Analyze projective dimensions using codimension, tangent rank, and cone relations. Check smoothness with local estimates. Plot, export, and study dimension behavior across algebraic inputs.
Use exact invariants when known. Otherwise, the calculator falls back to equation-count estimates inside projective space.
1. Codimension route: For a projective variety X ⊂ Pn, the global relation is dim(X) = n − codim(X).
2. Complete-intersection estimate: If X is cut by r independent homogeneous equations in general position, an expected dimension is dim(X) ≈ n − r. If this becomes negative, the generic intersection is treated as empty.
3. Tangent-space test: At a chosen point p, dim(TpX) = n − rank(J(p)). Equality with the chosen global dimension is consistent with smoothness at p.
4. Affine cone route: For nonempty X, dim(C(X)) = dim(X) + 1, so dim(X) = dim(C(X)) − 1.
5. Hilbert polynomial route: If PX(m) is the Hilbert polynomial, then dim(X) = deg(PX(m)).
6. Coordinate-ring route: If S/I is the homogeneous coordinate ring, then dim(X) = dim(S/I) − 1.
7. Expected generic intersection: If X and Y lie in Pn, then dim(X ∩ Y) ≥ dim(X) + dim(Y) − n, and the generic estimate uses that lower bound.
8. Expected join dimension: The projective join often satisfies dim(J(X,Y)) ≤ min(n, dim(X) + dim(Y) + 1).
| Variety | Ambient Space | Independent Equations | Codimension | Dimension |
|---|---|---|---|---|
| Quadric hypersurface | P4 | 1 | 1 | 3 |
| Twisted cubic | P3 | 3 | 2 | 1 |
| Veronese surface | P5 | Several generators | 3 | 2 |
| Segre P1 × P2 | P5 | 2 × 2 minors | 2 | 3 |
| Projective point | P4 | Many generators possible | 4 | 0 |
It measures the number of independent projective parameters needed locally on the variety. Geometrically, it is the longest chain length of irreducible subvarieties minus one.
Equation count works cleanly for generic complete intersections. If equations are dependent, redundant, or define singular schemes, the true dimension can differ from n − r.
Jacobian rank gives a local tangent-space dimension at a chosen point. Comparing that value with the global estimate helps detect smooth points, singular points, or inconsistent input data.
Passing from a projective variety to its affine cone adds the scaling direction through the origin. That increases dimension by exactly one for nonempty projective varieties.
Hilbert polynomial degree, coordinate-ring dimension, codimension, and affine cone dimension are stronger than raw equation count. The calculator prioritizes those exact-style invariants automatically.
Yes. In many computational settings, −1 is used to represent an empty generic intersection. It helps distinguish emptiness from zero-dimensional projective sets such as finite point collections.
No. Equality only supports smoothness at the tested point. Other points may still be singular, and global smoothness needs broader geometric or computational verification.
No. It is a dimension-analysis aid, not a full algebra system. Exact primary decomposition, Hilbert series, and Gröbner-basis computations still require dedicated symbolic tools.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.