Skip to content

Catalog

Browse quantum multiplication matrices by geometry, or use the table below to find an example. The catalog includes projective and Grassmannian complete intersections, other flag varieties, and supported Fano threefold realizations.

Reading an entry

Matrices describe multiplication by the first Chern class on the returned ambient sector. Basis Rank is the size of the displayed matrix. The Ambient Index field uses the software’s ambient Chern gcd; a conic can have displayed index 1 while its geometric index is 2, and this gcd does not define a geometric Fano index for a point. See Methods and conventions.

Entries labelled Catalog only are reference descriptions without a supported computation. To repeat a computed example, use its algebra, kept nodes and bundle in the user guide. Download matrices and inputs provides the symbolic results, ordered bases and gradings for all 131 saved computations.

All computed entries

Filter by a name or number, or select a column heading to sort the table.

Geometry Ambient Space Dimension Ambient Index Basis Rank
Gr(2,6) A5 8 6 15
P8 Fano index 9 A8 8 9 9
Fl(1,4,5) A4 7 4 20
Gr(2,6) / O(1) A5 7 5 12
P7 Fano index 8 A7 7 8 8
P8 / O(2) A8 7 7 8
P8 / O(3) A8 7 6 8
P8 / O(4) A8 7 5 8
P8 / O(5) A8 7 4 8
P8 / O(6) A8 7 3 8
P8 / O(7) A8 7 2 8
Gr(2,5) A4 6 5 10
Gr(2,6) / O(1)+O(1) A5 6 4 10
Gr(2,6) / O(1)+O(2) A5 6 3 10
Gr(3,5) A4 6 5 10
P6 Fano index 7 A6 6 7 7
P7 / O(2) A7 6 6 7
P7 / O(3) A7 6 5 7
P7 / O(4) A7 6 4 7
P7 / O(5) A7 6 3 7
P7 / O(6) A7 6 2 7
P8 / O(2)+O(2) A8 6 5 7
P8 / O(2)+O(3) A8 6 4 7
P8 / O(2)+O(4) A8 6 3 7
P8 / O(2)+O(5) A8 6 2 7
P8 / O(3)+O(3) A8 6 3 7
P8 / O(3)+O(4) A8 6 2 7
Fl(1,4,5) / O(1,1)+O(1,1) A4 5 2 12
G2/P1 G2 5 5 6
G2/P2 G2 5 3 6
Gr(2,5) / O(1) A4 5 4 8
Gr(2,5) / O(2) A4 5 3 8
Gr(2,5) / O(3) A4 5 2 8
Gr(2,6) / O(1)+O(1)+O(1) A5 5 3 8
Gr(2,6) / O(1)+O(1)+O(2) A5 5 2 8
Gr(3,5) / O(1) A4 5 4 8
Gr(3,5) / O(2) A4 5 3 8
Gr(3,5) / O(3) A4 5 2 8
LG(3,6) / O(1) C3 5 3 6
OG(3,7) / O(2) B3 5 4 6
OG(3,7) / O(3) B3 5 3 6
OG(3,7) / O(4) B3 5 2 6
P5 Fano index 6 A5 5 6 6
P6 / O(2) A6 5 5 6
P6 / O(3) A6 5 4 6
P6 / O(4) A6 5 3 6
P6 / O(5) A6 5 2 6
P7 / O(2)+O(2) A7 5 4 6
P7 / O(2)+O(3) A7 5 3 6
P7 / O(2)+O(4) A7 5 2 6
P7 / O(3)+O(3) A7 5 2 6
P8 / O(2)+O(2)+O(2) A8 5 3 6
P8 / O(2)+O(2)+O(3) A8 5 2 6
