where $B^{\prime}\in|2(s+4f)|$, $C_{11}\in|s+4f|$. The curve $B^{\prime}$ has
a node $q$ and the curves $C_{11}$ and $B^{\prime}$ meet at $q$ in such a way
that $C_{11}$ intersect one of the branches with multiplicity $3$ (so that the
singularity at $q$ of $B^{\prime}+C_{11}$ has type $\mathbf{d}_{8}$) ; the $4$
points $p_{12},\dots,,p_{15}$ nodes, they are the remaining intersection
points of $C_{11}$ and $B^{\prime}$ (so that $B^{\prime}C_{11}=8$),
The images of curves $A_{16},\dots,A_{19}$ are the fibers through
$p_{12},\dots,,p_{15}$,
The images of curves $A_{20},A_{22},A_{24},A_{26}$ by $\varphi$ are curves in
the linear system $|s+4f|$ passing through $3$ of the $4$ points
$p_{12},\dots,,p_{15}$, and through $q$ and infinitely near points of $q$ with
certain multiplicities,
The images of curves $A_{21},A_{23},A_{25},A_{27}$ by $\varphi$ are curves in
the linear system $|s+4f|$ passing through $1$ of the $4$ points
$p_{12},\dots,,p_{15}$, and through $q$ and infinitely near points of $q$ with
certain multiplicities.

##  15\. Rank $\geq 15$ lattices

###  15.1. The lattice
$U\oplus\mathbf{E}_{8}\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}$

The K3 surface $X$ contains $21$ $(-2)$-curves $A_{1},\dots,A_{21}$. The dual
graph of these curves is given in [23, Section 4], we reproduce it here:

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$A_{10}$$A_{9}$$A_{8}$$A_{7}$$A_{6}$$A_{5}$$A_{3}$$A_{11}$$A_{4}$$A_{20}$$A_{17}$$A_{18}$$A_{19}$$A_{21}$$A_{14}$$A_{15}$$A_{16}$$A_{1}$$A_{13}$$A_{12}$$A_{2}$$6$

The curves $A_{1},\dots,A_{15}$ generate the Néron-Severi lattice; in that
base the divisor

$D_{242}=(2,5,9,4,10,12,15,19,24,30,14,23,17,8,4)$

is ample, of square $242$, with $D_{242}A_{j}=1$ for
$j\in\\{1,\dots,14\\}\setminus\\{11\\}$.

The divisor

$D_{2}=(1,2,3,1,3,3,3,3,3,3,1,2,1,0,0)$

is nef, base point free with $D_{2}^{2}=2,$ $D_{2}A_{20}=D_{2}A_{21}=2$,
$D_{2}A_{j}=1$ for $j\in\\{4,11,14,\dots,19\\}$, else $D_{2}A_{j}=0$. We have

$2D_{2}\equiv A_{20}+A_{21}$

and

$D_{2}\equiv\sum_{k\in\\{1,\dots,13\\}\setminus\\{4,11\\}}A_{k}+A_{13+j}+A_{16+j},\,\,j\in\\{1,2,3\\}.$

By using the linear system $|D_{2}|$, we obtain:

######  Proposition 150.

The $K3$ surface $X$ is the double cover of $\mathbb{P}^{2}$ branched over a
sextic curve $C_{6}$ with an $\mathbf{a}_{11}$ singularity $q$: the curves
$A_{j}$ with $j\in\\{1,\dots,13\\}\setminus\\{4,11\\}$ are mapped to $q$. The
image of the curves $A_{4},A_{11}$ is the line $L$ tangent to the branch of
$C_{6}$ at $q$. The images of $A_{13+j},A_{16+j},\,\,j\in\\{1,2,3\\}$ are
lines $L_{1},L_{2},L_{3}$ through $q$ that are bitangent to the sextic at
other intersection points. The image of $A_{20},A_{21}$ is a conic which is
$6$-tangent to the sextic.

