Alladi Ramakrishnan Hall

Projective Bundle and Blow-up

Nabanita

TIFR

In the first part of the talk we see some examples of blow up of projective space along some projective subvariety, such that these blown up spaces are isomorphic to a projective bundle over some projective space.

In the second part of the talk we prove that $\mathbb{P}^{2}$ 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 $\mathbb{P}^1\times\mathbb{P}^2$. Conversely, any smooth surface in the complete linear system $\mid (2, 2) \mid$ of $\mathbb{P}^1\times\mathbb{P}^2$ 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 $\mathbb{P}^1\times\mathbb{P}^2$ has at most four $(-2)$ curves .