# Log K3 surfaces with irreducible boundary

János Kollár<EMAIL_ADDRESS>
###### Abstract.

We determine the automorphism group of an open log K3 surface with irreducible
boundary.

A log K3 surface over a field $k$ is a pair $(S,N)$, where $S$ is a projective
surface over $k$ such that $S_{\bar{k}}$ is rational, and $N\in|-K_{S}|$ is a
curve with only nodes as singularities. These are frequently called
anticanonical pairs or log Calabi-Yau surfaces. We allow $S_{\bar{k}}$ to have
Du Val singularities away from $N$. The field $k$ can be arbitrary, but some
minor issues are left to the reader in characteristic 2.

Our aim is to understand whether the open surface $S\setminus N$ determines
the pair $(S,N)$. As a first counterexample, let $s\in N$ be a node that is a
smooth point of $S$, and $\pi:S^{\prime}\to S$ its blow up with exceptional
curve $E$. Then $(S^{\prime},N^{\prime}+E)$ is another log K3 surface, and
$S\setminus N\cong S^{\prime}\setminus(N^{\prime}+E)$. Less obvious examples
are given in (10.1–4). These suggest that the best one can hope for is the
following.

######  Theorem 1.

Let $k$ be a field and $(S_{i},N_{i})$ smooth, log K3 surfaces over $k$ with
geometrically irreducible boundaries. Then

$None$ $S_{1}\setminus N_{1}\cong S_{2}\setminus
N_{2}\quad\Leftrightarrow\quad(S_{1},N_{1})\cong(S_{2},N_{2}).$

The natural approach to proving (1.1) would be to show that every isomorphism
$\phi^{\circ}:S_{1}\setminus N_{1}\cong S_{2}\setminus N_{2}$ extends to an
isomorphism $\phi:S_{1}\cong S_{2}$. This turns out to hold if
$(N_{i}^{2})\leq 2$, but not otherwise.

The failure of this isomorphism-extension property is measured by the
difference between $\operatorname{Aut}(S\setminus N)$ and
$\operatorname{Aut}(S,N)$. We show that there is a natural normal subgroup
$\operatorname{Aut}^{\circ}(S\setminus N)\subset\operatorname{Aut}(S\setminus
N)$ such that

$None$ $\operatorname{Aut}(S\setminus
N)\cong\operatorname{Aut}^{\circ}(S\setminus
N)\rtimes\operatorname{Aut}(S,N).$

It is easy to check that $\operatorname{Aut}^{\circ}(S\setminus N)$ is trivial
if either $N$ is smooth or $(N^{2})\leq 0$; see (18) and (19). We are thus
left with the cases when $S$ is a Del Pezo surface (11) and $N$ is a nodal
curve. We determine $\operatorname{Aut}^{\circ}(S\setminus N)$ for all of
them.

######  Theorem 2.

Let $k$ be a field, $S$ a Del Pezzo surface over $k$ and $N\in|-K_{S}|$ a
geometrically irreducible curve with a node $s\in N$, such that $S$ is smooth
at $s$. Then $\operatorname{Aut}^{\circ}(S\setminus N)$ is

  1. (2.1)

trivial if $\deg S=1,2$,

  2. (2.2)

${\mathbb{Z}}/2$ if $\deg S=3$,

  3. (2.3)

trivial if $4\leq\deg S\leq 9$ and the node is unsplit (12), and

  4. (2.4)

the infinite dihedral group $D_{\infty}$ (13) if $4\leq\deg S\leq 9$ and the
node is split.

Previous results. [Yos85, 4.2.2] describes an infinite order element of
$\operatorname{Aut}\bigl{(}{\mathbb{P}}^{2}\setminus(\mbox{nodal
cubic})\bigr{)}$. The same construction is used in [Ore02, Sec.6] to obtain
highly cuspidal rational curves in ${\mathbb{P}}^{2}$. It turns out that a
variant gives the whole $\operatorname{Aut}^{\circ}({\mathbb{P}}^{2}\setminus
N)$; most likely both authors suspected this. Versions for other Del Pezzo
surfaces are considered in [MS24, Sec.2.1]. A more detailed discussion of the
relationship with the maps considered in [Yos85, Ore02, MS24] is given in
(25).

######  Remarks 3.

If $S$ is a Del Pezzo surface, then $\operatorname{Aut}(S,N)$ is always
finite, and trivial in most cases; see (26) for some examples.

By contrast, $\operatorname{Aut}(S,N)$ can be infinite if $(N^{2})\leq 0$. A
comprehensive study is given in [Li22], which builds on the works of [GHK15b,
GHK15a, Fri16].

I do not know how to write down a priori the quotient map
$\operatorname{Aut}(S\setminus N)\to\operatorname{Aut}(S,N)$. Thus we
determine the structure of $\operatorname{Aut}(S\setminus N)$ in all cases,
and then observe that they have a semidirect product structure as claimed in
(1.2).

######  Acknowledgments.

The main impetus came from trying to understand the algebraic geometry side of
the paper [MS24]. I thank D. McDuff and K. Siegel for many e-mails answering
my questions and posing new problems. The many comments and corrections of P.
Hacking, J. Li and K. Oguiso were also very helpful.

Partial financial support was provided by the NSF under grant number
DMS-1901855.

##  1\. Outline of the proofs

For Theorem 2, the easy cases are (2.1–3); these are settled in (19–20). In
order to prove (2.4) we first construct some automorphisms.

######  4Geiser-type involutions.

Let $S$ be a Del Pezzo surface of degree $\geq 4$, and $N\in|-K_{S}|$ a
geometrically irreducible curve with a split node $s\in N$, such that $S$ is
smooth at $s$. In (23) we construct two Geiser-type involutions

$None$ $\sigma_{+},\sigma_{-}\in\operatorname{Aut}(S\setminus N),$

indexed by the local branches $B_{\pm}$ of $N$ at $s$. There is no natural way
to distinguish the branches from each other, so it is best to think of
$\\{\sigma_{+},\sigma_{-}\\}$ as a pair of involutions. Then we define

$None$ $\operatorname{Aut}^{\circ}(S\setminus
N):=\langle\sigma_{+},\sigma_{-}\rangle.$

We check in Paragraph 6 that $\operatorname{Aut}^{\circ}(S\setminus N)\cong
D_{\infty}$, and we see in (27) that its normalizer contains
$\operatorname{Aut}(S,N)$.

It takes longer to show that the Geiser involutions and
$\operatorname{Aut}(S,N)$ generate $\operatorname{Aut}(S\setminus N)$. The
approach is modeled on the Noether-Fano method, as implemented by Segre and
Manin for cubic surfaces; see [Seg42, Man72] or [KSC04, Sec.2.1].

The first step is a computation of how the basic numerical invariants of a
cuspidal curve transform under $\sigma_{+},\sigma_{-}$. The following is
equivalent to [MS24, 2.2.6]; in the first version the second case in (5.2) was
overlooked. The formulas are simplest using the normalized degree

$None$ $\operatorname{ndeg}(C):=\tfrac{1}{\deg S}(-K_{S}\cdot
C)\quad\mbox{and}\quad\operatorname{mult}(C):=\operatorname{mult}_{s}(C).$

######  Lemma 5.

Let $S$ be a Del Pezzo surface of degree $d\geq 4$, and $N\in|-K_{S}|$ an
irreducible curve with a split node $s\in N$, such that $S$ is smooth at $s$.
Let $C\subset S$ be an irreducible curve with only single, cuspidal point on
$N$. Then, for a suitable choice of the local branch $B_{+}$ of $N$,

$\begin{array}[]{ccl}(\ref{cusp.transf.cor}.1)\qquad\operatorname{ndeg}\bigl{(}\sigma_{+}(C)\bigr{)}&=&(d{-}1)\operatorname{ndeg}(C)-\operatorname{mult}(C),\quad\mbox{and}\\\\[4.30554pt]
(\ref{cusp.transf.cor}.2)\qquad\operatorname{ndeg}\bigl{(}\sigma_{-}(C)\bigr{)}&=&\left\\{\begin{array}[]{l}\operatorname{mult}(C)-\operatorname{ndeg}(C)\quad\mbox{if
$\tfrac{d}{d{-}1}\geq\tfrac{\operatorname{mult}C}{\operatorname{ndeg}C}$,
and}\\\\[4.30554pt]
(d{-}1)\operatorname{ndeg}(C)-(d{-}2)\operatorname{mult}(C)\quad\mbox{otherwise.}\end{array}\right.\end{array}$

The proof is given in (32).

######  6Bounds on the multiplicity.

Note first that $2\cdot\operatorname{mult}(C)\leq(N\cdot
C)=d\cdot\operatorname{ndeg}(C)$, thus, if $d\geq 4$ then

$None$
$\operatorname{ndeg}\bigl{(}\sigma_{+}(C)\bigr{)}>\operatorname{ndeg}(C),$

except when $d=4$ and $\operatorname{mult}(C)=2\cdot\operatorname{ndeg}(C)$.
Thus we can increase the degree by one of $\sigma_{+},\sigma_{-}$, hence they
generate an infinite dihedral subgroup of
$\operatorname{Aut}^{\circ}(S\setminus N)$.

By explicit computation we also see that

$None$
$\operatorname{ndeg}\bigl{(}\sigma_{-}(C)\bigr{)}<\operatorname{ndeg}(C)$

provided

$None$
$\operatorname{ndeg}(C)<\operatorname{mult}(C)<2\cdot\operatorname{ndeg}(C).$

The upper bound $\operatorname{mult}(C)<2\cdot\operatorname{ndeg}(C)$ holds
for almost all curves, see Corollary 34. The exceptional cases are when
$d\in\\{4,5\\}$ and $\operatorname{mult}(C)=2=2\cdot\operatorname{ndeg}(C)$.

Therefore, using Lemma 5 we can lower the degree of $C$, and get the
following.

