Adjunction Process for Surfaces

A surface in this section is a smooth projective surface over $\mathbb C$.

Blowing up a surface in a point means to replace the point by an exceptional curve. Each such curve $E$ is a smooth, rational curve with self-intersection number $E^{2}=-1$. We speak of a $(-1)$-curve. A surface is minimal if it contains no $(-1)$-curves. That is, the surface cannot be obtained by blowing up a point on another surface. A surface $X_{\text{min}}$ is called a minimal model of a surface $X$ if $X_{\text{min}}$ is minimal and $X$ can be obtained from $X_{\text{min}}$ by repeatedly blowing up a point. Each surface $X$ has a minimal model which is unique if $X$ has non-negative Kodaira dimension. The Enriques-Kodaira classification classifies surfaces according to their minimal models. See [BHPV-D-V04] for more on this.

Given a surface, we may apply the adjunction process of Van de Ven and Sommese [SV-D-V87] to discover a minimal model. To describe this process, consider a surface $X \subset \mathbb P^{n}$ of codimension $c$. Let $S$ and $S_{X}$ denote the homogeneous coordinate rings of $\mathbb P^{n}$ and $X$, respectively. Consider $\omega_{X}=\text{Ext}^{c}_{S}(S_{X},S(-n-1)),$ the graded dualizing module of $S_{X}$. A basis of the graded piece $(\omega_{X})_{{1}}$ corresponds to the linear system $|K_X +H|$, where $K_X$ is a canonical divisor on $X$ and $H$ is the hyperplane class. Except for some exceptional cases, this linear system defines a birational morphism $\varphi_{|K_X+H|}\colon X \to X'$ onto another smooth projective surface $X'$ such that $\varphi_{|K_X+H|}$ blows down precisely all $(-1)$-lines on $X$. As shown by Van de Ven and Sommese, in the exceptional cases,

• $X$ is a linearly or quadratically embedded $ \mathbb P^{2}$ or $X$ is ruled by lines, in which case $|K_X+H| = \emptyset$,
• $X$ is an anti-canonically embedded del Pezzo surface, in which case $\varphi_{|K_X+H|}$ maps $X$ to a point,
• $X$ is a conic bundle, in which case $\varphi_{|K_{X}+H|}\colon X \to B$ maps $X$ to a curve $B$ such that the fibers of $\varphi_{|K_{X}+H|}$ are the conics, or
• $X$ is a surface in one of four explicit families identified by Sommese and Van de Ven, and $\varphi_{|K_X+H|}\colon X \to X'$ is not birational, but finite to one.

If we are not in one of these cases, a $(-1)$-conic $C$ in $X$ is mapped to a $(-1)$-line in $X'$ since $(K_X+H)\;\!. \;\! C=-1+2=1$. Thus, the adjunction process, which consists of applying the adjunction maps $\varphi_{|K_X+H|}$, $\varphi_{|K_{X'}+H'|}$ and so on, yields finitely many surfaces $X \rightarrow X^{\prime} \rightarrow X^{\prime\prime} \rightarrow \dots$ which are called the adjoint surfaces of $X$. The last adjoint surface is either minimal or belongs to one of the exceptional cases. In particular, if $X$ has non-negative Kodaira dimension, the adjunction process yields the uniquely determined minimal model of $X$.

adjunction_process(X::AbsProjectiveVariety, steps::Int=0)

julia> X = bordiga()
[ Info: upgrading serialized data....
Projective variety
  in projective 4-space over GF(31991) with coordinates [x, y, z, u, v]
defined by ideal with 4 generators


julia> dim(X)
2

julia> codim(X)
2

julia> L = adjunction_process(X);

julia> L[1]
2-element Vector{NTuple{4, ZZRingElem}}:
 (4, 6, 3, 0)
 (2, 1, 0, 10)

julia> L[4]
Projective variety
  in projective 2-space over GF(31991) with coordinates [z[1], z[2], z[3]]
defined by ideal (0)

julia> L[3][1]
Projective algebraic set
  in projective 2-space over GF(31991) with coordinates [z[1], z[2], z[3]]
defined by ideal with 5 generators

julia> dim(L[3][1])
0

julia> degree(L[3][1])
10

julia> X = rational_d9_pi7();
[ Info: upgrading serialized data....

julia> L = adjunction_process(X);

julia> L[1]
3-element Vector{NTuple{4, ZZRingElem}}:
 (4, 9, 7, 0)
 (6, 9, 4, 6)
 (3, 3, 1, 3)