######  Remark 151.

The branch curve is irreducible, with geometric genus $4$.

The divisor $D^{\prime}_{2}=(0,1,2,1,2,2,2,2,3,4,2,3,2,1,0)$ is nef, base
point free of square $2$, with

$D_{2}^{\prime}A_{1}=D_{2}^{\prime}A_{8}=1,\,D_{2}^{\prime}A_{15}=D_{2}^{\prime}A_{16}=D_{2}^{\prime}A_{17}=2,\,D_{2}^{\prime}A_{20}=D_{2}^{\prime}A_{21}=4.$

Therefore the linear system $|D_{2}^{\prime}|$ induces a double cover
$\varphi^{\prime}:X\to\mathbb{P}^{2}$ which is ramified above the line
$L^{\prime}=\varphi^{\prime}(A_{8})$, the curves
$A_{2},A_{3},A_{5},A_{6},A_{7}$ are contracted to a singularity of the branch
curve, and the curves $A_{9},\dots,A_{14}$ are contracted to another
singularity. In that way we obtain a second involution acting on the K3
surface. By [23], we know that the automorphism group of the generic K3 is
$(\mathbb{Z}/2\mathbb{Z})^{2}$.

###  15.2. The lattice $U\oplus\mathbf{E}_{8}\oplus\mathbf{D}_{6}$

The K3 surface $X$ contains $19$ $(-2)$-curves $A_{1},\dots,A_{19}$. The dual
graph of these curves is (see [23, Section 4]):

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$A_{1}$$A_{2}$$A_{3}$$A_{5}$$A_{6}$$A_{7}$$A_{8}$$A_{9}$$A_{10}$$A_{11}$$A_{12}$$A_{14}$$A_{15}$$A_{19}$$A_{17}$$A_{18}$$A_{16}$$A_{4}$$A_{13}$

The curves $A_{1},\dots,A_{16}$ generate the Néron-Severi lattice; in that
base the divisor

$D_{304}=(20,41,63,31,55,48,42,37,33,30,28,27,13,14,2,-9)$

is ample, of square $304,$ with $D_{304}A_{j}=1$ for $j\leq 15$,
$D_{304}A_{16}=D_{304}A_{19}=20$ and $D_{304}A_{17}=D_{304}A_{18}=2$. The
divisor

$D_{2}=(1,2,3,1,3,3,3,3,3,3,3,3,1,2,1,0)$

is nef, base point free, of square $2$, with $D_{2}A_{j}=1$ for
$j\in\\{4,13,16,17,18,19\\}$, else $D_{2}A_{j}=0$. One has

$\begin{array}[]{c}D_{2}\equiv\left(-A_{4}-A_{13}+\sum_{j=1}^{15}A_{j}\right)+A_{16}+A_{17}\\\
D_{2}\equiv\left(-A_{4}-A_{13}+\sum_{j=1}^{15}A_{j}\right)+A_{18}+A_{19}\end{array}.$

By using the linear system $|D_{2}|$, we obtain:

######  Proposition 152.

The K3 surface is the double cover of $\mathbb{P}^{2}$ branched over a sextic
curve with an $\mathbf{a}_{13}$ singularity. The images of $A_{4}$ and
$A_{13}$ is a line, so are the images of $A_{16},A_{17}$ and $A_{18},A_{19}$.

The automorphism group of the generic K3 surface is
$(\mathbb{Z}/2\mathbb{Z})^{2}$ (see [23, Section 4]).

