On a plane section of an integral curve in positive characteristic

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If C ⊂ P³ₖ is an integral curve and k an algebraically closed field of characteristic 0, it is known that the points of the general plane section C ∩ H of C are in uniform position. From this it follows easily that the general minimal curve containing C∩ H is irreducible. If char k = p > 0, the points of C∩ H may not be in uniform position. However, we prove that the general minimal curve containing C∩ H is still irreducible.