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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
##### Keywords:
plane curve; finite field; rational point
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##### References:
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