###  15.3. The lattice $U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{7}$

The K3 surface $X$ contains $19$ $(-2)$-curves $A_{1},\dots,A_{19}$. The dual
graph of these curves is (see [23, Section 4]):

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$A_{2}$$A_{6}$$A_{7}$$A_{8}$$A_{9}$$A_{10}$$A_{11}$$A_{12}$$A_{13}$$A_{14}$$A_{1}$$A_{3}$$A_{4}$$A_{5}$$A_{15}$$A_{18}$$A_{19}$$A_{16}$$A_{17}$

The curves $A_{1},\dots,A_{17}$ generate the Néron-Severi lattice; in that
base the divisor

$D_{538}=(32,66,46,27,9,55,45,36,28,21,15,10,6,3,1,0,1)$

is ample, of square $538$, with $D_{538}A_{j}=1$ for $j\neq 1,4,17$, and
$D_{538}A_{j}=2,11,16$ for $j=1,4,17$.

The automorphism group of the generic K3 surface is
$(\mathbb{Z}/2\mathbb{Z})^{2}$ (see [23, Section 4]).

The coarse moduli $\mathcal{M}^{N}$ of
$N=U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{7}$-polarized K3 surfaces is studied
in [9], where it is proved that this moduli is can be seen naturally as the
open set

$\mathcal{M}^{N}=\\{[\alpha,\beta,\gamma,\delta]\in\mathbb{WP}^{3}(2,3,5,6)\,|\,\gamma\neq
0\text{ or }\delta\neq 0\\}$

of a weighted projective space, thus in particular that moduli is rational. In
[9] is also given a (singular) model of the K3 surfaces in $\mathcal{M}^{N}$
as a quartic surface in $\mathbb{P}^{3}$. The K3 surfaces with lattice
$U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{7}$ also belong among the “famous 95”
families discussed in Section 2.5.

###  15.4. The lattice $U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{8}$

The set of $(-2)$-curves $A_{1},...,A_{19}$ on the K3 surface $X$ and their
configuration had been determined in the classical work of Vinberg [42] (see
also [23, Section 4]):

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$A_{1}$$A_{2}$$A_{3}$$A_{5}$$A_{6}$$A_{7}$$A_{8}$$A_{19}$$A_{9}$$A_{10}$$A_{11}$$A_{12}$$A_{13}$$A_{14}$$A_{15}$$A_{17}$$A_{18}$$A_{4}$$A_{16}$

The curves $A_{1},\dots,A_{18}$ generate the Néron-Severi lattice ; in that
base, the divisor

$D_{620}=(-46,-91,-135,-68,-110,-84,-57,-29,30,61,93,126,160,195,231,115,153,76)$

is ample, of square $620$, with $D_{620}A_{j}=1$ for $1\leq j\leq 19$. The
divisor

$D_{2}=(-5,-10,-15,-8,-12,-9,-6,-3,3,6,9,12,15,18,21,10,14,7)$

is nef, of square $2$, with $D_{2}A_{4}=D_{2}A_{16}=1$ and $D_{2}A_{j}=0$ for
$j\neq 4,16$. Using $|D_{2}|$, we see that the K3 surface $X$ is the double
cover of $\mathbb{P}^{2}$ branched over a sextic curve with an
$\mathbf{a}_{17}$ singularity. We have

$D_{2}\equiv
A_{1}+2A_{2}+3A_{3}+A_{4}+3(\sum_{j=5}^{15}A_{j})+A_{16}+2A_{17}+A_{18}+3A_{19},$

thus the image by the double cover map of $A_{4},A_{16}$ is a line through the
singularity.

The automorphism group of the generic K3 surface is
$(\mathbb{Z}/2\mathbb{Z})^{2}$ (see [23, Section 4]).

The rich geometry of $U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{8}$-polarized K3
surfaces is studied in [10]. These K3 surfaces also belong among the “famous
95” families discussed in Section 2.5.

