let $\tilde{E}_{2}$, $\tilde{E}_{3+}$ and $\tilde{E}_{3-}$ denote the
exceptional curves lying over $P_{2},P_{3+}$ and $P_{3-}$ respectively. Then
for each $k\in\\{2,3+,3-\\}$, $\tilde{E}_{k}$ is an irreducible curve
isomorphic to $\mathbb{P}^{1}$, $\tilde{E}_{2}^{2}=-2$ and
$\tilde{E}_{3+}^{2}=\tilde{E}_{3-}^{2}=-3$. By applying Proposition 1.5.2 we
find

$K_{\tilde{S}}=\sigma^{*}K_{S}-\frac{1}{3}\tilde{E}_{3+}-\frac{1}{3}\tilde{E}_{3-}.$
(17)

Let $\tilde{\Delta}$ denote the proper transform of $\Delta$ on $\tilde{S}$.
Since $1=\operatorname{{\rm mult}}_{P_{2}}(\Delta)=\operatorname{{\rm
mult}}_{P_{3+}}(\Delta)=\operatorname{{\rm mult}}_{P_{3-}}(\Delta)$ we have
$\sigma^{*}\Delta=\tilde{\Delta}+\frac{1}{2}\tilde{E}_{2}+\frac{1}{3}\tilde{E}_{3+}+\frac{1}{3}\tilde{E}_{3-}$.
Since $2\Delta$ is an anti-canonical divisor of $S$, we obtain using (17) that

$2\tilde{\Delta}+\tilde{E}_{2}+\tilde{E}_{3+}+\tilde{E}_{3-}\text{ is an anti-
canonical divisor of $\tilde{S}$ and $K_{\tilde{S}}^{2}=0$. }$ (18)

The support of $\sigma^{*}(\Delta)$ is given by the following weighted graph

$\begin{array}[]{ccccc}&&(\tilde{E}_{2},-2)\\\ &&|\\\
(\tilde{E}_{3+},-3)&-&(\tilde{\Delta},-1)&-&(\tilde{E}_{3-},-3).\end{array}$

#### Auxiliary surfaces and birational morphisms

######  4.2.5.

As in Section 4.1, we describe some birational morphisms from $\tilde{S}$.
Recall that $S,\tilde{S}$ and $\sigma$ are already defined.

Let $\tau_{1}:\tilde{S}\to\hat{S}$ be the contraction of $\tilde{\Delta}$ onto
a smooth point $\hat{x_{0}}$ of $\hat{S}$. For each $i\in\\{2,3+,3-\\}$, let
$\hat{E}_{i}={\tau_{1}}_{*}(\tilde{E}_{i})$. The support of
${\tau_{1}}_{*}(\sigma^{*}\Delta)=\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ has
the weighted graph

$\textstyle{(\hat{E}_{2},-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\hat{E}_{3+},-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\hat{E}_{3-},-2).}$

Let $\tau_{2}:\hat{S}\to\breve{S}$ denote the contraction of $\hat{E}_{2}$
onto a smooth point $\breve{x_{0}}$ of $\breve{S}$ and for each
$i\in\\{3+,3-\\}$, let $\breve{E}_{i}={\tau_{2}}_{*}(\hat{E}_{i})$. Then
$\breve{E}_{3+}$ and $\breve{E}_{3-}$ are two $(-1)$-curves that intersect
tangentially at $\breve{x_{0}}$. We obtain the following diagram.

$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\tau_{1}}$$\textstyle{\hat{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau_{2}}$$\textstyle{\breve{S}}$$\textstyle{S}$

######  Proposition 4.2.6.

With notation as in 4.2.5,

  1. (a)

$\hat{S}$ is a weak del Pezzo surface of degree 1,

  2. (b)

$\breve{S}$ is a smooth del Pezzo surface of degree 2,

  3. (c)

the only two $(-2)$-curves on $\hat{S}$ are $\hat{E}_{3+}$ and $\hat{E}_{3-}$
and they intersect transversely.

