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Quartic del Pezzo surfaces without quadratic points

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arxiv 2408.08436 v2 pith:JQYUQYSV submitted 2024-08-15 math.NT math.AG

classification math.NTmath.AG
keywords degreeclosedpezzopointquarticsurfacesindexpoints
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abstract

Previous work of the authors showed that every quartic del Pezzo surface over a number field has index dividing $2$ (i.e., has a closed point of degree $2$ modulo $4$),, 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 $\mathbb{Q}$ 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.

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  1. Counting quadratic points on Fano varieties

    math.NT 2025-05 conditional novelty 7.0 of 10

    The count of quadratic point pairs on the non-split quadrics x^2 - d y^2 = z w matches the predicted c B log B, once a thin set of new flavour (contributing the same order) is removed.

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