# Projection of singular del Pezzo from the sigunlar point

Take $$\mathbb{P}^2$$ and blow up three points on the same line, let the resulting surfaces be $$S$$. Use $$H$$ to denote the hyperplane and $$E_i,i=1,2,3$$ to denote the exceptional divisors. Consider the complete linear system $$|3H-E_1-E_2-E_3|$$, gives a morphism $$f:S\to \mathbb{P}^6$$ where the image $$f(S)$$ is a degree $$6$$ singular del Pezzo obtained by contracting the $$-2$$ curve $$H-E_1-E_2-E_3$$ on $$S$$.

My question: if we consider the projection of $$\mathbb{P}^6$$ from the double point on $$f(S)$$, what would be a minimal compactification of the image of $$f(S)$$ in $$\mathbb{P}^5$$?

Clearly the three exceptional curves $$E_i$$ will be contracted by the projection since they are straight lines through the double point.

Thanks!

First you have to blowup the singular point of the sextic del Pezzo; then what you get is your surface $$S$$; the exceptional divisor of this blowup is the unique curve $$E$$ in the linear system $$H - E_1 - E_2 - E_3$$. Then the projection is given by the linear system $$(3H - E_1 - E_2 - E_3) - E = (3H - E_1 - E_2 - E_3) - (H - E_1 - E_2 - E_3) = 2H.$$ The result, of course, is just the Veronese surface in $$\mathbb{P}^5$$.
• Thank you very much for the answer. Just to make sure I understand this fully, in a similar situation if $S$ is blow up $\mathbb{P}^2$ at two infinitesimally near points, let $f(S)$ be the image of the $|3H-E_1-E_2|$ ($E_1,E_2$ exceptional and $E_1-E_2$ is the $-2$ curve.), then (compatification) of the image of projection from the double point of $f(S)$ would be the same as the image in $|3H-E_1-E_2-(E_1-E_2)|=|3H-2E_1|$? – Rust Q Feb 26 at 20:30