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Publisher's Note: “Cohomology of line bundles: Proof of the algorithm” [J. Math. Phys. 51, 103520 (2010)]
1. R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of line bundles: A computational algorithm” (2010) e-print arXiv:1003.5217 [hep-th].
2.The speed-optimized implementation in C++ can be downloaded from http://wwwth.mppmu.mpg.de/members/blumenha/cohomcalg/ and is regularly updated. To get a first experience of the calculations possible, one can also have a quick start with a short Mathematica script that is also available there.
3. R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of line bundles: Applications” (unpublished).
4. D. Grayson and M. Stillman, “Macaulay 2, a software system for research in algebraic geometry,” Available by ftp at http://www.math.uiuc.edu/Macaulay2/.
5.In order to do sheaf cohomology computations on general toric varieties, the additional package “NORMALTORICVARIETIES.M2” written by G. Smith is needed. Since this is still work in progress, it is not yet included in the official distribution, but the package content can be copied from his homepage, http://www.cs.amherst.edu/~dac/toric.html, and then separately loaded into MACAULAY2.
6. D. A. Cox, J. B. Little, and H. Schenck, Toric Varieties (unpublished), available at http://www.cs.amherst.edu/~dac/toric.html.
7. M. Cvetic, I. Garcia-Etxebarria, and J. Halverson, “Global F-theory models: Instantons and gauge dynamics” (2010) e-print arXiv:1003.5337 [hep-th].
8.Note that it can be shown that Čechcohomology on an open cover of a toric variety can be shown to be isomorphic to sheaf cohomology, see Theorem 9.0.4 in Ref. 6.
9. S.-Y. Jow, “Cohomology of toric line bundles via simplicial Alexander duality” e-print arXiv:1006.0780 [math.AGJ].
10.In the sense of Chap. 3 of Re. 6.
11.A condensed introduction to simplicial complexes meeting our requirements is given, e.g., by the first chapter of Ref. 15.
12.For a short review of sheaf theory and sheaf cohomology have a look at the appendix of Ref. 1.
13.Note that we always identiy Picard group and class group of X, since in the smooth case all Weil divisors are already Cartier.
14.The shift in the rank comes from a shift between the ordinary and the local Čech complex, see also Theorem 9.5.7 in Ref. 6.
15. E. Miller and B. Strumfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics Vol. 227 (Springer, New York, 2005).
16.Here, the term “power set of an ideal” stands for taking all possible unions of the generators. In fact, the sequences for remnant cohomology in the algorithm of Ref. 1 come from the combinatorics of this power set and the connection with the full Taylor resolution of S/I will be important for the proof.
17.For example, the Taylor resolution of the Stanley-Reisner ring of X = dP3 is not minimal, since the subset consisting of is among the generators of its Stanley-Reisner ideal, cf. the examples in Ref. 1.
18.See Ref. 25 for more details on these categorical issues.
19. D. Eisenbud, M. Mustată, and M. Stillman, J. Symb. Comput. 29, 583 (2000).http://dx.doi.org/10.1006/jsco.1999.0326
20.We want to note that the set (α, σ) corresponds to all so-called rationoms x
u with f(u) = α and precisely the coordinates x i with i ∈ σ standing in the denominator. Intuitively, these rationoms can be interpreted as “representatives” of Čech cohomology on intersections of open sets in the toric variety, cf. Sec. 2.2 of Ref. 1.
21.This corresponds to the sequences for “remnant cohomology” in Ref. 1. By counting the number of times that a fixed denominator xσ appears in rank r of the Stanley-Reisner power set, one gets the number of (r−1)-faces of the complex Γ
σ. If one also takes notice of the different combinations of Stanley-Reisner generators that lead to this denominator, one can write down the maps in Eq. (40) and gets a well-defined complex.
22. D. MacLagan and G. Smith, J. Reine Angew. Math. 571, 179 (2004).
23. D. Bayer, H. Charalambous, and S. Popescu, J. Algebra 221, 497 (1999).http://dx.doi.org/10.1006/jabr.1999.7970
24.For X = dP3 the divisor D = −3H − X − Y − Z that we took as an example in Ref. 1 is such a boundary divisor with the same charge as .