###### Proof.

We prove (a). First, observe that $\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ is
an effective anti-canonical divisor of $\hat{S}$ and
$(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})^{2}=1$. Thus it suffices to show that
$\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ is nef and big. Let $I=\\{2,3+,3-\\}$.
Let $\hat{C}$ be an irreducible curve in $\hat{S}$. If $\hat{C}=\hat{E}_{i}$
for some $i\in I$, then
$\hat{C}\cdot(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})\geq 0$. Otherwise, since
$\Delta\in\operatorname{{\rm Div}}(S)$ is ample,
$\hat{C}={\tau_{1}}_{*}(\tilde{C})$ for some curve $\tilde{C}$ in $\tilde{S}$
intersecting $\sigma^{-1}(\Delta)$. In particular, $\hat{C}$ must intersect
$\hat{E}_{i}$ for some $i\in I$ and so
$\hat{C}\cdot(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})>0$. This shows that
$\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ is nef. Since $\hat{S}$ is smooth,
Proposition 1.3.7 implies that $\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ is big.
This proves (a).

For (b), observe that $\breve{E}_{3+}+\breve{E}_{3-}$ is an effective anti-
canonical divisor on $\breve{E}$ satisfying
$(\breve{E}_{3+}+\breve{E}_{3-})^{2}=2$. To prove
$\breve{E}_{3+}+\breve{E}_{3-}$ is ample, by Theorem 1.3.6 it suffices to show
that $\breve{C}\cdot(\breve{E}_{3+}+\breve{E}_{3-})>0$ for every irreducible
curve $\breve{C}$ in $\breve{S}$. If
$\breve{C}\in\\{\breve{E}_{3+},\breve{E}_{3-}\\}$, then
$\breve{C}\cdot(\breve{E}_{3+}+\breve{E}_{3-})=1$. Otherwise, since $\Delta$
is ample, $\breve{C}$ is the image of some curve intersecting
$\sigma^{-1}(\Delta)$ and so
$\breve{C}\cdot(\breve{E}_{3+}+\breve{E}_{3-})>0$. This proves (b).

We prove (c). Suppose $\hat{E}$ is a $(-2)$-curve on $\hat{S}$ other than
$\hat{E}_{3+}$ and $\hat{E}_{3-}$. Then, by adjunction

$0=\hat{E}\cdot(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})$

and so $\hat{E}$ is disjoint from
$\hat{E}_{2}\cup\hat{E}_{3+}\cup\hat{E}_{3-}$ and consequently,
${\tau_{2}}_{*}(\hat{E})$ is a $(-2)$-curve on the smooth del Pezzo surface
$\breve{S}$ (by part (b)). This contradicts Remark 1.5.6. ∎

######  4.2.7.

Since by Proposition 4.2.6, $\breve{S}$ is a smooth del Pezzo surface of
degree 2, there exist 7 disjoint $(-1)$-curves
$\breve{F}_{1},\dots,\breve{F}_{7}$ on $\breve{S}$. For any $(-1)$-curve
$\breve{F}_{i}$ on $\breve{S}$ ($i\in\\{1,\dots,7\\}$), the adjunction formula
implies

$-2=\breve{F}_{i}^{2}+\breve{F}_{i}\cdot K_{\breve{S}},$

so
$\breve{F}_{i}\cdot(-K_{\breve{S}})=\breve{F}_{i}\cdot(\breve{E}_{3+}+\breve{E}_{3-})=1$.
Consequently, every curve $\breve{F}_{i}$ intersects either $\breve{E}_{3+}$
or $\breve{E}_{3-}$ at a point other than the singular point of
$\breve{E}_{3+}\cup\breve{E}_{3-}$. Let $u_{+}$ and $u_{-}$ denote the number
of curves $\breve{F}_{i}$ intersecting $\breve{E}_{3+}$ and $\breve{E}_{3-}$
and let $\mu:\breve{S}\to\mathbb{P}^{2}$ denote the contraction of
$\cup_{i=1}^{7}\breve{F}_{i}$. Then

