# 2024-10-03-12-49-19

## Del Pezzo surfaces

:::{.definition}
A **del Pezzo surface** is a smooth projective surface over $\CC$ such that $-K_S$ is ample.
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:::{.remark}
Recall that if $S$ is a surface then $\T_S$ is a rank 2 vector bundle with dual $\Omega_S$.
We define the canonical line bundle $\omega_S \da \det \Omega_S = \wedge^2 \Omega_S$.
This is spanned by $dx \wedge dy$ and is thus rank 1.
Since this is a line bundle, it is of the form $\OO_S(K_S)$ for some divisor $K_S$, denoted that canonical divisor.
The anticanonical bundle is $\omega_S\dual$, the dual of the canonical, with associated divisor $-K_S$, the anticanonical divisor.
Note that any toric boundary divisor is anticanonical.
This divisor is not unique.
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:::{.remark}
Recall that a line bundle $L$ is **very ample** if the associated rational map $\phi: S\to \PP H^0(L)\dual$ (where $x\mapsto \PP(s\mapsto s(x))$) is well-defined everywhere. Equivalently, $L$ is basepoint free, i.e. for every $x\in S$ there is a section $s$ where $s(x)\neq 0$.
This follows because $[0:\cdots :0]$ is not a point in $\PP^N$, and this condition guarantees that at least one coordinate is nonzero.
In this case, $\phi$ is an embedding.
A divisor $L$ is very ample iff $\OO_S(L)$ is very ample.
We say $L$ is ample if $L^{\tensor n}$ is very ample for some $n\in \ZZ_{>0}$.
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:::{.remark}
In higher dimensions, any $X$ with $-K_X$ ample is called a **Fano variety**.
Thus a Fano surface is by definition a del Pezzo surface.
This condition roughly describes having positive curvature -- determining precisely when such metrics exist uses K-stability.
An interesting research question: can Fano varieties be classified?
Some classification results:

- $\dim X = 1$: $X = \PP^1$ is the only Fano.
- $\dim X = 2$: del Pezzo surfaces, which we will classify today.
- $\dim X \geq 3$: there are finitely many Fanos up to deformation (theorem, 1960s).
    In $\dim X = 3$, there are 105. 
    In $\dim X = 4$, the number is unknown.
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:::{.remark}
Claim: $\PP^2$ is a del Pezzo surface.
Check that $-K_{\PP^2} = \OO_{\PP^2}(3)$, which is very ample.
Sections intersect the zero section along a cubic, and this has 10 global sections since these biject with degree 3 homogeneous polynomials in 3 variables $x,y,z$ on $\PP^2$.
Note that $\PP^2$ is toric, so choosing this as a polarization yields a moment polytope with 10 integral points: a triangle with side lengths 3.
$-K_{\PP^2}$ defines an embedding $\PP^2\embeds \PP H^0(\OO(3))$ which sends $[x:y:z]$ to $[x^3:,x^2 y:,\cdots,]$, all monomials of degree 3.
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:::{.theorem}
A surface $S$ is a del Pezzo if it either obtained from $\PP^2$ by blowing up $k$ points in general position where $0\leq k\leq 8$, or $S \cong \PP^1\times \PP^1$.
Note that *general position* means

- No 3 points are on a line, and
- No 6 points are on a conic.

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:::{.remark}
If $S$ is a del Pezzo surface, then

- $K_S^2 > 0$, and
- $-K_S.C > 0$ for any algebraic curve $C$ on $S$.

Note that $(-K_S)^2 = K_S^2$.
Generally, if $L$ is ample then $L^2 > 0$.
We define the **degree** of a del Pezzo surface to be $K_S^2$.
Note that conversely, any surface satisfying these properties is a del Pezzo by the Nakai-Moishezon criterion.
This is proved in Hartshorne chapter 5.
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:::{.remark}
Some non-obvious del Pezzo surfaces:

- $(\PP^2, \OO(3))$
- $(\PP^1\times \PP^1, \OO(2,2))$.
- $(S = \Bl_k \PP^2, \OO(-K_S))$ where $-K_S = 3H - \sum E_i$ with $E_i^2=-1$ the exceptional curves and $H\in \Pic(S)$ is the class of a general line in $\PP^2$ (avoiding the blowup points).

The first two are very ample, which is easy to check.
Note that $\Pic(S) = H_2(S; \ZZ)$ in the third case, and that the $k$ points must be in general position.
Different configurations of points yield deformation-equivalent surfaces.
In $\dim X = 2$, there are thus 10 possible del Pezzo surfaces up to deformation equivalence.
To compute $(-K_S)^2$, one needs to know

- $H^2 = 1$
- $H.E_i = 0$
- $E_i E_j = 0$ for $i\neq j$
- $E_i^2 = -1$ 

Distributing and multiplying, one obtains $K_S^2 = 9-k$ which is positive iff $k\leq 8$.

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:::{.remark}
A counterexample where the points are not in general position: let $p_1,p_2,p_3 \in l$ be three points contained in a line in $\PP^2$.
The strict transform of $l$ is of the form $H - E_1 - E_2 - E_3$.
Intersect 
\[
-K_S.(H-E_1-E_2-E_3) = (3H-E_1-E_2-E_3)(H-E_1-E_2-E_3) = 3-1-1-1=0
,\] 
which violates the second condition in the definition of a del Pezzo.
Similarly, taking 6 points on a conic, the strict transform is $2H-E_1-\cdots-E_6$.
Intersecting as above yields $6-1-1-1-1-1-1 = 0$.
One could formulate similar conditions on higher numbers of points, but we already require $k
\leq 8$ and the next condition would be on 9 points.
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:::{.remark}
Denote by $S_d$ a del Pezzo of degree $d$, obtained by blowing up $k=9-d$ points in general position in $\PP^2$.
How to remember: $d=9$ yields $k=0$ points, and $\deg \OO_{\PP^2}(3) = 3^2=9$.
By assumption $-K_{S_d}$ is ample, and it will be very ample if $3\leq d\leq 9$.
This corresponds to blowing up $k\leq 6$ points.
One can calculate $h^0(-K_{S_d}) = 10-(9-d) = d+1$, so $S_d \embeds \PP^d$ where $3\leq d\leq 9$.
This is referred to as the anticanonical embedding of $S_d$ into $\PP^d$.
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:::{.example}
For $d=3$, we have $S_3 = \Bl_6 \PP^2 \embeds \PP^3$, and in fact any smooth projective cubic surface can be embedded in $\PP^3$.
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:::{.theorem}
Any smooth projective cubic surface is isomorphic to the del Pezzo surface $S_3$.
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:::{.example}
$S_4 = \Bl_5 \PP^2 \embeds \PP^4$, which is true since $S_4$ is a complete intersection of two quadrics in $\PP^4$.
Note that a complete intersection is a transverse intersection of hypersurfaces.
Note that $S_5$ is no longer a complete intersection.
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:::{.corollary}
Any del Pezzo surface is birational to $\PP^2$.
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:::{.remark}
One could ask if this is true for higher dimensional Fano varieties.
It fails in dimension 3 -- not every Fano threefold is rational.
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