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Is it well known what happens if one blows-up $\mathbb{P}^2$ at points in non-general position (ie. 3 points on a line, 6 on a conic etc)? Are these objects isomorphic to something nice?

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2 Answers 2

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I'll just add to Francesco's answer by saying that general position of the points on the plane is equivalent to ampleness of the anticanonical sheaf $\omega_X^{\otimes -1}$.

The key observation is that on a del Pezzo surface, an irreducible negative curve ($C^2 < 0$) must be an exceptional curve (i.e. $C^2 = C\cdot K_X = -1$). This follow from the adjunction formula and the Nakai-Moishezon criterion.

If you blow up 3 colinear points, then the strict transform of the line containing these points will have self-intersection $-2$, which is not allowed by the key observation. Similiarly for the strict transform of a conic through 6 blown-up points. There is one more condition you have to impose: if you blow up 8 points, they cannot lie on a singular cubic with one of the points at the singularity.

If you relax the requirement that $\omega_X^{\otimes -1}$ be ample to just big and nef, then you can have some degenerate point configurations: this time $C^2 = -2$ is allowed, so 3 colinear points ar OK. However, 4 colinear points would not be OK.

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In both examples you are considering, the anticanonical model is a singular del Pezzo surface.

In fact, let $X$ be the blow-up of $\mathbb{P}^2$ at three points lying on a line $L$. By Bezout's theorem, the birational map associated with the linear system of cubics through the three points contracts $L$. Since $L^2=1$, the blow-up of $L$ at three points gives a $(-2)$-curve. Therefore, the anticanonical model of $X$ is a Del Pezzo of degree $6$ in $\mathbb{P}^6$ with an ordinary double point (i.e., a node).

Analogously, let $Y$ be the blow-up of $\mathbb{P}^2$ at six points lying on a conic $C$. By Bezout's theorem, the birational map associated with the linear system of cubics through the six points contracts $C$. Since $C^2=4$ and we are blowing-up six points over $C$, we obtain again a $(-2)$-curve. Therefore, the anticanonical model of $Y$ is a Del Pezzo surface of degree $3$ in $\mathbb{P}^3$ with an ordinary double point, i.e. a cubic surface with a node.

Of course, if you blow-up more than $8$ points then the result is not a Del Pezzo anymore. For instance, the blow-up of $\mathbb{P}^2$ at nine points which are the base locus of a pencil of cubics is an elliptic fibration $X \to \mathbb{P}^1$ with nine sections; in general, such fibration has exactly $12$ nodal fibres, corresponding to the singular elements of the pencil.

When the number of points increases the situation becomes more and more complicated, and I guess that a satisfatory description is out of reach.

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