## Biregular models of log Del Pezzo surfaces with rigid singularities

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Muhammad Imran Qureshi
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## Abstract:

We construct biregular models of families of log Del Pezzo surfaces with rigid cyclic quotient singularities such that a general member in each family is wellformed and quasismooth. Each biregular model consists of infinite series of such families of surfaces; parameterized by the natural numbers $\mathbb {N}$. Each family in these biregular models is represented by either a codimension 3 Pfaffian format modelled on the Plücker embedding of $\mathrm {Gr}(2,5)$ or a codimension 4 format modelled on the Segre embedding of $\mathbb {P}^2 \times \mathbb {P}^2$. In particular, we show the existence of two biregular models in codimension 4 which are biparameterized, giving rise to an infinite series of models of families of log Del Pezzo surfaces. We identify those biregular models of surfaces which do not admit a $\mathbb {Q}$-Gorenstein deformation to a toric variety.## References

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## Additional Information

**Muhammad Imran Qureshi**- Affiliation: Department of Mathematics, SBASSE Lahore University of Management Sciences (LUMS), Lahore, Pakistan; and Mathematisches Institut, Universität Tübingen, Germany
- MR Author ID: 948483
- Email: i.qureshi@maths.oxon.org
- Received by editor(s): November 28, 2017
- Received by editor(s) in revised form: September 16, 2018
- Published electronically: April 3, 2019
- Additional Notes: This research was supported by a Higher Education Commission (HEC)’s NRPU grant 5906/Punjab/NRPU/RD/HEC/2016 and a fellowship of the Alexander von Humboldt Foundation.
- © Copyright 2019 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 2497-2521 - MSC (2010): Primary 14J10, 14M07, 14J45, 14Q10
- DOI: https://doi.org/10.1090/mcom/3432
- MathSciNet review: 3957903