######  Corollary 7.

Let $(S,N)$ and $C\subset S$ be as in Lemma 5. Then there is a
$\phi\in\langle\sigma_{+},\sigma_{-}\rangle$ such that

  1. (7.1)

either
$\operatorname{mult}\bigl{(}\phi(C)\bigr{)}\leq\operatorname{ndeg}\bigl{(}\phi(C)\bigr{)}$,

  2. (7.2)

or $d\in\\{4,5\\}$, $C$ is rational, and $\phi(C)\in|-K_{S}|$ has a $(2,3)$
cusp at $s$.

Moreover, for (7.1),

  1. (7.3)

either $\phi$ is unique and the inequality in (7.1) is strict,

  2. (7.4)

or there are two such $\phi$ and equality holds in (7.1) for both.

The inequality (7.1) is exactly the one needed for the Noether-Fano method.
(The Noether-Fano method usually works with the base points of birational
transforms of very ample linear systems. In the current situation, studying
the cusps of curves works better.)

To complete the proof of Theorem 2 we start with an elliptic curve $E\subset
S$ that meets $N$ at a single smooth point, see (38). Given
$\psi\in\operatorname{Aut}(S\setminus N)$, we apply Corollary 7 to $\psi(E)$.
Since $E$ is not rational, we are in case (7.1). Thus we have a
$\phi\in\langle\sigma_{+},\sigma_{-}\rangle$ such that

$\operatorname{mult}\bigl{(}\phi\circ\psi(E)\bigr{)}\leq\operatorname{ndeg}\bigl{(}\phi\circ\psi(E)\bigr{)}.$

The Noether-Fano method now says that $\phi\circ\psi$ is an automorphism of
$(S,N)$, as needed; see Section 8 for details.∎

Another application to log K3 surfaces is the study of curves $C\subset
S\setminus N$ that are isomorphic to ${\mathbb{A}}^{1}$. These are called
affine lines on $S\setminus N$. A series of papers [MM23, MMW23, MS24] studies
those affine lines $C\subset S$ over ${\mathbb{C}}$ that are ‘meaningful’ in
symplectic geonetry. That is, a certain Fredholm index is 0; see [MS24,
Sec.1.2]. These are the affine lines for which the birational transform of $C$
on the minimal log resolution of $(S,N+C)$ is a $(-1)$-curve. [MM23, MMW23,
MS24] give a complete answer for ${\mathbb{P}}^{2}$ and for its 1-point blow-
up.

Combining Corollary 7 with the multiplicity inequality of [MS89] gives all
affine lines for $S={\mathbb{C}}{\mathbb{P}}^{2}$.

######  Corollary 8.

The affine lines ${\mathbb{A}}^{1}\cong
C\subset\bigl{(}{\mathbb{C}}{\mathbb{P}}^{2}\setminus(xyz=x^{3}+y^{3})\bigr{)}$
form 3 orbits under
$\operatorname{Aut}\bigl{(}{\mathbb{C}}{\mathbb{P}}^{2}\setminus(xyz=x^{3}+y^{3})\bigr{)}$.
Representative curves in these orbits are:

  1. (8.1)

$(x=0)$, tangent to a branch of $N$ at the node,

  2. (8.2)

$(3x+3y+z=0)$, flex tangent of $N$, and

  3. (8.3)

$(21x^{2}-22xy+21y^{2}-6xz-6yz+z^{2}=0)$, conic 6-tangent to $N$.

The proof is given in Section 9. All these affine lines were known to [Ore02,
MS24], and the series (8.1) gives the lines that have Fredholm index 0. The
proof relies on [MS89], which is known only over ${\mathbb{C}}$, and does not
seem to have an analog for other Del Pezzo surfaces.

On the 1-point blow-up of ${\mathbb{P}}^{2}$ there are infinitely many orbits
of lines with Fredholm index 0 [MM23, MMW23, MS24]; and also infinitely many
orbits of other lines, see (49).

######  9Related works.

Isomorphisms between affine surfaces of the form
${\mathbb{P}}^{2}\setminus(\mbox{curve})$ are considered in [Yos85], giving a
fairly detailed classification. Remaining cases involving some high degree
curves are studied in [Bla09].

Affine lines in ${\mathbb{A}}^{2}={\mathbb{P}}^{2}\setminus(\mbox{line})$ are
classified in [AM75], and affine lines in
${\mathbb{P}}^{2}\setminus(\mbox{smooth conic})$ are described in [DD18]. In
these cases, a suitable automorphism transforms any affine line into a line
(resp. a line or a conic).

The papers [Tak96, Tak01] consider curves in
${\mathbb{P}}^{2}\setminus(\mbox{smooth cubic})$ whose normalization is
${\mathbb{A}}^{1}$; with strongest answers in degrees $\leq 8$. A more general
counting problem involving smooth cubics and rational curves is considered in
[CC08].

Using birational transformations to study cuspidal curves has been a recurring
theme; see for example [MS89, AB97, Ore02]. [FdBLMHN07] gives a topological
classification of rational curves in ${\mathbb{P}}^{2}$ that have a unique
singularity, that is a cusp with one Puiseux pair.

######  Examples 10.

These examples show that the assumptions in Theorem 1 are mostly necessary.

(10.1) A boundary $N=\cup_{i}N_{i}$ could be called minimal if none of its
irreducible components are $(-1)$-curves. If $(N_{i}^{2})\leq-2$ for every $i$
then it is easy to see that $S\setminus N$ determines $(S,N)$. However,
Theorem 1 does not extend to all minimal boundaries. For example,
${\mathbb{G}}_{m}\times{\mathbb{G}}_{m}$ can be compactified to
$\bigl{(}{\mathbb{P}}^{2},(xyz=0)\bigr{)}$ and to
$\bigl{(}{\mathbb{P}}^{1}_{x}\times{\mathbb{P}}^{1}_{y},(x_{0}x_{1}y_{0}y_{1}=0)\bigr{)}$,
neither boundary contains a curve with $-1$ self-intersection.

(10.2) A more sophisticated version is the following. Let $S$ be a nontrivial
Severi-Brauer surface and $N\subset S$ the union of 3 conjugate lines. Note
that $N$ is irreducible, but not geometrically irreducible.

Blowing up the 3 nodes of $N$ and then contracting the birational transform of
$N$ gives the dual Severi-Brauer surface $(S^{\prime},N^{\prime})$. Thus
$S\setminus N\cong S^{\prime}\setminus N^{\prime}$, but $S\not\cong
S^{\prime}$.

(10.3) Let $(S,N)$ be a log K3 surface such that $d:=(N^{2})\leq 2$. Assume
that $S$ is smooth at the node $s\in N$. Blowing up $s$, the birational
transform $\tilde{N}$ of $N$ is a smooth rational curve with self-intersection
$d-4$. Contracting it we get a log K3 surface $(S^{\prime},N^{\prime})$ such
that $S\setminus N\cong S^{\prime}\setminus N^{\prime}$ but $S\not\cong
S^{\prime}$. Here $S^{\prime}$ is singular at the node $s^{\prime}\in
N^{\prime}$. If $d=2$ then $S^{\prime}$ is also a Del Pezzo surface with an
$A_{1}$ singularity at the node $s^{\prime}$.

(10.4) If $(N^{2})<0$ then $N$ is (analytically) contractible
$(S,N)\to(S^{*},s^{*})$. Then $S^{*}$ is the one-point compactification of
$S\setminus N$, and $S$ is the minimal resolution of the singularity at
$s^{*}$. Thus the isomorphism-extension property holds. The same holds if $N$
is reducible with a negative definite intersection matrix.

##  2\. Notation and definitions

######  11Del Pezzo surfaces.

Let $k$ be a field. I follow the terminology of [Dol12, 8.1.12], according to
which a Del Pezzo surface is a projective $k$-surface $S$ such that $-K_{S}$
is ample, and $S_{\bar{k}}$ is irreducible with at worst Du Val singularites.

Du Val singularites are also called rational double points or ADE
singularites. In our discussions only the $A_{n}$ singularities $(xy=z^{n+1})$
will appear.

The degree of $S$ is the self-intersection number of $-K_{S}$.

A weak Del Pezzo surface is a projective surface $S$ such that $(-K_{S}\cdot
C)\geq 0$ for every curve, the self-intersection number of $-K_{S}$ is
positive, and $S_{\bar{k}}$ has at worst Du Val singularites.

For such an $S$ there is a unique birational morphism $m_{S}:S\to S^{*}$ to a
Del Pezzo surface $S^{*}$ such that $K_{S}=m_{S}^{*}K_{S^{*}}$. $S$ and
$S^{*}$ have the same degrees, and $m_{S}$ contracts all the curves $C\subset
S$ for which $(-K_{S}\cdot C)=0$. If $S$ is smooth, these are the smooth
rational curves with self-intersection $-2$. $S^{*}$ is singular at the images
of these curves.

Note on terminology. Many authors require (sometimes implicitly) that a Del
Pezzo surface be smooth. For us the more encompassing definition is more
convenient. In the construction of the Geiser involutions, the most important
surface is a singular Del Pezzo surface; see Example 24.

A normal, projective surface $S$ with $-K_{S}$ ample is called a log Del Pezzo
surface. Some authors assume that the singularities are quotient
singularities, some allow a boundary, and some drop the modifier ‘log.’ So
caveat lector.

######  Definition 12 (Split nodes).

Let $C$ be a curve over a field $k$ and $c\in C(k)$ a point. $C$ has a node at
$c$ if, after completion, it is isomorphic to
$\operatorname{Spec}k[[x,y]]/(q)$ where $q$ is a homogeneous degree 2
polynomial with 2 simple roots.

The node is split if $q$ has 2 roots in $k$. Then we can use the standard form
$\operatorname{Spec}k[[x,y]]/(xy)$. The node is unsplit if the roots of $q$
are conjugate over $k$. If $\operatorname{char}k\neq 2$ then we can use the
normal form $\operatorname{Spec}k[[x,y]]/(x^{2}-ay^{2})$, where $a$ is not a
square in $k$.

######  Definition 13.

Let $D_{\infty}$ denote the infinite dihedral group. It can be given as
$\langle a,b:a^{2}=1,aba=b^{-1}\rangle$, as $\langle a,c:a^{2}=c^{2}=1\rangle$
(setting $c=ab$), as ${\mathbb{Z}}/2\ast{\mathbb{Z}}/2$, or as
${\mathbb{Z}}\rtimes{\mathbb{Z}}/2$ with the ${\mathbb{Z}}/2$-action $r\to-r$
on ${\mathbb{Z}}$.

