previously known theoretic method to compute A-resultants for suitable monomial support sets due to Weyman.

Abstract

We improve a previously known theoretic method to compute A-resultants for suitable monomial support sets due to Weyman to the extent that it becomes computationally feasible and effective. This is achieved by introducing a new algorithm for the computation of direct images of complexes of coherent sheaves on toric varieties. The procedure does not rely on Gröbner basis computations at any stage.
Paper Structure (18 sections, 5 theorems, 77 equations, 1 figure, 4 tables)

This paper contains 18 sections, 5 theorems, 77 equations, 1 figure, 4 tables.

Table of Contents

  1. Results
  2. Methods from the Green Book
  3. Broadening the scope of applicability
  4. The Weyman functor
  5. Synopsis to related work
  6. An algorithmic construction of the Weyman complex
  7. Toric Varieties and Čech Cohomology
  8. Gauß reduction for complexes
  9. Simplification of total complexes
  10. Detailed construction of the Weyman functor
  11. Differentials in the Weyman complex
  12. The Weyman functor on morphisms of complexes
  13. Examples
  14. Sturmfels' example
  15. Parametric images of rational curves
  16. ...and 3 more sections

Key Result

Proposition 1.1

(GelfandKapranovZelevinsky94) When $n = m+1$ and all line bundles $\mathcal{L}_i$ are very ample, the variety $\nabla_{\mathcal{L}_1,\dots, \mathcal{L}_{m+1}}$ is a multihomogeneous irreducible hypersurface.

Figures (1)

  • Figure 1: Runtimes for the computation of the morphism of the direct image from Example \ref{['exp:CotantengSheafExample']} and Table \ref{['tab:TimingsCotangentSheafExample']}

Theorems & Definitions (13)

  • Proposition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • proof
  • Theorem 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 3 more