###  15.5. The lattice
$U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{1}$

The K3 surface $X$ contains $24$ $(-2)$-curves $A_{1},\dots,A_{24}$. The dual
graph of these curves is (see [23, Section 4]):

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$A_{22}$$A_{23}$$A_{24}$$\bullet$$\bullet$$\bullet$$A_{19}$$A_{20}$$A_{21}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$A_{1}$$A_{2}$$A_{3}$$A_{4}$$A_{5}$$A_{6}$$A_{7}$$A_{12}$$A_{11}$$A_{10}$$A_{9}$$A_{8}$$A_{13}$$A_{14}$$A_{15}$$A_{16}$$A_{17}$$A_{18}$

Here the dotted thick segment indicates an intersection number equal to $6$
between the curves. The automorphism group of the generic K3 surface is
$\mathfrak{S}_{3}\times\mathbb{Z}/2\mathbb{Z}$ (see [23, Section 4]).

##  16\. Tables : Number of $(-2)$-curves

Rank $3$ lattice | $\\#{\rm(-2)}\,$ |  | ${\rm Aut}$ |  |  Rank $3$ lattice | $\\#{\rm(-2)}$ |  | ${\rm Aut}$ |
---|---|---|---|---|---|---|---|---|---
$S_{1}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{1,3,1}$ | $3$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |
$S_{2}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |  | $S_{1,4,1}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$S_{3}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |  | $S_{1,5,1}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$S_{4}$ | $4$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{1,6,1}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |
$S_{5}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{1,9,1}$ | $9$ |  | ${\scriptscriptstyle\\{1\\}}$ |
$S_{6}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{4,1,1}$ | $3$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |
$S_{1,1,1}$ | $3$ |  $\,\aleph\,$$\,\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $S_{5,1,1}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$S_{1,1,2}$ | $3$ |  $\,\aleph\,$$\,\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $S_{6,1,1}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |
$S_{1,1,3}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{7,1,1}$ | $6$ |  | ${\scriptscriptstyle\\{1\\}}$ |
$S_{1,1,4}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |  | $S_{8,1,1}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |
$S_{1,1,6}$ | $6$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{10,1,1}$ | $8$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$S_{1,1,8}$ | $8$ |  | ${\scriptscriptstyle\\{1\\}}$ |  | $S_{12,1,1}$ | $6$ |  | ${\scriptscriptstyle\\{1\\}}$ |
$S_{1,2,1}$ | $3$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $S_{4,1,2}^{\prime}$ | $4$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$