######  14Moduli of log K3 pairs.

Let $S$ be a Del Pezzo surface and $N\in|-K_{S}|$ a geometrically irreducible,
nodal curve.

If $S={\mathbb{P}}^{2}$ and $N$ has a split node, then, up to isomorphism
$\bigl{(}{\mathbb{P}}^{2},(xyz=x^{3}+ay^{3})\bigr{)}$ are the only ones over
any field $k$. If $a$ is a cube in $k$ then we get
$\bigl{(}{\mathbb{P}}^{2},(xyz=x^{3}+y^{3})\bigr{)}$.

For $d<9$ the nodal pairs $(S,N_{S})$ with $\deg S=(N_{S}^{2})=d$ form a
$(9-d)$-dimensional family, see [GHK15b] for a detailed treatment of their
moduli theory.

A naive argument is the following. If $S$ is smooth then we can write
$\pi:S\to{\mathbb{P}}^{2}$ as the blow-up of ${\mathbb{P}}^{2}$ at $c$ points.
Each $\pi$-exceptional curve meets $N_{S}$ with multiplicity 1, so $(S,N_{S})$
is the blow-up of $({\mathbb{P}}^{2},N)$ at $c$ smooth points on $N$. Thus the
moduli space is roughly the same as the space of $c$ smooth points in $N$.

Note that if $c=1$ or $c=2$, then $\pi:S\to{\mathbb{P}}^{2}$ is unique, but
not for $c\geq 3$. We have finitely many choices for $\pi$ for $c\leq 8$, and
at most countably many for $c\geq 9$.

Note that the Torelli theorem of [GHK15b, §5] suggests that the ‘best’ choice
may be repeatedly blowing up the point
$(1{:}{-}1{:}0)\in(xyz=x^{3}+y^{3})\bigr{)}$. This gives the unique pair where
the mixed Hodge structure on $H_{2}(S\setminus N_{S})$ is split. However, this
is only a weak Del Pezzo surface for $8\geq c\geq 2$.

We need a few results on discrepancies. For general introductions see [KM98,
Sec.2.3] or [Kol13, Sec.2.1]. However, we need only 3 simple facts, which are
easy to state (and prove) for smooth surfaces.

######  15Discrepancies for surface pairs.

Let $S$ be a smooth surface and $\pi:T\to S$ a proper, birational morphism. We
are interested in the local behavior of $\pi$ over a point $s\in S$, so assume
that $\pi$ is an isomorphism over $S\setminus\\{s\\}$, and $T$ is normal.

We can write $K_{T}\sim\pi^{*}K_{S}+E$ where $E$ is effective and
$\pi$-exceptional.

Next let $C$ be a (possibly reducible) curve on $S$ and $C_{T}$ its birational
transform on $T$. Then $C_{T}\sim\pi^{*}C-D$ where $D$ is effective and
$\pi$-exceptional. For any (say rational) $c>0$ we can formally write

$None$ $K_{T}+cC_{T}\sim_{{\mathbb{Q}}}\pi^{*}(K_{S}+cC)+E-cD.$

The coefficients in $E-cD$ are called the discrepancies of $(S,cC)$.

The claims below can be established by checking them for 1 blow-up and using
induction. For precise references see [Kol13, 2.29] for (15.2–3) and [KM98,
2.31] or [Kol13, 2.7] for (15.4).

(15.2) $E-cD$ is effective for every $\pi$ iff
$c\cdot\operatorname{mult}_{s}C\leq 1$. (Such pairs $(S,cC)$ are called
canonical.)

(15.3) $E-cD$ is effective and
$\operatorname{Supp}(E-cD)=\operatorname{Ex}(\pi)$ for every $\pi$ iff
$c\cdot\operatorname{mult}_{s}C<1$. (Such pairs $(S,cC)$ are called terminal.)

(15.4) If $C$ has a node at $s$ and $c\leq 1$ then all coefficients in $E-cD$
are $\geq-1$. (The latter is the log canonical property.)

##  3\. Easy cases of Theorem 1

######  16Preliminary remarks for the proof of Theorem 1.

Let $(S,N)$ be a log K3 surface with irreducible boundary. Then
$\operatorname{Cl}(S\setminus
N)\cong\operatorname{Cl}(S)/{\mathbb{Z}}[-K_{S}]$. In particular, if $k$ is
algebraically closed and $S$ is smooth, then the class group of $S\setminus N$
is

  1. (16.1)

${\mathbb{Z}}/3$ if $S\cong{\mathbb{P}}^{2}$,

  2. (16.2)

${\mathbb{Z}}+{\mathbb{Z}}/2$ if
$S\cong{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, and

  3. (16.3)

${\mathbb{Z}}^{9-d}$ in all other cases, where $d=\deg S$.

In particular, if $\phi:S_{1}\setminus N_{1}\cong S_{2}\setminus N_{2}$, then
$\deg S_{1}=\deg S_{2}$.

Let $C\in|-m_{1}K_{S_{1}}|$ be an irreducible curve. Then the restriction of
$C$ to $S_{1}\setminus N_{1}$ is linearly equivalent to $0$. Thus the
restriction of $\phi_{*}(C)$ to $S_{2}\setminus N_{2}$ is also linearly
equivalent to $0$, hence $\phi_{*}(C)\in|-m_{2}K_{S_{2}}|$ for some $m_{2}$.

######  17Graphs of birational maps.

Let $(S_{i},N_{i})$ be log K3 surfaces and $\phi^{\circ}:S_{1}\setminus
N_{1}\cong S_{2}\setminus N_{2}$ an isomorphism that does not extend to an
isomorphism $\phi:(S_{1},N_{1})\cong(S_{2},N_{2})$. Let $T$ be the
normalization of the closure of the graph of $\phi^{\circ}$ with projections
$p_{i}:T\to S_{i}$.

Let $E_{i}\subset T$ be the reduced exceptional curve of $p_{i}$. Then $E_{i}$
is not contracted by $p_{3-i}$, so

$p_{3-i}:E_{i}\dasharrow N_{3-i}\quad\mbox{is birational.}\quad$

So $E_{i}$ is the birational transform of $N_{r-i}$ on $T$, let us thus write
$N^{\prime}_{3-i}:=E_{i}$.

Next we use (15.4), thus

$None$
$\begin{array}[]{ccl}K_{T}+N^{\prime}_{1}+N^{\prime}_{2}-D_{1}&\sim&p_{1}^{*}(K_{S_{1}}+N_{1}),\quad\mbox{and}\\\
K_{T}+N^{\prime}_{1}+N^{\prime}_{2}-D_{2}&\sim&p_{2}^{*}(K_{S_{2}}+N_{2}),\end{array}$

where the $D_{i}$ are effective and
$\operatorname{Supp}D_{i}\subset\operatorname{Supp}N^{\prime}_{3-i}$. Since
$K_{S_{i}}+N_{i}\sim 0$, we conclude that $D_{1}\sim D_{2}$. Thus in fact
$D_{1}=D_{2}=0$, hence

$None$ $p_{1}^{*}(K_{S_{1}}+N_{1})\sim K_{T}+N^{\prime}_{1}+N^{\prime}_{2}\sim
p_{2}^{*}(K_{S_{2}}+N_{2}).$

Note that even if the $S_{i}$ are smooth, $T$ is usually singular. For
example, for $S={\mathbb{P}}^{2}$ and $\phi=\sigma_{+}\circ\sigma_{-}$, one
can compute that the singularities of $T$ are
${\mathbb{A}}^{2}/\frac{1}{7}(1,1)$ and ${\mathbb{A}}^{2}/\frac{1}{48}(1,41)$.

######  Lemma 18.

Let $(S_{i},N_{i})$ be log K3 surfaces with geometrically irreducible
boundaries, and $\phi^{\circ}:S_{1}\setminus N_{1}\cong S_{2}\setminus N_{2}$
an isomorphism that does not extend to an isomorphism
$\phi:(S_{1},N_{1})\cong(S_{2},N_{2})$. Then the $N_{i}$ are nodal and
$p_{i}:T\to S_{i}$ contracts $N^{\prime}_{3-i}$ to the node $s_{i}\in N_{i}$.

Proof. Set $w_{i}:=p_{i}(N^{\prime}_{3-i})$. If $w_{i}$ is a smooth point of
$N_{i}$ then (15.2) applies at $w_{i}$, which contradicts (17.2). ∎

######  19Proof of Theorem 1 for $d\leq 3$.

Let $T^{r}\to T$ be the minimal resolution. By (18) it dominates the blow-up
$B_{s_{i}}S_{i}\to S_{i}$. Let $\bar{N}_{i}\subset B_{s_{i}}S_{i}$ denote the
birational transform of $N_{i}$. Then the self intersection of $\bar{N}_{i}$
is $d-4$, and this can only decrease with further blow-ups. For $T^{r}\to
S_{3-i}$ there must be at least one exceptional $(-1)$ curve, and the only
candidate is the birational transform of $N_{i}$. So $d\geq 3$ and if $d=3$
then $B_{s_{1}}S_{1}=T=B_{s_{2}}S_{2}$. ∎

######  20Proof of Theorem 1 in the unsplit case.

Assume that $(S_{1},N_{1})$ has an unsplit node.

As we noted in (19), $T^{r}\to S_{1}$ dominates the blow-up $B_{s_{1}}S_{1}\to
S_{1}$. The boundary of $B_{s_{1}}S_{1}$ is $F_{1}+\bar{N}_{1}$, where $F_{1}$
is the exceptional curve. $F_{1}+\bar{N}_{1}$ has 2 singular points, and these
are conjugate if the node is unsplit. Thus the image of $N^{\prime}_{3-i}$
must be $F_{1}$. Thus again we conclude that $B_{s_{1}}S_{1}=T$ and $d=3$. ∎

##  4\. Geiser involutions

Geiser involutions are constructed using a natural automorphism of degree 2
Del Pezzo surfaces. We start by recalling their properties; see [Dol12,
Chap.8] for details. We assume for simplicity that the characteristic is $\neq
2$.

