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Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing 2, and asked whether such surfaces always have a closed point of degree 2. We resolve this by constructing infinitely many quartic del Pezzo surfaces over Q without degree 2 points. These are the first examples of smooth intersections of two quadrics with index strictly less than the minimal degree of a closed point.
Creutz et al. (Thu,) studied this question.
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