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| | Description and Representative | Tensor Rank |
| | | Rank Distr. |
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| o1 | e1 ⊗ e1 ⊗ e1 | 1 |
| PG(A1) : | Point on S3,3 | [1,0,0] |
| PG(A2) : | Point on S2,3 | [1,0,0] |
| PG(A3) : | Point on S2,3 | [1,0,0] |
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| o2 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2) | 2 |
| PG(A1) : | Point of rank 2 | [0,1,0] |
| PG(A2) : | Line on S2,3 | [q + 1,0,0] |
| PG(A3) : | Line on S2,3 | [q + 1,0,0] |
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| o3 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3) | 3 |
| PG(A1) : | Point of rank 3 | [0,0,1] |
| PG(A2) : | Plane on S2,3 | [q2 + q + 1,0,0] |
| PG(A3) : | Plane on S2,3 | [q2 + q + 1,0,0] |
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| o4 | e1 ⊗ e1 ⊗ e1 + e2 ⊗ e1 ⊗ e2 | 2 |
| PG(A1) : | Line on S3,3 | [q + 1,0,0] |
| PG(A2) : | Point of rank 2 | [0,1,0] |
| PG(A3) : | Line on S2,3 | [q + 1,0,0] |
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| o5 | e1 ⊗ e1 ⊗ e1 + e2 ⊗ e2 ⊗ e2 | 2 |
| PG(A1) : | Secant line | [2,q - 1,0] |
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| PG(A2) : | Secant line | [2,q - 1,0] |
| PG(A3) : | Secant line | [2,q - 1,0] |
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| o6 | e1 ⊗ e1 ⊗ e1 + e2 ⊗ (e1 ⊗ e2 + e2 ⊗ e1) | 3 |
| PG(A1) : | Tangent line contained in an < S2,2 > | [1,q,0] |
| PG(A2) : | Tangent line contained in an < S2,2 > | [1,q,0] |
| PG(A3) : | Tangent line contained in an < S2,2 > | [1,q,0] |
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| o7 | e1 ⊗ e1 ⊗ e3 + e2 ⊗ (e1 ⊗ e1 + e2 ⊗ e2) | 3 |
| PG(A1) : | Tangent line contained in an < S2,3 >, | [1,q,0] |
| | not contained in an < S2,2 > | |
| PG(A2) : | Tangent line, not contained in an < S2,2 > | [1,q,0] |
| PG(A3) : | Plane containing 2 lines of an S2,2 | [2q + 1,q2 - q,0] |
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| o8 | e1 ⊗ e1 ⊗ e1 + e2 ⊗ (e2 ⊗ e2 + e3 ⊗ e3) | 3 |
| PG(A1) : | Tangent line not contained in an < S2,3 >, | [1,1,q - 1] |
| | containing a point of rank 2 | |
| PG(A2) : | Plane containing a line and a point of S2,3 | [q + 2,q2 - 1,0] |
| | not contained in an < S2,2 > | |
| PG(A3) : | Plane containing a line and a point of S2,3 | [q + 2,q2 - 1,0] |
| | not contained in an < S2,2 > | |
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| o9 | e1 ⊗ e3 ⊗ e1 + e2 ⊗ (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3) | 4 |
| PG(A1) : | Tangent line not contained in an < S2,3 >, | [1,0,q] |
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| | not containing a point of rank 2 | |
| PG(A2) : | Plane containing a line of S2,3, | [q + 1,q2,0] |
| | not contained in an < S2,2 > | |
| PG(A3) : | Plane containing a line of S2,3 not contained in an < S2,2 > | [q + 1,q2,0] |
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| o10 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2 + ue1 ⊗ e2) + e2 ⊗ (e1 ⊗ e2 + ve2 ⊗ e1) | 3 |
| | vλ2 + uvλ - 1≠0 for all λ ∈ Fq | |
| PG(A1) : | Line of constant rank 2, contained in an < S2,2 >, | [0,q + 1,0] |
| PG(A2) : | Line of constant rank 2, contained in an < S2,2 >, | [0,q + 1,0] |
| PG(A3) : | Line of constant rank 2, contained in an < S2,2 >, | [0,q + 1,0] |
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| o11 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2) + e2 ⊗ (e1 ⊗ e2 + ve2 ⊗ e3) | 3 |
| PG(A1) : | Line of constant rank 2, contained in an < S2,3 >, | [0,q + 1,0] |
| | but not in an < S2,2 > | |
| PG(A2) : | Line of constant rank 2, contained in an < S2,3 >, | [0,q + 1,0] |
| | but not in an < S2,2 > | |
| PG(A3) : | Plane in an < S2,2 >, meeting in a conic | [q + 1,q2,0] |
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| o12 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2) + e2 ⊗ (e1 ⊗ e3 + e3 ⊗ e2) | 4 |
| PG(A1) : | Line of constant rank 2, not contained in an < S2,3 >, | [0,q + 1,0] |
| PG(A2) : | Plane containing a line of S2,3 | [q + 1,q2,0] |
| PG(A3) : | Plane containing a line of S2,3 | [q + 1,q2,0] |
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| o13 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2) + e2 ⊗ (e1 ⊗ e2 + e3 ⊗ e3) | 4 |
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| PG(A1) : | Line with 2 points of rank 2 | [0,2,q - 1] |
| PG(A2) : | Plane containing 2 points of S2,3 | [2,q2 + q - 1,0] |
| PG(A3) : | Plane containing 2 points of S2,3 | [2,q2 + q - 1,0] |
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| o14 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2) + e2 ⊗ (e2 ⊗ e2 + e3 ⊗ e3) | 3 |
| PG(A1) : | Line with 3 points of rank 2 | [0,2,q - 1] |
| PG(A2) : | Plane contain ing 3 points of S2,3 | [0,3,q - 2] |
| PG(A3) : | Plane containing 3 points of S2,3 | [0,3,q - 2] |
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| o15 | e1 ⊗ (e + ue1 ⊗ e2) + e2 ⊗ (e1 ⊗ e2 + ve2 ⊗ e1); | 4 |
| | vλ2 + uvλ - 1≠0 for all λ ∈ Fq | |
| | and e = e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3. | |
| PG(A1) : | Line having one point of rank 2 | [0,1,q] |
| PG(A2) : | Plane containing one point of S2,3 | [1,q2 + q,0] |
| PG(A3) : | Plane containing one point of S2,3 | [1,q2 + q,0] |
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| o16 | e1 ⊗ (e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3) + e2 ⊗ (e1 ⊗ e2 + e2 ⊗ e3) | 4 |
| PG(A1) : | Line having one point of rank 2 | [0,1,q] |
| PG(A2) : | Plane containing one point of S2,3 | [1,q2 + q,0] |
| PG(A3) : | Plane containing one point of S2,3 | [1,q2 + q,0] |
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| o17 | e1 ⊗ (e) + e2 ⊗ (e1 ⊗ e2 + e2 ⊗ e3 + e3 ⊗ (αe1 + βe2 + γe3)); | 4 if q ≥ 3 |
| | λ3 + γλ2 - βλ + α≠0 for all λ ∈ Fq | 5 if q = 2 |
| | and e = e1 ⊗ e1 + e2 ⊗ e2 + e3 ⊗ e3. | |
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| PG(A1) : | Line of constant rank 3 | [0,0,q + 1] |
| PG(A2) : | Plane disjoint from S2,3 | [0,q2 + q + 1,0] |
| PG(A3) : | Plane disjoint from S2,3 | [0,q2 + q + 1,0] |
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