######  21Degree 2 Del Pezzo surfaces.

Let $T$ be a degree 2 Del Pezzo surface. The linear system $|-K_{T}|$ gives a
morphism $p:T\to{\mathbb{P}}^{2}$, which is a double cover, branched along a
quartic curve $R\subset{\mathbb{P}}^{2}$. We let $\tau$ or $\tau_{T}$ denote
the covering involution.

The anticanonical class is the pull-back of the line class $L$ on
${\mathbb{P}}^{2}$.

$T$ is smooth iff $R$ is smooth. The singularities of $T$ correspond to the
singular points of $R$, and they have the same name in the ADE classification.

A curve $E\subset T$ is called a line if $(-K_{T}\cdot E)=1$. Since
$-K_{T}=p^{*}L$, we see that $E$ is a line iff $p(E)\subset{\mathbb{P}}^{2}$
is a line and $p^{-1}(p(E))=E+E^{\prime}$ is a sum of 2 lines. This happens
over the bitangents of $R$. If $R$ is smooth, there are 28 bitangents, giving
56 lines on $T$. (If $p(E)\subset R$ then $p^{-1}(p(E))=2E$.)

If $E,E^{\prime}$ are 2 lines and $p(E)\neq p(E^{\prime})$, then $E\cap
E^{\prime}$ is at most 1 point. Thus, if $E\cap E^{\prime}$ consists of 2
points, then $p(E)=p(E^{\prime})$, hence $\tau(E)=E^{\prime}$.

Let $D\subset T$ be a curve. Then $D+\tau_{T}(D)=p^{-1}(p(D))$, hence

$None$ $D+\tau_{T}(D)\in|-rK_{T}|\quad\mbox{where}\quad r=(-K_{T}\cdot D).$

Let $T$ be a smooth, weak Del Pezzo surface of degree 2 and $m_{T}:T\to T^{*}$
the corresponding Del Pezzo surface of degree 2. Then the covering involution
$\tau^{*}$ on $T^{*}$ lifts to an involution $\tau$ on $T$.

21.2 Note on characteristic 2. In the cases that we consider later,
$T\to{\mathbb{P}}^{2}$ is always separable, so the above descriptions work.

######  22Geiser involutions.

These were discovered in [Gei1867]; see [Dol12, Sec.8.7] for a modern
treatment.

Start with $S={\mathbb{P}}^{2}$ and blow up $7$ points
$P=\\{p_{1},\dots,p_{7}\\}$ to get $\pi_{P}:T_{P}\to S$. If no 3 points are on
a line and no 6 on a conic, then $T_{P}$ is a smooth Del Pezzo surface of
degree 2. Thus $T_{P}$ is a double cover of (another copy of)
${\mathbb{P}}^{2}$, branched along a smooth quartic curve. Let
$\tau_{P}:T_{P}\to T_{P}$ be the covering involution of
$T_{P}\to{\mathbb{P}}^{2}$. The classical Geiser involutions are given as

$\sigma_{P}:=\pi_{P}\circ\tau_{P}\circ\pi_{P}^{-1}:{\mathbb{P}}^{2}\dasharrow{\mathbb{P}}^{2}.$

More generally, let $S$ be a Del Pezzo surface and $\pi_{T}:T\to S$ a
birational morphism from a smooth, degree 2 Del Pezzo surface to $S$. Let
$\tau_{T}:T\to T$ be the covering involution. As above, we get

$\sigma_{T}:=\pi_{T}\circ\tau_{T}\circ\pi_{T}^{-1}:S\dasharrow S.$

I also call these Geiser involutions. (Again, the terminology is not uniform.
If $S={\mathbb{P}}^{2}$ but $T\neq T^{*}$, then [BB00] considers $\sigma_{T}$
to be a de Jonquières involution. For the current purposes, the representation
$\sigma_{T}=\pi_{T}\circ\tau_{T}\circ\pi_{T}^{-1}$ shows the underlying
geometry much more clearly.)

Next we look at a special case where the blow-up centers of $\pi_{T}$ are in a
rather degenerate position.

######  23Geiser involutions from split nodes.

Assume now that $S$ is a Del Pezzo surface of degree $r+2\geq 3$, and
$N\in|-K_{S}|$ a geometrically irreducible curve with a split node at a smooth
point $s\in S$.

Pick a local branch $s\in B_{+}\subset N$ and let $\pi_{+}:T_{+}\to S$ denote
the blow-up of $s\in B_{+}$ successively $r$ times. We check below that
$T_{+}$ is a weak Del Pezzo surface of degree 2. Thus we get a Geiser
involution $\sigma_{+}:S\dasharrow S$.

Starting with the other branch $B_{-}\subset N$, the first blow-up is the
same, but the next $r-1$ blow-ups are different. For $r\geq 2$ this gives
another Geiser involution $\sigma_{-}:S\dasharrow S$. (If $r=1$ then we blow
up $s$ only once, hence the 2 involutions coincide.)

As we noted after (4.1), the branches $B_{\pm}$ are indistinguishable from
each other, so one should think of $\\{\sigma_{+},\sigma_{-}\\}$ as a pair of
involutions.

For future computations, let us give a detailed description of
$\pi_{\pm}:T_{\pm}\to S$ and the action of the covering involution $\tau$.

More generally, let $S$ be a surface and $N\subset S$ a curve with a split
node $s\in N$. Assume that $S$ is smooth at $s$. We represent this as
$B{\mbox{ --- }}B$ where the $B$ are the local branches at the node. In
suitable local analytic coordinates we can identify the branches with the
coordinate axes. This is very covenient for explicit computations.

All diagrams below are arranged so that we blow-up points on the branch $B$
that is on the right hand side.

For example, 3 blow-ups give the following sequence of dual graphs, where we
denote an exceptional curve by $E$, and the subscript tells the number of
blow-ups needed for that curve to appear.

$B{\mbox{ --- }}B\qquad B{\mbox{ --- }}E_{1}{\mbox{ --- }}B\qquad B{\mbox{ ---
}}E_{1}{\mbox{ --- }}E_{2}{\mbox{ --- }}B\qquad B{\mbox{ --- }}E_{1}{\mbox{
--- }}E_{2}{\mbox{ --- }}E_{3}{\mbox{ --- }}B$

After $r$ blow-ups a more symmetric version is

$None$ $\begin{array}[]{ccccc}E_{1}&-&E_{2}\ -\cdots-\ E_{r-2}&-&E_{r-1}\\\
|&&&&|\\\ N_{T}&&\rule[1.0pt]{40.0pt}{0.4pt}&&{\ }E_{r}\end{array}$

where $N_{T}$ denotes the birational transform of $N$.

Since we repeatedly blow up nodes, the sum of all curves in (23.1) is a member
of $|-K_{T}|$. Thus $T$ is a weak Del Pezzo surface of degree 2. Note that
$(-K_{T}\cdot E_{i})=0$ for $1\leq i<r$, $(-K_{T}\cdot E_{r})=1$ and
$(-K_{T}\cdot N_{T})=1$.

Let $m_{T}:T\to T^{*}$ be the birational morphism to a degree 2 Del Pezzo
surface $T^{*}$ as in (21). Then

  1. (23.2)

$m_{T}$ contracts the curves $E_{1},\dots E_{r-1}$,

  2. (23.3)

$m_{T}(N_{T})$ and $m_{T}(E_{r})$ are lines that meet at 2 points, and so

  3. (23.4)

$\tau^{*}$ interchanges $m_{T}(N_{T})$ and $m_{T}(E_{r})$.

  4. (23.5)

Therefore $\tau(N_{T})=E_{r},\tau(E_{r})=N_{T}$, and $\tau(E_{i})=E_{r-i}$ for
$i=1,\dots,r-1$.

23.6 Note on characteristic 2. Since $\tau^{*}$ interchanges $m_{T}(N_{T})$
and $m_{T}(E_{r})$, the anticanonical map $T\to{\mathbb{P}}^{2}$ is separable.

######  Example 24.

If $S={\mathbb{P}}^{2}$ then $r=7$, hence $m_{T}:T\to T^{*}$ contracts the
curves $E_{1}-\cdots-E_{6}$. Thus $T^{*}$ has an $A_{6}$ singularity. The
branch curve of $T^{*}\to{\mathbb{P}}^{2}$ is then a plane quartic with an
$A_{6}$ singularity. There is a unique such curve up to isomorphism, which can
be given by an affine equation $(y-x^{2})^{2}-xy^{3}=0$. The corresponding
affine equation of $T^{*}$ is $u^{2}=(y-x^{2})^{2}-xy^{3}$.

There are 4 lines on $T^{*}$ that pass through the singular point. $L_{\pm}$
lie over $(x=0)$ and $M_{\pm}$ lie over $(y=0)$. Note that $M_{+}$ and $M_{-}$
intersect only over the origin $(x=y=0)$, while $L_{+}$ and $L_{-}$ intersect
over $(x=y=0)$ and at infinity. Thus the $L_{\pm}$ are the images of
$N_{T}+E_{7}$.

Note also that, if $k$ contains a 3rd root of unity $\epsilon\neq 1$, then $T$
and $T^{*}$ have an order 3 automorphism $(x,y,u)\mapsto(\epsilon
x,\epsilon^{2}y,\epsilon^{2}u).$

######  25Geiser involutions and Yoshihara twists.

A version of the Geiser involution (23) for ${\mathbb{P}}^{2}$ was considered
in [Yos85], rediscovered in [Ore02] and then generalized to other Del Pezzo
surfaces in [MS24].

The constructions in these papers start with the same $r$ blow-ups as in (23),
and then contract the curves $N_{T},E_{1},\dots,E_{r-1}$, starting with
$N_{T}$.

The end result is then a birational map
$\phi:(S,N)\dasharrow(S^{\prime},N^{\prime})$, where $S^{\prime}$ is a Del
Pezzo surface and $N^{\prime}$ is the image of $E_{r}$. It is clear that $\deg
S=\deg S^{\prime}$ and that $N^{\prime}$ is also a nodal cubic.

