The complete system of quadratic forms maps to a smooth surface of degree four in . The six coordinates are . It occurs as a degree-two image of a K3 surface and, after taking a projective cone, in the plane-quintic exception to the K3 linear system dichotomies.
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The Veronese surface is a well-known example in algebraic geometry, and it is often studied in relation to the projective geometry of higher-dimensional spaces. Specifically, it is defined as a two-dimensional algebraic surface that can be embedded in projective space. The Veronese surface can be constructed by considering the image of the projective plane under the Veronese embedding.