Lattice | $\\#{\rm(-2)}$ |  | ${\rm Aut}$ |  | Lattice | $\\#{\rm(-2)}$ |  | ${\rm Aut}$ |
---|---|---|---|---|---|---|---|---|---
Rank $4$ |  |  |  |  |  Rank $6$ |  |  |  |
$L(24)$ | $6$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U(3)\oplus\mathbf{A}_{2}^{\oplus 2}$ | $12$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$L(27)$ | $8$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U(4)\oplus\mathbf{D}_{4}$ | $9$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$[8]\oplus\mathbf{A}_{1}^{\oplus 3}$ | $12$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{4}$ | $6$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U\oplus\mathbf{A}_{1}^{\oplus 2}$ | $5$ |  $\,\angle\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{A}_{1}\oplus\mathbf{A}_{3}$ | $7$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U(2)\oplus\mathbf{A}_{1}^{\oplus 2}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{A}_{2}^{\oplus 2}$ | $7$ | $\angle\star$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U(3)\oplus\mathbf{A}_{1}^{\oplus 2}$ | $8$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{1}^{\oplus 2}\oplus\mathbf{A}_{2}$ | $8$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U(4)\oplus\mathbf{A}_{1}^{\oplus 2}$ | $8$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |  | $U(2)\oplus\mathbf{A}_{1}^{\oplus 4}$ | $16$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{A}_{2}$ | $4$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{1}^{\oplus 4}$ | $9$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U(2)\oplus\mathbf{A}_{2}$ | $4$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U(2)\oplus\mathbf{D}_{4}$ | $6$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U(3)\oplus\mathbf{A}_{2}$ | $4$ |  $\,\angle\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |  | $U\oplus\mathbf{D}_{4}$ | $6$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U(6)\oplus\mathbf{A}_{2}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\\{1\\}}$ |  |  |  |  |  |
$L_{12}=S_{0}\oplus\mathbf{A}_{2}$ | $6$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
$[4]\oplus[-4]\oplus\mathbf{A}_{2}$ | $6$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
$[4]\oplus\mathbf{A}_{3}$ | $5$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
|  |  |  |  |  |  |  |  |
Rank $5$ |  |  |  |  |  Rank $7$ |  |  |  |
$U\oplus\mathbf{A}_{1}^{\oplus 3}$ | $7$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}$ | $8$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U(2)\oplus\mathbf{A}_{1}^{\oplus 3}$ | $10$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{A}_{1}^{\oplus 5}$ | $12$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U(4)\oplus\mathbf{A}_{1}^{\oplus 3}$ | $24$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U(2)\oplus\mathbf{A}_{1}^{\oplus 5}$ | $27$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{A}_{1}\oplus\mathbf{A}_{2}$ | $6$ | $\angle\star$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{1}\oplus\mathbf{A}_{2}^{\oplus 2}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U\oplus\mathbf{A}_{3}$ | $5$ | $\angle\star$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{1}^{\oplus 2}\oplus\mathbf{A}_{3}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$[4]\oplus\mathbf{D}_{4}$ | $5$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{2}\oplus\mathbf{A}_{3}$ | $8$ | $\angle\star$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$[8]\oplus\mathbf{D}_{4}$ | $7$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{1}\oplus\mathbf{A}_{4}$ | $8$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$[16]\oplus\mathbf{D}_{4}$ | $8$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{5}$ | $7$ | $\angle\star$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$[6]\oplus\mathbf{A}_{2}^{\oplus 2}$ | $10$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{D}_{5}$ | $7$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$

