Geometry of $\mathbb{P}²$ blown up at seven points

Year, pages: 2019, 1279 - 1292

In this paper, we prove that blown up at seven general points admits a conic bundle structure over $\mathbb{P}¹$and it can be embedded as $(2, 2)$ divisor in $\mathbb{P}¹×\mathbb{P}²$. Conversely, any smooth surface in the complete linear system $\vert (2, 2) \vert$ of $\mathbb{P}¹×\mathbb{P}²$ can be obtained as an embedding of blowing up $\mathbb{P} 2$ at seven points. We also show that smooth surface linearly equivalent to $(2, 2)$ in $\mathbb{P}¹×\mathbb{P}²$ has at most four $(-2)$ curves.

Ray, N., , . 2019. Geometry of $\mathbb{P}²$ blown up at seven points. In Mathematica Slovaca, vol. 69, no.6, pp. 1279-1292. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0308

Ray, N., , . (2019). Geometry of $\mathbb{P}²$ blown up at seven points. Mathematica Slovaca, 69(6), 1279-1292. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0308

Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava