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Real Pfaffian representations of real cubic surfaces

Consider the following classical construction (which is called Pfaffian representation as Sasha indicates): Let $ V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of ...
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Nonsingular zeroes are algebraic?

I'm getting started in Real Algebraic Geometry (from a model-theory perspective), and a paper makes the following assertion (here $K\subset L$ are real-closed fields): Suppose $Q\in L^n$, $f_1, ...