Lattice | $\\#{\rm(-2)}$ |  | ${\rm Aut}$ |  | Lattice | $\\#{\rm(-2)}$ |  | ${\rm Aut}$ |
---|---|---|---|---|---|---|---|---|---
Rank $8$ |  |  |  |  |  Rank $11$ |  |  |  |
$U\oplus\mathbf{D}_{6}$ | $8$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{1}$ | $12$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}^{\oplus 2}$ | $10$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{D}_{8}\oplus\mathbf{A}_{1}$ | $14$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{A}_{1}^{\oplus 6}$ | $19$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{D}_{4}^{\oplus 2}\oplus\mathbf{A}_{1}$ | $22$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U(2)\oplus\mathbf{A}_{1}^{\oplus 6}$ | $56$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}^{\oplus 5}$ | $90$ |  | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{A}_{2}^{\oplus 3}$ | $10$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
$U\oplus\mathbf{A}_{3}^{\oplus 2}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
$U\oplus\mathbf{A}_{2}\oplus\mathbf{A}_{4}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  Rank $12$ |  |  |  |
$U\oplus\mathbf{A}_{1}\oplus\mathbf{A}_{5}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{1}^{\oplus 2}$ | $14$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{A}_{6}$ | $8$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{D}_{8}\oplus\mathbf{A}_{1}^{\oplus 2}$ | $19$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{A}_{2}\oplus\mathbf{D}_{4}$ | $9$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{D}_{4}^{\oplus 2}\oplus\mathbf{A}_{1}^{\oplus 2}$ | $59$ |  | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{A}_{1}\oplus\mathbf{D}_{5}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{A}_{2}\oplus\mathbf{E}_{8}$ | $13$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U\oplus\mathbf{E}_{6}$ | $8$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
|  |  |  |  |  |  |  |  |
Rank $9$ |  |  |  |  |  Rank $13$ |  |  |  |
$U\oplus\mathbf{E}_{7}$ | $9$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{1}^{\oplus 3}$ | $17$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{D}_{6}\oplus\mathbf{A}_{1}$ | $10$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{D}_{8}\oplus\mathbf{A}_{1}^{\oplus 3}$ | $39$ |  $\aleph$$\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}^{\oplus 3}$ | $15$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{3}$ | $14$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$
$U\oplus\mathbf{A}_{1}^{\oplus 7}$ | $37$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ |  |  |  |  |
$U(2)\oplus\mathbf{A}_{1}^{\oplus 7}$ | $240$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$ |  |  |  |  |
$U\oplus\mathbf{A}_{7}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  Rank $14$ |  |  |  |
$U\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{3}$ | $10$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{D}_{4}$ | $15$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$
$U\oplus\mathbf{D}_{5}\oplus\mathbf{A}_{2}$ | $10$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ | $U\oplus\mathbf{D}_{8}\oplus\mathbf{D}_{4}$ | $20$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{D}_{7}$ | $9$ | $\,\star\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{1}^{\oplus 4}$ | $27$ |  | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $t$
$U\oplus\mathbf{A}_{1}\oplus\mathbf{E}_{6}$ | $10$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |
|  |  |  |  |  |  |  |  |
Rank $10$ |  |  |  |  |  Rank $\geq 15$ |  |  |  |
$U\oplus\mathbf{E}_{8}$ | $10$ |  $\aleph$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}$ | $21$ |  | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{D}_{8}$ | $10$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{D}_{6}$ | $19$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{E}_{7}\oplus\mathbf{A}_{1}$ | $11$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{7}$ | $19$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{D}_{4}^{\oplus 2}$ | $11$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{8}$ | $19$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$
$U\oplus\mathbf{D}_{6}\oplus\mathbf{A}_{1}^{\oplus 2}$ | $14$ |  | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ | $U\oplus\mathbf{E}_{8}\oplus\mathbf{E}_{8}\oplus\mathbf{A}_{1}$ | $24$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathfrak{S}_{3}\times\mathbb{Z}/2\mathbb{Z}}$ | $tn$
$U(2)\oplus\mathbf{D}_{4}^{\oplus 2}$ | $18$ |  $\,\aleph\,$$\,\mathbf{u}\,$ | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$ |  |  |  |  |
$U\oplus\mathbf{D}_{4}\oplus\mathbf{A}_{1}^{\oplus 4}$ | $27$ | $\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $t$ |  |  |  |  |
$U\oplus\mathbf{A}_{1}^{\oplus 8}$ | $145$ |  | ${\scriptscriptstyle(\mathbb{Z}/2\mathbb{Z})^{2}}$ | $tn$ |  |  |  |  |
$U\oplus\mathbf{A}_{2}\oplus\mathbf{E}_{6}$ | $11$ |  $\aleph\star$$\,\mathbf{u}\,$ | ${\scriptscriptstyle\mathbb{Z}/2\mathbb{Z}}$ | $n$ |  |  |  |  |

These tables give the number of $(-2)$-curves. A $\,\aleph\,$ means that the
lattice is among the $95$ famous families, a $\,\angle\,$ means that the
lattice is a mirror of one of the $95$ famous, but not one of these $95$ (see
Section 2.5). A $\star$ means that their ${\rm(-2)}$-curve configuration is
predicted by Theorem 9. A $\,\mathbf{u}\,$ means unirationality of the moduli
space. The column ${\rm Aut}$ gives the automorphism group of the general K3
surface. A $t$ mean that the action on the Néron-Severi lattice of an
hyperelliptic involution is trivial, an $n$ means that the action of an
hyperelliptic involution is not trivial, $tn$ means that both cases exists;
from that data one can recover the kernel of the map ${\rm Aut}(X)\to O({\rm
NS(}X))$.

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Xavier Roulleau,
Aix-Marseille Université, CNRS, Centrale Marseille,
I2M UMR 7373,
13453 Marseille,
France

http://www.i2m.univ-amu.fr/perso/xavier.roulleau/Site_Pro/Bienvenue.html


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