For $S={\mathbb{P}}^{2}$ there is a unique pair $({\mathbb{P}}^{2},N)$ up to
isomorphism over ${\mathbb{C}}$, hence then
$({\mathbb{P}}^{2},N^{\prime})\cong({\mathbb{P}}^{2},N)$. There are 6 distinct
isomorphisms $\rho:({\mathbb{P}}^{2},N^{\prime})\cong({\mathbb{P}}^{2},N)$.
The maps in [Yos85, Ore02] are the Geiser involutions composed with an
automorphism of $({\mathbb{P}}^{2},N)$ that switches the 2 branches at the
node, giving an infinite order automorphism of ${\mathbb{P}}^{2}\setminus N$.

For $d\in\\{5,6,7\\}$ there is a unique Del Pezzo surface of degree $d$ over
${\mathbb{C}}$, up to isomorphism. Thus, in these cases, $S^{\prime}\cong S$,
and it is easy to see that the same holds for $d=8$. However, for
$d\in\\{5,6,7,8\\}$ the pairs $(S,N)$ have moduli (14), and it is not clear
from the descriptions in [Yos85, Ore02, MS24] that
$(S,N)\cong(S^{\prime},N^{\prime})$.

The main advantage of working with Geiser involutions is that we get
automorphisms of $S\setminus N$, and the construction works over arbitrary
fields.

Note that [MS24] studies deformation invariant properties of the pairs
$(S,N)$, so knowing that $(S,N)\cong(S^{\prime},N^{\prime})$ is not important
for their purposes.

######  26Automorphisms of log K3 surfaces.

The automorphism group of $(S,N)$ acts on $N$, and
$\operatorname{Aut}(N)\cong{\mathbb{G}}_{m}\rtimes{\mathbb{Z}}/2$. The action
also fixes ${\mathcal{O}}_{S}(K_{S})|_{N}\in\operatorname{Pic}^{d}(N)$ where
$d=\deg S$. Note that
$\lambda\in{\mathbb{G}}_{m}=\operatorname{Aut}^{\circ}(N)$ acts on
$\operatorname{Pic}^{d}(N)$ by multiplication by $\lambda^{d}$. Thus, for
$d\neq 0$, the subgroup of $\operatorname{Aut}(N)$ that leaves
${\mathcal{O}}_{S}(K_{S})|_{N}\in\operatorname{Pic}^{d}(N)$ invariant is the
dihedral group $D_{2d}\cong{\mathbb{Z}}/d\rtimes{\mathbb{Z}}/2$. Thus we get a
homomorphism $\operatorname{Aut}(S,N)\to D_{2d}$. If $d\geq 1$ then the kernel
is cyclic.

If $k$ contains a 3rd root of unity $\epsilon\neq 1$, then
$\operatorname{Aut}\bigl{(}{\mathbb{P}}^{2},(xyz=x^{3}+y^{3})\bigr{)}\cong
S_{3}$, generated by the involution $\rho_{2}:(x:y:z)\mapsto(y:x:z)$ and the
order 3 element $\rho_{3}:(x:y:z)\mapsto(\epsilon x:\epsilon^{2}y:z)$. As
another example, if $\operatorname{char}k\neq 2$ then
$\operatorname{Aut}\bigl{(}{\mathbb{P}}^{1}\times{\mathbb{P}}^{1},(x_{0}^{2}y_{1}^{2}+x_{1}^{2}y_{0}^{2}=x_{0}^{2}y_{0}^{2})\bigr{)}$
is the dihedral group $D_{8}$. For $d\leq 8$ the general pair $(S,N)$ has no
automorphisms.

Let $T_{\pm}\to S$ be the blow-ups involved in the construction of the Geiser
involutions. If $\rho\in\operatorname{Aut}(S,N)$ interchanges the branches
$B_{\pm}$, then $\rho$ lifts to an isomorphism $\bar{\rho}:T_{+}\cong T_{-}$.
Otherwise $\rho$ lifts to an isomorphism $\bar{\rho}:T_{+}\cong T_{+}$.

Therefore, if $\rho\in\operatorname{Aut}(S,N)$ interchanges the branches
$B_{\pm}$, then $\rho^{-1}\sigma_{\pm}\rho=\sigma_{\mp}$, and
$\rho^{-1}\sigma_{\pm}\rho=\sigma_{\pm}$ otherwise. This gives the following.

######  Corollary 27.

Let $S$ be a Del Pezzo surface of degree $\geq 4$, and $N\in|-K_{S}|$ with a
split node at a smooth point $s\in S$. Then the subgroup of
$\operatorname{Aut}(S\setminus N)$ generated by the $\sigma_{\pm}$ and
$\operatorname{Aut}(S,N)$ is a semidirect product
$\langle\sigma_{+},\sigma_{-}\rangle\rtimes\operatorname{Aut}(S,N)$. ∎

Note that $\operatorname{Aut}(S,N)$ can be infinite if $(N^{2})\leq 0$; see
[Li22] for a description of these groups and applications to the Morrison cone
conjecture for log K3 surfaces.

##  5\. Cusps

######  Definition 28 (Cusp).

Let $C$ be a geometrically reduced curve over a field $k$ with normalization
$\pi:\bar{C}\to C$. For a point $c\in C(k)$ set
$\bar{c}:=\operatorname{red}\pi^{-1}(c)$. We say that $C$ has a cusp or is
unibranch at $c\in C$ if $k(\bar{c})=k(c)=k$.

Let $S$ be a smooth surface germ and fix local (analytic or formal)
coordinates $(x,y)$ at $s\in S$. Then a cusp $s\in C\subset S$ can be
parametrized as

$None$ $t\mapsto\bigl{(}x(t):=t^{p}+(\mbox{higher
terms}),y(t):=t^{q}+(\mbox{higher terms})\bigr{)},$

(after scaling $x,y$ if necessary). The values $p,q$ are the contact orders of
$C$ with the axes: $(C\cdot(\mbox{$x$-axis}))_{s}=q$ and
$(C\cdot(\mbox{$y$-axis}))_{s}=p$.

The multiplicity of $C$ is $\min\\{p,q\\}$, independent of the choice of the
coordinates. If $q\nmid p$ and $p\nmid q$ then both $p,q$ are invariants of
$C$; we then call $C$ a $(p,q)$-cusp.

Note that if $p=rq$ then the coordinate change $x^{\prime}:=x-y^{r}$ gives
$\operatorname{mult}x^{\prime}(t)>\operatorname{mult}x(t)$. Newton proved that
if $C$ is singular, then after finitely many steps we reach a

$t\mapsto\bigl{(}x(t):=t^{p^{\prime}}+(\mbox{higher
terms}),y(t):=t^{q}+(\mbox{higher terms})\bigr{)},$

such that $q\nmid p^{\prime}$; see [BK86, Sec.8.3] or [Kol07, Sec.1.1]. Then
$C$ is a $(p^{\prime},q)$-cusp.

After blowing up the origin, the parametrization (28.1) changes to

$None$
$\begin{array}[]{lcll}(x/y,y)&=&\bigl{(}t^{p-q}+(\mbox{higher}),t^{q}+(\mbox{higher})\bigr{)}&\mbox{if
$p>q$, and}\\\\[4.30554pt]
(x,y/x)&=&\bigl{(}t^{p}+(\mbox{higher}),t^{q-p}+(\mbox{higher})\bigr{)}&\mbox{if
$p<q$.}\end{array}$

######  29Blowing up cusps.

Let $s\in N\subset S$ be a nodal curve; call one branch $B_{+}$ and the other
$B_{-}$. Using these branches as local coordinate axes, for a cusp $s\in
C\subset S$ the key invariants are the contact orders with $N$. We write these
as $p:=(B_{-}\cdot C)_{s}$ and $q:=(B_{+}\cdot C)_{s}$, and represent this
with the dual graph $B_{-}\stackrel{{\scriptstyle p,q}}{{{\mbox{ ---
}}}}B_{+}$.

After blowing up $s$, the dual graphs change as

$None$ $\circ\stackrel{{\scriptstyle p,q}}{{{\mbox{ ---
}}}}\circ\qquad\rightsquigarrow\qquad\begin{array}[]{ll}\circ\stackrel{{\scriptstyle
p-q,q}}{{{\mbox{ --- }}}}E_{1}{\mbox{ --- }}\circ&\mbox{if $p\geq q$,
and}\\\\[4.30554pt] \circ{\mbox{ --- }}E_{1}\stackrel{{\scriptstyle
p,q-p}}{{{\mbox{ --- }}}}\circ&\mbox{if $p\leq q$.}\end{array}$

Note that if $p=q$ then the 2 versions mean the same thing: the transform of
$C$ meets $E_{1}$ away from the transform of $N$.

By induction, after $r$ blow-ups of $s\in B_{+}$ we get the following dual
graphs.

$\begin{array}[]{ll}(\ref{a.catb}.2)\quad B\stackrel{{\scriptstyle
p-q,q}}{{{\mbox{ --- }}}}E_{1}{-}\cdots{-}E_{r}{\mbox{ --- }}B&\mbox{if $q\leq
p$,}\\\\[4.30554pt] (\ref{a.catb}.3)\quad
B{-}\cdots{-}E_{i}\stackrel{{\scriptstyle(i+1)p-q,q-ip}}{{{\mbox{ ---
}}}}E_{i+1}{-}\cdots{-}B&\mbox{if $p\leq ip\leq q<(i+1)p\leq
rp$,}\\\\[4.30554pt] (\ref{a.catb}.4)\quad B{\mbox{ ---
}}E_{1}{-}\cdots{-}E_{r}\stackrel{{\scriptstyle p,q-rp}}{{{\mbox{ ---
}}}}B&\mbox{if $rp\leq q$.}\end{array}$

##  6\. Geiser involutions and cusps

######  Proposition 30.

Let $S$ be a Del Pezzo surface of degree $r+2\geq 4$. Let $N\in|-K_{S}|$ be a
geometrically irreducible curve with a split node $s\in N$ such that $S$ is
smooth at $s$, and $\sigma_{\pm}$ the corresponding Geiser involutions (23).
Let $s\in C^{\circ}\subset S$ be a (germ of a) cusp represented with the dual
graph $B\stackrel{{\scriptstyle p,q}}{{{\mbox{ --- }}}}B$. Then
$\sigma_{+}(C^{\circ})$ is represented with the dual graph

$None$ $\begin{array}[]{ll}B\stackrel{{\scriptstyle p,rp-q}}{{{\mbox{ ---
}}}}B&\mbox{if $q\leq rp$, and}\\\\[4.30554pt] B\stackrel{{\scriptstyle
q-rp,p+r(q-rp)}}{{{\mbox{ --- }}}}B&\mbox{if $q\geq rp$.}\end{array}$

Note that if $q=rp$ then both forms mean that intersection point is different
from $s$. Since $N\setminus\\{s\\}$ is connected, it does not make sense to
ask on which branch of $N$ the point is.

