 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2014/3/b/solution

A projective quadric is the zero locus of a nonzero homogeneous quadratic polynomial

q (v) = v^{T} B v

, with

B

symmetric over a field of characteristic different from two. A projective point on it is singular when all first partial derivatives vanish. Since

\nabla q (v) = 2 B v

, the quadric is nonsingular exactly when

B

Over the complex numbers, a nonzero quadratic form in three variables can be diagonalized by an invertible change of basis. Its rank classifies a projective conic:

Rank three gives a nonsingular conic.
Rank two gives two distinct projective lines, since the reduced form $X^{2} + Y^{2}$ factors as $(X + iY) (X - iY)$ . Their intersection is the singular point.
Rank one gives a double projective line, defined by a squared linear form. The gradient vanishes along that line.

The zero polynomial is excluded from the definition of a conic. Thus these last two cases are precisely the singular conics.

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