Conics with a Hermitian curve

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A Hermitian curve X is a plane curve of degree q +1 which is projectively equivalent to the plane curve with the inhomogeneous equation yq +y = xq+1 over the finite field Fq2 of q2 elements, which has q3+1 Fq2-rational points. The geometry of lines over Fq2 harmonizes with those points, that is to say, a line over Fq2 either tangents to X at an Fq2-rational point with multiplicity q + 1 or meets X in exactly q+1 Fq2-rational points. For the conics over Fq2, we can not expect them to behave well with the Fq2-rational points of X, however, in the joint research with Seon Jeong Kim on the two-point codes on X, we met a certain family of conics over Fq2 whose behavior on the Fq2-rational points of X seemed interesting.

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