Proof. After $r$ blow-ups we have the dual graphs (29.2–4). The action of
$\tau_{B}$ is described in (23.2–5). Thus the cases (29.2–4) are transformed
into the following.

$\begin{array}[]{ll}(\ref{a.catbt}.2)\quad
B{-}\cdots{-}E_{r-1}\stackrel{{\scriptstyle q,p-q}}{{\mbox{ ---
}}}E_{r}{\mbox{ --- }}B&\mbox{if $q<p$,}\\\\[4.30554pt] (\ref{a.catbt}.3)\quad
B{-}\cdots{-}E_{r-i-1}\stackrel{{\scriptstyle q-ip,(i+1)p-q}}{{{\mbox{ ---
}}}}E_{r-i}{-}\cdots{-}B&\mbox{if $p\leq ip<q<(i+1)p\leq rp$,}\\\\[4.30554pt]
(\ref{a.catbt}.4)\quad B{\mbox{ ---
}}E_{1}{-}\cdots{-}E_{r}\stackrel{{\scriptstyle q-rp,p}}{{{\mbox{ ---
}}}}B&\mbox{if $rp<q$.}\end{array}$

Next we contract the curves $E_{r},\dots,E_{1}$. In cases (30.2–3) the first
contractions that involve the curve $C$ are

$\begin{array}[]{lcll}E_{r-1}\stackrel{{\scriptstyle q,p-q}}{{\mbox{ ---
}}}E_{r}&\rightsquigarrow&E_{r-1}\stackrel{{\scriptstyle p,p-q}}{{\mbox{ ---
}}}B&\mbox{and}\\\\[4.30554pt] E_{r-i-1}\stackrel{{\scriptstyle
q-ip,(i+1)p-q}}{{{\mbox{ ---
}}}}E_{r-i}&\rightsquigarrow&E_{r-i-1}\stackrel{{\scriptstyle
p,(i+1)p-q}}{{{\mbox{ --- }}}}B.&\end{array}$

These correspond to the first case in (29.1). After that all contractions are
as in the second case in (29.1). Thus $p$ stays fixed and we add $(r-i-1)p$ to
the second component, to end with $(p,rp-q)$.

In case (29.1) we add $r(q-rp)$ to $p$. ∎

######  Corollary 31.

Let $S$ be a Del Pezzo surface of degree $r+2\geq 4$. Let $N\in|-K_{S}|$ be a
geometrically irreducible curve with a split node $s\in N$ such that $S$ is
smooth at $s$. Let $\sigma_{+},\sigma_{-}$ the corresponding Geiser
involutions.

Let $C^{\circ}\subset S$ be a germ of a cusp that has contact orders $(p,q)$
with the branches of $N$. Then $\sigma_{+}(C^{\circ})$ and
$\sigma_{-}(C^{\circ})$ have contact orders

  1. (31.1)

$(p,rp-q)$ and $(rq-p,q)$ if $\tfrac{1}{r}<\tfrac{p}{q}<r$,

  2. (31.2)

$\bigl{(}q-rp,p+r(q-rp)\bigr{)}$ and $(rq-p,q)$ if
$\tfrac{p}{q}\leq\tfrac{1}{r}$,

  3. (31.3)

$(p,rp-q)$ and $\bigl{(}q+r(p-rq),p-rq\bigr{)}$ if $r\leq\tfrac{p}{q}$. ∎

######  32Proof of Lemma 5.

Using the notation of Lemma 5, let $C\subset S$ be an irreducible curve with
only 1 point on $N$ having contact orders $(p,q)$ at $s$. Then
$\operatorname{mult}(C)=\min\\{p,q\\}$, and
$\operatorname{ndeg}(C)=\tfrac{1}{d}(-K_{S}\cdot C)$. The rest follows by
rewriting the formulas (31.1–3) in terms of $\operatorname{mult}(C)$ and
$\operatorname{ndeg}(C)$. ∎

##  7\. Upper bound for the multiplicity

We check that the upper bound
$\operatorname{mult}(C)<2\cdot\operatorname{ndeg}(C)$ needed in (6.3) holds
for almost all curves.

######  33Max Noether’s genus formula.

(See [BK86, Sec.9.2] or [Kol07, p.38].) The genus of a planar curve
singularity is $\delta(0\in C):=\sum\frac{m_{i}(m_{i}-1)}{2}$, where $m_{i}$
are the multiplicities of all points that appear after some blow-ups (called
infinitely near points). If $C$ is an irreducible, projective curve on a
smooth surface $S$ with normalization $\bar{C}$, then Noether’s formula is

$None$
$g(\bar{C})=\tfrac{1}{2}\bigl{(}C\cdot(C+K_{S})\bigr{)}+1-\textstyle{\sum}_{c\in
C}\delta(c\in C).$

Let $S$ be a smooth, weak Del Pezzo surface of degree $d$ and $C\subset S$ an
irreducible curve. Write $C\sim_{{\mathbb{Q}}}-mK_{S}+F$ where $(K_{S}\cdot
F)=0$. Then

$\bigl{(}C\cdot(C+K_{S})\bigr{)}=dm(m-1)+(F^{2}).$

Thus we get that

$None$ $2\cdot\textstyle{\sum}_{c\in C}\delta(c\in
C)=dm(m-1)+2-2g(\bar{C})+(F^{2}).$

Using (28.2) and induction we see that

$None$ $2\cdot\delta\bigl{(}\mbox{$(p,q)$-cusp}\bigr{)}\geq(p-1)(q-1),$

and equality holds iff $(p,q)=1$. Putting these together gives the following.

Claim 33.4. Let $S$ be a smooth, weak Del Pezzo surface of degree $d$ and
$C\subset S$ an irreducible curve such that $C\sim_{{\mathbb{Q}}}-mK_{S}+F$
where $(K_{S}\cdot F)=0$. If $C$ has a $(p,q)$-cusp then

$(p-1)(q-1)\leq dm(m-1)+2,$

with equality holding iff $(p,q)=1$, $C$ is rational with at most 1
singularity, and $C\sim_{{\mathbb{Q}}}-mK_{S}$ for some $m$. ∎

######  Corollary 34.

Let $S$ be a smooth, weak Del Pezzo surface of degree $d\geq 4$, and $C\subset
S$ an irreducible curve with a cusp at $\\{s\\}=C\cap N$. Then one of the
following holds.

  1. (34.1)

$\operatorname{mult}_{s}C<\frac{2}{d}(-K_{S}\cdot C)$, or

  2. (34.2)

$d\in\\{4,5\\}$, and $C\in|-K_{S}|$ has a $(2,3)$ cusp at $s$.

Proof. If $C$ and $N$ have contact orders $(p,q)$ at $s$, then $p+q=dm$ where
$m=\frac{1}{d}(-K_{S}\cdot C)$, and $(p-1)(q-1)\leq dm(m-1)+2$ by (33.4). Thus

$None$ $(dm-q-1)(q-1)\leq dm(m-1)+2.$

The left hand side reaches its maximum at $q_{0}=\frac{1}{2}(dm+2)$, and is
monotone decreasing as a function of $|q-q_{0}|$. So it is enough to check
what happens at $q=2m$. Then

$(dm-2m-1)(2m-1)=2dm^{2}-4m^{2}-dm+1=dm(m-1)+1+(d-4)m^{2}.$

This is bigger than $dm(m-1)+2$, except when $d=5$ and $m=1$, or when $d=4$.

If $d=4$ then $p+q=4m$ gives (34.1) except when $p=q=2m$. Then we have a
$(2m,2m+1)$ cusp or worse. (33.4) then becomes $(2m-1)2m\leq 4m(m-1)+2$ which
implies that $m=1$. ∎

##  8\. Noether-Fano method

Here we study curves with a low multiplicity cusp at the node of $N$, closely
following [KSC04, Sec.5.1].

######  Theorem 35.

Let $S_{i}$ be Del Pezzo surfaces, $s_{i}\in N_{i}\in|-K_{S_{i}}|$
geometrically irreducible, nodal curves and $\phi^{\circ}:(S_{1}\setminus
N_{1})\cong(S_{2}\setminus N_{2})$ an isomorphism. Let $N_{1}\neq C_{1}\subset
S_{1}$ be an irreducible curve and set $C_{2}:=\phi^{\circ}_{*}(C_{1})$.
Assume that

$None$ $\operatorname{mult}_{s_{i}}C_{i}\leq\tfrac{1}{\deg
S_{i}}(-K_{S_{i}}\cdot C_{i})\quad\mbox{for $i=1,2$.}\quad$

Then

  1. (35.2)

$(K_{S_{1}}\cdot C_{1})=(K_{S_{2}}\cdot C_{2})$, and

  2. (35.3)

if one of the inequalities in (35.1) is strict, then $\phi^{\circ}$ extends to
an isomorphism $\phi:S_{1}\cong S_{2}$.

Proof. Set $m_{i}:=\frac{1}{\deg S_{i}}(-K_{S_{i}}\cdot C_{i})$. We may assume
that $m_{1}\leq m_{2}$.

