#### 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 .

Done