Degree of irrationality of properly elliptic surfaces

@misc{lee2026degree,
  title = {{Degree of irrationality of properly elliptic surfaces}},
  author = {Yongnam Lee and De-Qi Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.27895},
  eprint = {2608.27895},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27895},
}

TY  - UNPB
TI  - Degree of irrationality of properly elliptic surfaces
AU  - Yongnam Lee
AU  - De-Qi Zhang
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.27895
UR  - https://arxiv.org/abs/2608.27895
AB  - In this paper, we study the degree of irrationality of properly elliptic surfaces with a section. We prove min{χ(𝒪_S), 2gon(C)} ≤ irr(S) ≤ 2gon(C). The lower bound is obtained from the canonical bundle formula and the Cayley--Bacharach property. We show that this bound is sharp. We also study the behavior of the degree of irrationality in moduli. A very general properly elliptic surface with a section over a curve of genus at least two has degree of irrationality at least four, whereas special families with χ(𝒪_S)=1 or $2$ have degree two. Finally, we investigate properly elliptic surfaces with χ(𝒪_S)=0, proving a generic lower bound of four and showing that irr(C× E)=4 for every hyperelliptic curve C of genus at least two and every elliptic curve E. Our paper also includes special properly elliptic surfaces without a section. For Dolgachev surfaces, we exclude degree two for a very general member and construct special examples of degrees two and three.
ER  -