Q. How do you prove hyperplane section is in linearly general position, so the free pencil exists? A. C in PP^n irreducible curve, not in a hyperplane. Write phi: C[n] - -> (PP^n)^dual for the rational map taking a general n-tuple of points to the hyperplane that is their linear span. If C is singular, take C[n] the projective closure of the nth symmetric power of NonsSing C. Since C spans PP^n, a Zariski open set of (n+1)-tuples of NonSing C consists of linearly independent points, so a general set of n points spans a hyperplane. Now over a field k of characteristic zero, the rational map phi: C[n] - -> (PP^n)^dual is separable. The same conclusion holds also if char k > deg(phi), and one sees that deg(phi) <= (d choose n) with d = deg C. If phi is separable, then it is a finite unramified cover over a dense Zariski open set of (PP^n)^dual. For a hyperplane H of PP^n in this open, every inverse image of H in C[n] is a set of n points spanning H, therefore H intersect C is in linearly general position. (Think of this as a kind of Sard's theorem.) In small characteristic (and large degree), there are counterexamples. The main result is due to Pierre Samuel and treated in Hartshorne, 4.3, p. 310-316. If the general section of C in PP^n (not in a hyperplane) is not a set of d points in general position then C is _strange_: the tangent lines T_{C,P} at all P in NonSing C pass through a common point Q in PP^n. Examples are C1 : y^p = x^{p-1}*z in PP^2, image of v |-> (1, v, v^p). project from P_y gives a purely inseparable cover of PP^1. C2 : codim 2 c.i. x*t=z^2, x^(p-1)*z = y^p, image of PP^1 under (u^(2*p), u^(2*p-1)*v, u^p*v^p, v^(2*p). project from P_y = (0,1,0,0) is inseparable cover of conic. Then projection from Q gives an inseparable map C - -> PP^{n-1}. For a canonical curve C_{2g-2} in PP^{g-1} this is impossible for reasons of degree. I think there exist curves C in PP^n for which every hyperplane section is a configuration of points with symmetry by a product (FF_p^+d), so not linearly general. Exercise: figure this out. This note still needs to be written out correctly. Recent preprint Qile Chen and Ryan Contreras arXiv:2108.09024 Title: Plane A^1-curves on the complement of strange rational curves