(joint work with Dino Lorenzini) Let K be a field and let L/K be a finite extension. Let X be an algebraic variety over K. A point of X(L) is said to be new if it does not belong to the union of the X(F)'s, where F runs over all proper subextensions F of L. Fix now a finite separable extension L/K. In this talk I will investigate the existence of a smooth projective curve with a new point in X(L) and with "small" genus with respect to [L:K].
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