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consider the blow-up of the plane in one point. Let $E$ the exceptional divisor. We know that $(E,E)=-1$. Which is the geometrical reason for which the auto-intersection of $E$ is $-1$? In general what does it means, geometrically, that a divisor has negative self-intersection or that the "right" number of divisors gives rise to negative intersection?

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4 Answers

If you take a line (isomorphic to $\mathbb P^1$) in $\mathbb P^2$ its self intersection is $1$. This is a manifestation of the fact that if you slightly move it, the unmoved copy will intersect the moved copy in $1$ point.Great you say, I'll do the same with the exceptional divisor $E$ in the blow-up. Let's see: I'll slightly move $E$ and the moved copy will intersect the fixed copy in...-1 points?! But this is utter nonsense! Yes, it is: the way out of this absurdity is to realize that you can't move $E$ ! . In slightly more technical terms, the normal bundle of $E$ in the blown-up plane has degree (-1) and this shows that it cannot be moved.So one intuition could be: negative self-intersection= rigidity.

Another intuition could be that negative self-intersection smells of blow-up. A basic result in that direction is Castelnuevo's criterion: if $S$ is a projective surface containing a curve $E$ isomorphic to $\mathbb P^1$ of self-intersection -1, then $E$ can be blown down to a point $e$. This means that there exists a surface $S_0$ containing a point $e$, which when blown-up in $S_0$will become the curve $E$ in $S$. Grauert and others has proved very profound generalizations of Castelnuevo's theorem.

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Let me complement PseudoNeo's excellent answer by actually writing down a motion of $E$ and showing that it intersects itself with multiplicity $-1$. Recall that $\mathbb{C}^2$ blown up at $(0,0)$ can be thought of as the space of ordered pairs $(z, \ell)$ where $z$ is a point of $\mathbb{C}^2$ and $\ell$ is a line through $z$ and through $(0,0)$. The exceptional fiber $E$ is the set of pairs of the form $((0,0),\ \ell)$ where $\ell$ can be any line through $(0,0)$.

In order to perturb this, I want to move to a set of the form $\{ \zeta(\ell), \ell \}$ where $\zeta$ is some continuous function which, to a line through the origin in $\mathbb{C}^2$, assigns a point on that line. We'll write $\ell$ in homogenous coordinates as $(x : y)$. So we need a continuous function $\zeta$ which, given $(x, y) \in \mathbb{C}^2 \setminus \{ (0,0) \}$, chooses some point on the line through $(x,y)$, so that $\zeta(x, y) = \zeta(\lambda x, \lambda y)$ for any nonzero $\lambda$.

Some experimentation produces $$\zeta(x,y) := \left( \frac{x \overline{x}}{|x|^2+|y|^2},\ \frac{y \overline{x}}{|x|^2+|y|^2} \right).$$ Let $E_{moved}$ be the set of points $\left( \zeta(x,y), (x:y) \right)$. (If you want to reassure yourself that $E_{moved}$ is homotopic to $E$, consider the homotopy $\left( t\cdot \zeta(x,y), (x:y) \right)$, as $t$ goes from $1$ to $0$.)

Now $E$ intersects $E_{moved}$ when $\zeta(x,y)=(0,0)$, which happens when $(x:y) = (0:1)$. Near $\left( (0,0), (0:1) \right)$, local complex coordinates are $u$ and $v$, corresponding to the point $\left( (uv,u), (v:1) \right)$ in the blow up. In this chart, $E$ is given by $u=0$ and $E_{moved}$ is given by $$u = \frac{\overline{v}}{1+|v|^2}.$$

The intersection takes place at $(u,v) = (0,0)$. Writing $u = u_1 + i u_2$ and $v = v_1 + i v_2$, the Jacobian of $v \mapsto \overline{v}/(1+|v|^2)$ is $\left( \begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right)$. The fact that this has determinant $-1$ indicates that the intersection with $u=0$ counts with sign $-1$.

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I'll just add a remark to Georges's (excellent) answer.

The real plane $\mathbb R^2$ has two freely substitutable orientations but the complex plane $\mathbb C$ has a canonical one (as a real vector space). Essentially, when you chose $i$ and not $-i$ as your favorite root of $-1$, you chose at the same time an orientation for the real plane. As a consequence, when two (complex) lines intersect in the (complex) plane, all the orientations are compatible and intersection is positive. This is more generally always the case when two complex manifolds intersect. (My Geometric Algebrish isn't that good, so I won't try to translate this statement for algebraic varieties, but I'm confident a simple translation exists).

Now, when Georges say: "the way out of this absurdity is to realize that you can't move E", he is merely admitting that algebraists are powerless. I'm a topologist and I can move E. But I only obtain two real surfaces $S_1 = E$ and $S_2 = E_{\text{slightly moved}}$ in a real 4-manifold $W$ having a total intersection number of -1. That means that at least one of the intersection points $x$ comes with a negative sign: putting together an orientation of direct basis of $T_x S_1$ and a direct basis of $T_x S_2$ gives an indirect basis of $T_x W$. And the facts I stated above about intersections of complex submanifolds prove that the deformation I've just made has absolutely no counterpart in the complex/algebraic world.

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A curve $\gamma$ that has self-intersection $n$ means that the surface in which the curve is embedded admits a vector field that is everywhere transverse or vanishing on $\gamma$. The singularities of the vector field that are lying the curve are the self-intersection points of $\gamma$ and the sum of the indices of these singularities is the self-intersection number $n$ of $\gamma$.

Thus, if an exceptional divisor $E$ having self-intersection $-1$ then this means that it is possible to place a transverse vector field on $E$ with a saddle type singularity at one point and that this vector field extends into a valid vector field of the embedding space. If you flow the divisor along the vector field for a small time this results in a copy of the divisor slightly moved away from the original divisor. At the point where the vector field vanishes the copy and the original divisor intersects showing that this point is a self-intersection point.

A projective line of the projective plane has self-intersection $+1$ since the projective plane admits a rotational vector field with the center placed on the line. But, that said, it also admits a vector field with two rotational centers and one sadle point. If the saddle point is placed on the line that would mean that the line in this case has self-intersection $-1$. This seems to agree with the possibility to blow-down a line of the projective plane resulting in a sphere? But obviously there is more to the story here and I have maybe (probably) misunderstood some basic stuff.

The case of self-intersection of a manifold in the toplogical sense, that is the intrinsic self-intersection without regard to any embedding is somewhat more straight-forward. Then it is only possible to flow the manifold within itself. Then of course every point can be said to be trivially intersection points. So instead we consider those points that are intersecting in a higher order sense to be self-intersection points. This immediately translates into meaning that the singularities of a tangent vector field are self-intersection points and the topological self-intersection number of a manifold is the index of its vector fields.

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