Fl(1,2,5) / O(0,1)+O(0,2)+O(1,0) (GM-20) A4 4 1 9
Fl(1,3,4) / O(1,1) A3 4 2 9
G2/P2 / O(1) G2 4 2 5
Gr(2,4) A3 4 4 6
Gr(2,5) / O(1)+O(1) A4 4 3 6
Gr(2,5) / O(1)+O(2) A4 4 2 6
Gr(2,5) / O(1)+O(3) A4 4 1 6
Gr(2,5) / O(2)+O(2) A4 4 1 6
Gr(3,5) / O(1)+O(1) A4 4 3 6
Gr(3,5) / O(1)+O(2) A4 4 2 6
Gr(3,5) / O(1)+O(3) A4 4 1 6
Gr(3,5) / O(2)+O(2) A4 4 1 6
Gr(4,5) A4 4 5 5
LG(3,6) / O(1)+O(1) C3 4 2 5
OG(3,7) / O(2)+O(2) B3 4 2 5
P4 Fano index 5 A4 4 5 5
P5 / O(2) A5 4 4 5
P5 / O(3) A5 4 3 5
P5 / O(4) A5 4 2 5
P5 / O(5) A5 4 1 5
P6 / O(2)+O(2) A6 4 3 5
P6 / O(2)+O(3) A6 4 2 5
P6 / O(2)+O(4) A6 4 1 5
P6 / O(3)+O(3) A6 4 1 5
P7 / O(2)+O(2)+O(2) A7 4 2 5
P7 / O(2)+O(2)+O(3) A7 4 1 5
B2/B / O(1,1) B2 3 1 6
Fanography 1-10 A6 3 1 4
Fanography 1-13 A4 3 2 4
Fanography 1-14 A5 3 2 4
Fanography 1-15 A4 3 2 4
Fanography 1-16 A4 3 3 4
Fanography 1-17 A3 3 4 4
Fanography 1-2 A4 3 1 4
Fanography 1-3 A5 3 1 4
Fanography 1-4 A6 3 1 4
Fanography 1-5 A4 3 1 4
Fanography 1-6 D5 3 1 4
Fanography 1-7 A5 3 1 4
Fanography 1-8 C3 3 1 4
Fanography 1-9 G2 3 1 4
Fanography 2-24 A2xA2 3 1 6
Fanography 2-32 A2 3 2 6
Fanography 2-34 A1xA2 3 1 6
Fanography 3-11 A1xA1xA2 3 1 8
Fanography 3-27 A1xA1xA1 3 2 8
Fanography 4-1 A1xA1xA1xA1 3 1 10
Fl(1,2,3) A2 3 2 6
Fl(1,3,4) / O(1,1)+O(1,1) A3 3 1 6
Gr(2,4) / O(1) A3 3 3 4
Gr(2,4) / O(2) A3 3 2 4
Gr(2,4) / O(3) A3 3 1 4
Gr(2,5) / O(1)+O(1)+O(1) A4 3 2 4
Gr(2,5) / O(1)+O(1)+O(2) A4 3 1 4
Gr(3,4) A3 3 4 4
Gr(3,5) / O(1)+O(1)+O(1) A4 3 2 4
Gr(3,5) / O(1)+O(1)+O(2) A4 3 1 4
Gr(4,5) / O(1) A4 3 4 4
Gr(4,5) / O(2) A4 3 3 4
Gr(4,5) / O(3) A4 3 2 4
Gr(4,5) / O(4) A4 3 1 4
P3 Fano index 4 A3 3 4 4
P4 / O(2) A4 3 3 4
P4 / O(3) A4 3 2 4
P4 / O(4) A4 3 1 4
P5 / O(2)+O(2) A5 3 2 4
P5 / O(2)+O(3) A5 3 1 4
P6 / O(2)+O(2)+O(2) A6 3 1 4
Gr(2,4) / O(1)+O(1) A3 2 2 3
Gr(2,4) / O(1)+O(2) A3 2 1 3
Gr(3,4) / O(1) A3 2 3 3
Gr(3,4) / O(2) A3 2 2 3
Gr(3,4) / O(3) A3 2 1 3
Gr(4,5) / O(1)+O(1) A4 2 3 3
Gr(4,5) / O(1)+O(2) A4 2 2 3
Gr(4,5) / O(1)+O(3) A4 2 1 3
Gr(4,5) / O(2)+O(2) A4 2 1 3
P2 Fano index 3 A2 2 3 3
P3 / O(2) A3 2 2 3
P3 / O(3) A3 2 1 3
P4 / O(2)+O(2) A4 2 1 3
Gr(2,4) / O(1)+O(1)+O(1) A3 1 1 2
Gr(3,4) / O(1)+O(1) A3 1 2 2
Gr(3,4) / O(1)+O(2) A3 1 1 2
Gr(4,5) / O(1)+O(1)+O(1) A4 1 2 2
Gr(4,5) / O(1)+O(1)+O(2) A4 1 1 2
P2 / O(2) A2 1 1 2
Gr(3,4) / O(1)+O(1)+O(1) A3 0 1 1