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Fragments of plane filling curves of degree \(q + 2\) over the finite field of \(q\) elements, and of affine-plane filling curves of degree \(q + 1\). (English) Zbl 1439.14082
Summary: Nonsingular plane curves over a finite field \(\mathbb{F}_q\) of degree \(q + 2\) passing through all the \(\mathbb{F}_q\)-points of the plane admit a representation by \(3 \times 3\) matrices over \(\mathbb{F}_q\). We classify their degenerations by means of the matrix representation. We also discuss the similar problem for the affine-plane filling projective curves of degree \(q + 1\).
MSC:
14G15 Finite ground fields in algebraic geometry
14H50 Plane and space curves
14G05 Rational points
11G20 Curves over finite and local fields
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