Tangent spaces of a product of two nodal line pairs

For

X = Z (x_{1}^{2} - x_{2}^{2}, x_{3}^{2} - x_{4}^{2}) \subset A^{4}

(39)

in characteristic other than two,

T_{P} X = {v : x_{1} v_{1} - x_{2} v_{2} = 0, x_{3} v_{3} - x_{4} v_{4} = 0} .

(40)

The variety has dimension two. Its tangent dimension is two when both coordinate pairs are nonzero, three when exactly one pair vanishes, and four at the origin.

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Zariski tangent space
Algebraic geometry
Geometry and topology
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Solution