PreprintCC BY 4.0HAL
Closed points on del Pezzo surfaces of degree 4 and index 2
Abstract
Let X be a del Pezzo surface of degree $4$ over a field~k, and let ρ(X) denote its Picard rank. We prove that if the index of X, that is, the greatest common divisor of the degrees of its closed points, is equal to $2$, then X has a closed point of degree $2$ or $6$. This result is optimal, as shown by examples due to Creutz and Viray. Moreover, if ρ(X)≥ 2, then X has a closed point of degree $2$.
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BibTeX
@misc{colliotthelene2026closed,
title = {{Closed points on del Pezzo surfaces of degree 4 and index 2}},
author = {Jean-Louis Colliot-Thélène and Federico Scavia},
howpublished = {HAL},
year = {2026},
month = nov,
url = {https://hal.science/hal-05852639v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Closed points on del Pezzo surfaces of degree 4 and index 2
AU - Jean-Louis Colliot-Thélène
AU - Federico Scavia
PY - 2026
DA - 2026/11/04
PB - HAL
UR - https://hal.science/hal-05852639v1
AB - Let X be a del Pezzo surface of degree $4$ over a field~k, and let ρ(X) denote its Picard rank. We prove that if the index of X, that is, the greatest common divisor of the degrees of its closed points, is equal to $2$, then X has a closed point of degree $2$ or $6$. This result is optimal, as shown by examples due to Creutz and Viray. Moreover, if ρ(X)≥ 2, then X has a closed point of degree $2$.
ER -