Using the notation in (17), let $p_{i}:T\to S_{i}$ be the projections and
$C_{T}$ the birational transform of the $C_{i}$. Write
$C_{i}\sim_{{\mathbb{Q}}}-m_{i}K_{S_{i}}+F_{i}$.

$K_{T}\sim_{{\mathbb{Q}}}p_{i}^{*}K_{S_{i}}+E_{i}$ and
$C_{T}=p_{i}^{*}C_{i}-D_{i}$. As in (15.1) for any $c$ we have

$K_{T}+cC_{T}\sim_{{\mathbb{Q}}}p_{i}^{*}\bigl{(}K_{S_{i}}+c(C_{i}-F_{i})\bigr{)}+E_{i}-cD_{i}+cp_{i}^{*}F_{i}.$

Now choose $c=\frac{1}{m_{1}}$. Then

$K_{S_{1}}+c(C_{1}-F_{1})\sim_{{\mathbb{Q}}}0\quad\mbox{and}\quad
K_{S_{2}}+c(C_{2}-F_{2})\sim_{{\mathbb{Q}}}\tfrac{m_{1}-m_{2}}{m_{1}}K_{S_{2}}.$

Hence

$p_{2}^{*}\bigl{(}\tfrac{m_{1}-m_{2}}{m_{1}}K_{S_{2}}\bigr{)}+E_{2}-cD_{2}+cp_{2}^{*}F_{2}\sim_{{\mathbb{Q}}}E_{1}-cD_{1}+cp_{1}^{*}F_{1}.$

Take intersection number with $-p_{2}^{*}K_{S_{2}}$. We get that

$\tfrac{m_{1}-m_{2}}{m_{1}}\deg
S_{2}=\bigl{(}-K_{S_{2}}\cdot(p_{2})_{*}(E_{1}-cD_{1})\bigr{)}.$

Assumption (35.1) and (15.2) imply that $E_{1}-cD_{1}$ is effective. Therefore
$m_{1}\geq m_{2}$, hence $m_{2}=m_{1}$.

If strict inequality holds in (35.1) then $(p_{2})_{*}(E_{1}-cD_{1})$ is a
positive multiple of $N_{2}$ by (15.3), proving (35.3). ∎

######  Theorem 36.

Let $S_{i}$ be Del Pezzo surfaces, $s_{i}\in N_{i}\in|-K_{S_{i}}|$
geometrically irreducible, nodal curves and $\phi^{\circ}:(S_{1}\setminus
N_{1})\cong(S_{2}\setminus N_{2})$ an isomorphism. Assume that $N_{2}$ has a
split node. Then there is a
$\psi\in\langle\sigma_{+},\sigma_{-}\rangle\subset\operatorname{Aut}(S_{2}\setminus
N_{2})$ such that $\psi\circ\phi^{\circ}$ extends to an isomorphism
$S_{1}\cong S_{2}$.

Proof. Let $C_{1}\in|-K_{S_{1}}|$ be the curve $E$ in (38), which is defined
over $\bar{k}$. Set $C_{2}:=\phi^{\circ}_{*}(C_{1})$.

Choose $\psi\in\langle\sigma_{+},\sigma_{-}\rangle$ such that
$\operatorname{ndeg}\bigl{(}\psi_{*}(C_{2})\bigr{)}$ is the smallest possible.
Since $E$ is not rational, we are in the first case of Corollary 7, hence

$None$
$\operatorname{mult}\bigl{(}\psi_{*}(C_{2})\bigr{)}\leq\operatorname{ndeg}\bigl{(}\psi_{*}(C_{2})\bigr{)}.$

Since $\sigma_{+},\sigma_{-}$ are defined over $k$, $\psi\circ\phi^{\circ}$ is
defined over $k$.

We claim that $\psi\circ\phi^{\circ}$ extends to an isomorphism $S_{1}\cong
S_{2}$. A map is an isomorphism over $k$ iff it is an isomorphism over
$\bar{k}$. Thus we can now work over $\bar{k}$ and apply (35) to
$\psi\circ\phi^{\circ}$. Note that $s_{1}\notin C_{1}$, so (35.1) is a strict
inequality for $i=1$. Next (35.1) for $i=2$ is equivalent to (36.1).

Thus $\psi\circ\phi^{\circ}$ extends to an isomorphism $S_{1}\cong S_{2}$. ∎

######  Corollary 37.

Let $S$ be a Del Pezzo surface of degre $\geq 4$, and $N\in|-K_{S}|$ a
geometrically irreducible, nodal curve with a split node at a smooth point
$s\in S$. Then the $\sigma_{\pm}$ and $\operatorname{Aut}(S,N)$ generate
$\operatorname{Aut}(S\setminus N)$. ∎

######  Lemma 38.

Let $(S,N)$ be a log K3 surface with geometrically irreducible boundary. Then,
over $\bar{k}$, there is an irreducible curve $E\in|-K_{S}|$ such that $N\cap
E=\\{b\\}$ is a single point and $E$ is smooth at $b$.

Proof. There is a point $b\in N(\bar{k})$ such that
${\mathcal{O}}_{N}(d[b])\cong{\mathcal{O}}_{S}(-K_{S})|_{N}$. (If
$\operatorname{char}k\nmid d$, then there are $d^{2}$ such points if $N$ is
smooth and $d$ such points if $N$ is nodal.) From the sequence

$0\to{\mathcal{O}}_{S}\to{\mathcal{O}}_{S}(-K_{S})\to{\mathcal{O}}_{S}(-K_{S})|_{N}\to
0$

and $H^{1}(S,{\mathcal{O}}_{S})=0$ we get a pencil of curves
$|E|\subset|-K_{S}|$ meeting $N$ only at $b$. Since $N\in|E|$ and $N$ is
smooth at $b$, general members of $|E|$ are smooth at $b$. ∎

Remark 38.1. Note that $E$ is automatically smooth in characteristics $\neq
2,3$, but a more careful inspection shows that $E$ is actually smooth even in
characteristics $2,3$.

##  9\. Affine lines in $({\mathbb{P}}^{2},N)$

We aim to describe all embeddings
${\mathbb{A}}^{1}\lhook\joinrel\to({\mathbb{P}}^{2}\setminus N)$. The closure
of the image is then a cuspidal curve $C\subset{\mathbb{P}}^{2}$ that meets
$N$ at a single point.

In this section we work over a field of characteristic 0. The main reason for
this is the use of [MS89] in (40), but there are also other problems in
characteristics 2 and 3.

######  Theorem 39.

Let $N\subset{\mathbb{P}}^{2}$ be a geometrically irreducible cubic with a
split node, and $C\subset{\mathbb{P}}^{2}$ a cuspidal, rational curve that
meets $N$ at a single point $c$. Assume that all cusps of $C\setminus\\{c\\}$
have multiplicity $\leq 2$.

Then there is a $\phi\in\langle\sigma_{+},\sigma_{-}\rangle$ such that
$\phi_{*}(C)\subset{\mathbb{P}}^{2}$ is a line or a conic.

Lines and conics meeting $N$ at a single point are listed in (44).

The theorem implies that $C\setminus\\{c\\}$ is actually smooth. It is quite
possible that the extra assumption (multiplicity $\leq 2$) is not needed.

Proof. By Corollary 7 there is a $\phi\in\langle\sigma_{+},\sigma_{-}\rangle$
such that
$\operatorname{mult}\bigl{(}\phi(C)\bigr{)}\leq\operatorname{ndeg}\bigl{(}\phi(C)\bigr{)}$.
Note that $\operatorname{ndeg}$ is the normalized degree, so for the usual
degree this says that

$None$
$\operatorname{mult}_{s}\bigl{(}\phi(C)\bigr{)}\leq\tfrac{1}{3}\deg\bigl{(}\phi(C)\bigr{)}.$

If $\operatorname{mult}_{s}\bigl{(}\phi(C)\bigr{)}\geq 2$ then $s$ is a
maximal multiplicity point of $\phi(C)$, and then (39.1) contradicts (40).

If $s$ is a smooth point of $\phi(C)$ then $\deg C\leq 2$ by (41).

We are left with the case when $s\notin\phi(C)$. Set
$c^{\prime}:=N\cap\phi(C)$. If $c^{\prime}$ is a smooth point of $\phi(C)$
then again $\deg C\leq 2$ by (41). The last case is when $c^{\prime}$ is a
smooth point of $N$, but a singular point of $C^{\prime}$. Then $c^{\prime}$
is a maximal multiplicity point of $C^{\prime}$, which is impossible by (43).
∎

######  Theorem 40.

[MS89] Let $C\subset{\mathbb{P}}^{2}$ be a rational curve, all of whose
singularities are cusps. Then there is a point $c\in C$ such that
$\operatorname{mult}_{c}C>\frac{1}{3}\deg C$. ∎

[Ore02] proves a form where $\frac{1}{3}$ is asymptotically replaced by
$\frac{2}{3+\sqrt{5}}\approx\frac{1}{2.62}$.

######  Lemma 41.

Let $C\subset{\mathbb{P}}^{2}$ be a cuspidal, rational curve such that $C\cap
N=\\{s\\}$. If $s$ is a smooth point of $C$ then $\deg C\leq 2$.

Proof. The cases $\deg C\geq 4$ are excluded by (42). If $\deg C=3$, then $N$
lies in the pencil $|C,3L|$ by (42.2). A cuspidal cubic has a unique flex, so
this pencil is $\lambda(y^{2}z-x^{3})+\mu z^{3}=0$. It has only 1 member that
is singular at $s$, namely $(z^{3}=0)$. ∎

######  Lemma 42.

Let $C\subset{\mathbb{P}}^{2}$ be a cuspidal, rational curve of degree $d$,
and $B\subset{\mathbb{P}}^{2}$ a (possibly reducible) curve. Assume that
$c:=C\cap B$ is a single smooth point of $C$ with tangent line $L$. Then one
of the following holds.

  1. (42.1)

$\deg B>\deg C$,

  2. (42.2)

$\deg B=\deg C$ and $B\in|C,dL|$, or

  3. (42.3)

$B=eL$ for some $e<d$.

