CStarSurfaces.jl
A computer algebra package for rational $\mathbb{C}^*$-surfaces in the Julia programming language. This package makes use of the OSCAR Computer Algebra System.
The approach to $\mathbb{C}^*$-surfaces relies on the general combinatorial theory of varieties with finitely generated Cox ring developed in [1], [2] and its specialization to varieties with torus action initiated in [3] and [4]. As an introductory reference, we mention [5].
Features
The following invariants can be computed for $\mathbb{C}^*$-surfaces and toric surfaces:
- divisor class group, local class groups, (anti)canonical divisor class,
- Cox Ring, Gorenstein index, Picard index,
- intersection numbers, anticanonical self intersection,
- resolution of singularities, log canonicity, resolution graphs, ADE singularities,
- normal form of defining data, admissible operations, isomorphy test.
Furthermore, some functionality to save and retrieve $\mathbb{C}^*$-surfaces from a database is provided, see also the ldp-database
.
Installation
CStarSurfaces.jl is available in the General Registry, hence can be installed by typing ]add CStarSurfaces
into a Julia REPL.
Quick start
We work in the notation of [5, Section 5.4].
Import both Oscar and CStarSurfaces.jl to get started:
julia> using Oscar, CStarSurfaces
There are essentially two constructors for C-Star surfaces: The first takes the integral vectors $l_i=(l_{i1}, \dots, l_{in_i})$ and $d_i=(d_{i1}, \dots, d_{in_i})$ and one of the four symbols :ee, :pe, :ep, :pp
. The second takes the generating matrix P
of the correct shape:
julia> X = cstar_surface([[1, 1], [4], [4]], [[0, -2], [3], [3]], :ee)
C-star surface of type (e-e)
julia> Y = cstar_surface(ZZ[-1 -1 4 0 ; -1 -1 0 4 ; 0 -2 3 3])
C-star surface of type (e-e)
julia> X == Y
true
gen_matrix
returns the generating matrix (P-Matrix) of a C-star surface:
julia> gen_matrix(X)
[-1 -1 4 0] [-1 -1 0 4] [ 0 -2 3 3]
canonical_toric_ambient
returns the canonical toric ambient variety of a C-star surface, as an Oscar type:
julia> Z = canonical_toric_ambient(X)
Normal toric variety
We compute some geometric invariants of $X$:
julia> class_group(X)
GrpAb: Z/8 x Z
julia> cox_ring(X)
Quotient of graded multivariate polynomial ring in 4 variables over QQ by ideal(T[0][1]*T[0][2] + T[1][1]^4 + T[2][1]^4)
julia> gorenstein_index(X)
3
julia> picard_index(X)
48
julia> K = anticanonical_divisor(X)
CStarSurfaceDivisor{EE}(C-star surface of type (e-e), [0, 0, 1, 1], #undef)
julia> K * K # the anticanonical self intersection
2//3
julia> (Y, exceptional_divisors, discrepancies) = canonical_resolution(X);
julia> gen_matrix(Y)
[-1 -1 -1 4 1 3 2 1 0 0 0 0 0 0 0] [-1 -1 -1 0 0 0 0 0 4 1 3 2 1 0 0] [ 0 -2 -1 3 1 2 1 0 3 1 2 1 0 1 -1]
julia> log_canonicity(X)
1//3
References
- [1]
- F. Berchtold and J. Hausen. Cox rings and combinatorics. Trans. Amer. Math. Soc. 359, 1205–1252 (2007).
- [2]
- J. Hausen. Cox rings and combinatorics. II. Mosc. Math. J. 8, 711–757, 847 (2008).
- [3]
- J. Hausen and E. Herppich. Factorially graded rings of complexity one. In: Torsors, étale homotopy and applications to rational points, Vol. 405 of London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 2013); pp. 414–428.
- [4]
- J. Hausen and H. Süß. The Cox ring of an algebraic variety with torus action. Adv. Math. 225, 977–1012 (2010).
- [5]
- I. Arzhantsev, U. Derenthal, J. Hausen and A. Laface. Cox rings. Vol. 144 of Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2015); p. viii+530.
- [6]
- D. Hättig, J. Hausen and H. Süß. Log del Pezzo ``\mathbb{C}^``-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics* (2023), arXiv:2306.03796 [math.AG].