Proof. Set $e:=\deg B$ and let $H$ be the line class. By assumption $de[c]$
and $e[H|_{C}]$ are linearly equivalent on $C$. Since $C$ is cuspidal and
rational, $\operatorname{Pic}(C)$ is torsion free; see, for example, [Har77,
Exrc.II.6.9]. Thus $d[c]\sim[H|_{C}]$. That is, $L$ meets $C$ with
multiplicity $d$. If $e<d$ then
$H^{0}\bigl{(}{\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}}(B-C)\bigr{)}=0$,
thus $B$ is unique, so $B=eL$.

If $e=d$ then
$H^{0}\bigl{(}{\mathbb{P}}^{2},{\mathcal{O}}_{{\mathbb{P}}^{2}}(B-C)\bigr{)}=1$,
so $B$ lies in the pencil $|C,dL|$. ∎

######  Lemma 43.

Let $C^{\prime}\subset{\mathbb{P}}^{2}$ be a cuspidal, rational curve. Assume
that $c^{\prime}:=C^{\prime}\cap N$ is a single smooth point on $N$ and a
maximal multiplicity point of $C^{\prime}$. Then $C^{\prime}$ is a line or a
conic.

Proof. Set $r^{\prime}:=\deg C^{\prime}$ and let $C^{\prime}$ have a
$(p^{\prime},q^{\prime})$-cusp at $c^{\prime}$. Note that
$q^{\prime}>\frac{1}{3}r^{\prime}$ by (40). As we noted in (28),
$p^{\prime}\geq(C\cdot N)=3r^{\prime}$. Using (33.4) we get that

$r^{\prime
2}-\tfrac{7}{3}r^{\prime}+\tfrac{2}{3}=(3r^{\prime}-1)\bigl{(}\tfrac{1}{3}r^{\prime}-\tfrac{2}{3}\bigr{)}\leq(r^{\prime}-1)(r^{\prime}-2),$

thus $\tfrac{2}{3}r^{\prime}\leq\tfrac{4}{3}$. ∎

######  Example 44.

There are 4 types of smooth, rational plane curves $C$ such that $C\cap N$ is
a single point. Here $N:=(xyz=x^{3}+y^{3})$ with parametrization
$t\mapsto(t{:}t^{2}{:}1+t^{3})$.

(44.1) $C_{1}:=(x=0)$ and $(y=0)$, meeting $N$ at the node. These have contact
orders $(2,1)$ or $(1,2)$ with $N$. If $C$ is a conic with contact orders
$(5,1)$, then $\sigma_{-}(C)$ is a line with contact orders $(2,1)$.

These conics are $(x-y^{2}=0)$ and $(y-x^{2}=0)$

(44.2) Flex tangents. To find these we are looking for a line $ax+by+z=0$ such
that $at+bt^{2}+(1+t^{3})=(\mbox{cube})$. The unique real solution is with
$(1+t)^{3}$, giving $C_{2}:=(3x+3y+z=0)$, and $C_{2}\cap N=(1{:}{-}1{:}0)$.
Acting by $\rho_{3}$ gives 2 more.

(44.3) 5-tangent conics. We need to solve

$at^{2}+bt^{3}+ct^{4}+dt(1+t^{3})+et^{2}(1+t^{3})+(1+t^{3})^{2}=(\mbox{6th
power}),$

but we want to avoid the 3 cases that give the square of a flex tangent
equation. So the unique real solution is with $(1-t)^{6}$, giving
$C_{3}:=\bigl{(}Q(x,y,z)=0\bigr{)}$ where

$Q(x,y):=21x^{2}-22xy+21y^{2}-6xz-6yz+z^{2},$

and $C_{3}\cap N=(1{:}1{:}2)$. See [Ore02, p.670] for a different derivation
of this. Acting by $\rho_{3}$ gives 2 more.

In order to complete the proof of Corollary 8 it remains to show that the
curves $C_{1},C_{2},C_{3}$ are in different
$\operatorname{Aut}({\mathbb{P}}^{2}\setminus N)$-orbits.

######  45.

Let $N\neq C:=(g=0)\subset{\mathbb{P}}^{2}$ be an irreducible curve of degree
$d$. The group of non-constant units on ${\mathbb{P}}^{2}\setminus(N\cup C)$
is generated by $(xyz-x^{3}-y^{3})^{d}/g^{3}$ if $3\nmid d$, and by
$(xy-x^{3}-y^{3})^{d/3}/g$ if $3\mid d$. This gives a morphism

$m_{C}:{\mathbb{P}}^{2}\setminus(N\cup
C)\to{\mathbb{P}}^{1}\setminus\\{0,\infty\\}$

that is canonically associated to $N$ and $C$. We describe the fibers in the
cases (44.1–3).

(45.1) For $C_{1}$ the fibers are $\lambda(xyz-x^{3}-y^{3})+\mu x^{3}=0$;
these are rational curves.

(45.2) For $C_{2}$ the fibers are
$\lambda(xyz-x^{3}-y^{3})+\mu(3x+3y+z)^{3}=0$, these are genus 1 curves.

(45.3) For $C_{3}$ the fibers are $\lambda(xyz-x^{3}-y^{3})^{2}+\mu Q^{3}=0$.
At the base point $c$ we can choose local coordinates such that $N=(u=0)$ and
$Q=(v-u^{5}=0)$. Thus the fibers have a singularity with equation $\lambda
u^{2}+\mu(v-u^{5})^{3}=0$, which is analytically equivalent to
$u^{2}+v^{15}=0$. By (33.3) the genus of this singularity is $7$, thus the
geometric genus of the general fiber is $\binom{5}{2}-7=3$.

In particular, the pairs $\bigl{(}{\mathbb{P}}^{2}\setminus N,C_{i}\bigr{)}$
are not isomorphic to each other for different values of $i\in\\{1,2,3\\}$.

######  Example 46 (Cubics with contact order $(1,8)$).

For these we need to solve

$g_{3}(t,t^{2})+g_{2}(t,t^{2})(1+t^{3})+g_{1}(t,t^{2})(1+t^{3})^{2}+(1+t^{3})^{3}=t^{8},$

where $g_{i}$ is homogeneous of degree $i$. The solutions form a pencil

$None$ $\lambda(xyz-x^{3}-y^{3})+\mu(-xy^{2}-x^{2}z+yz^{2})=0,$

whose general member is smooth.

Remarkable properties of this pencil are studied in [MPS24].

######  Example 47 (Cubics with contact order $(1,7)$).

For these we need to solve

$g_{3}(t,t^{2})+g_{2}(t,t^{2})(1+t^{3})+g_{1}(t,t^{2})(1+t^{3})^{2}+(1+t^{3})^{3}=at^{7}+bt^{8},$

where $g_{i}$ is homogeneous of degree $i$. Besides the pencil (46.1), we also
have $y(yz-x^{2})$. The resulting linear system

$\lambda(xyz-x^{3}-y^{3})+\mu(-xy^{2}-x^{2}z+yz^{2})+\nu(-x^{2}y+y^{2}z)=0.$

gives the degree 2 map

${\mathbb{P}}^{2}=S\stackrel{{\scriptstyle\pi_{+}}}{{\leftarrow}}T_{+}\stackrel{{\scriptstyle
p}}{{\rightarrow}}{\mathbb{P}}^{2}$

in the definition (23) of the Geiser involutions for
$\bigl{(}{\mathbb{P}}^{2},(xyz=x^{3}+y^{3})\bigr{)}$.

######  Example 48.

Let $S$ be a smooth Del Pezzo surface and $C\subset S$ a smooth, rational
curve with $(C^{2})\geq 0$. Set $L:={\mathcal{O}}_{S}(C)$.

Let $N\in|-K_{S}|$ be a nodal curve. Consider the sequence

$0\to\omega_{S}(L)\to L\to L|_{N}\to 0.$

By the adjunction formula, $\deg L|_{N}=2+(C^{2})$. Thus there are $2+(C^{2})$
sections $s_{i}\in H^{0}(N,L|_{N})$ such that $(s_{i}=0)$ is a single smooth
point $c_{i}\in N$. Since $H^{1}\bigl{(}S,\omega_{S}(L)\bigr{)}=0$, the
$s_{i}$ lift back to sections of $L$. Thus we get divisors $C_{i}\in|C|$ such
that $C_{i}\cap N=\\{c_{i}\\}$.

######  Example 49.

Let $N\subset{\mathbb{P}}^{2}$ be a nodal cubic and $x\in N$ a smooth point.
Blow up $x$ to get $S:=B_{x}{\mathbb{P}}^{2}$ and $N\subset|-K_{S}|$. Let
$E\subset S$ be the $(-1)$-curve and $F\subset S$ a fiber of the projection
$\pi:S\to{\mathbb{P}}^{1}$ from $x$.

For any $r\geq 1$, the general members of $|E+rF|$ are smooth rational curves
of self-intersection $2r-1$. Thus by (48) there are $2r+1$ different points
$c_{i,r}\in N$ and divisors $C_{i,r}\in|E+rF|$ such that $C_{i,r}\cap
N=\\{c_{i,r}\\}$.

If $C_{i}$ is reducible, then it is of the form $F_{i,r}+C^{\prime}_{i,r}$
where $F_{i,r}$ is the fiber through $c_{i,r}$. Thus $2[c_{i,r}]\sim F|_{N}$.
Subtracting this from $(2r+1)[c_{i,r}]\sim(E+rF)|_{N}$ we get that
$[c_{i,r}]\sim[E|_{N}]=[E\cap N]$. So $E\cap N$ is a branch point of the
projection of $\pi|_{N}$. This means that $x\in N$ is a flex point. Then
$C_{i,r}=E+rF_{i,r}$ is reducible. Thus we conclude the following.

For each $r\geq 1$, the curves $C_{i,r}\in|E+rF|$ give

  1. (49.1)

$2r+1$ different $\operatorname{Aut}(S\setminus N)$-orbits of affine lines on
$S\setminus N$ if $x\in N$ is not a flex point, but only

  2. (49.2)

$2r$ different $\operatorname{Aut}(S\setminus N)$-orbits of affine lines if
$x\in N$ is a flex point.

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