  * •

$u_{+}+u_{-}=7$

  * •

$-1+u_{+}=(\mu_{*}\breve{E}_{3+})^{2}$

  * •

$-1+u_{-}=(\mu_{*}\breve{E}_{3-})^{2}$

  * •

$-1+u_{+}$ and $-1+u_{-}$ must be perfect squares

where the last bullet is because after blowing down all seven curves
$\breve{F}_{i}$, $\mu_{*}\breve{E}_{3+}$ and $\mu_{*}\breve{E}_{3-}$ are
irreducible curves in $\mathbb{P}^{2}$. It follows
$\\{u_{+},u_{-}\\}=\\{2,5\\}$ and so
$\mu_{*}(\tilde{E}_{3_{+}}\cup\breve{E}_{3-})$ is the union of a smooth conic
and its tangent line at the point of intersection
$\mu_{*}(\breve{E}_{3+})\cap\mu_{*}(\breve{E}_{3-})$.

For each $i\in\\{1,\dots,7\\}$, let $\hat{F}_{i}$ denote the proper transform
of $\breve{F}_{i}$ in $\hat{S}$. For each $i$, we obtain the following
commutative diagram

$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\tau_{1}}$$\textstyle{\hat{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\gamma}_{i}}$$\scriptstyle{\tau_{2}}$$\scriptstyle{\eta}$$\textstyle{\dot{S}}$$\textstyle{S}$$\textstyle{\bar{S}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{\breve{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ddot{\gamma}_{i}}$$\scriptstyle{\mu}$$\textstyle{\ddot{S}_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{2}}$

where

  * •

$\eta:\hat{S}\to\dot{S}$ is the contraction of the two $(-2)$-curves
$\hat{E}_{3+}$ and $\hat{E}_{3-}$ on $\hat{S}$,

  * •

$\bar{\gamma}_{i}:\hat{S}\to\bar{S}_{i}$ is the contraction of $\hat{F}_{i}$,

  * •

$\ddot{\gamma}_{i}:\breve{S}\to\ddot{S}_{i}$ is the contraction of
$\breve{F}_{i}$,

  * •

$\beta:\bar{S}_{i}\to\ddot{S}_{i}$ is the contraction of
$\bar{\gamma_{i}}_{*}(\hat{E}_{2})$.

######  Proposition 4.2.8.

With the notation of 4.2.7, the following hold:

  1. (a)

$\bar{S_{i}}$ is a smooth weak del Pezzo surface of degree $2$;

  2. (b)

$\ddot{S_{i}}$ is a smooth del Pezzo surface of degree $3$;

  3. (c)

$\dot{S}$ is a singular del Pezzo surface of degree 1 with a unique $A_{2}$
singularity.

###### Proof.

We prove (a). Since by Proposition 4.2.6, $\hat{S}$ is a smooth weak del Pezzo
surface of degree 1, and $\bar{\gamma_{i}}$ contracts a $(-1)$-curve it is
clear that $(-K_{\bar{S}_{i}})^{2}=2$ and that $\bar{S}_{i}$ is smooth. For
each $j\in\\{2,3+,3-\\}$, let $\bar{E}_{j}=\bar{\gamma_{i}}_{*}{\hat{E}_{j}}$
so that $\bar{E}_{2}+\bar{E}_{3+}+\bar{E}_{3-}$ is an anti-canonical divisor
of $\bar{S}_{i}$. It suffices to show that
$\bar{E}_{2}+\bar{E}_{3+}+\bar{E}_{3-}$ is nef and big. Let $\bar{C}_{i}$ be
an irreducible curve in $\bar{S}_{i}$. If $\bar{C}_{i}=\bar{E}_{i}$ for some
$i\in\\{2,3+,3-\\}$ then
$\hat{C}\cdot(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})\geq 0$. Otherwise, since
$\Delta$ is ample on $S$,
$\bar{C}_{i}={\gamma_{i}\circ\tau_{1}}_{*}(\tilde{C})$ for some curve
$\tilde{C}$ in $\tilde{S}$ intersecting $\sigma^{-1}(\Delta)$. In particular,
$\bar{C}_{i}$ must intersect $\bar{E}_{i}$ for some $i$ and so
$\hat{C}\cdot(\bar{E}_{2}+\bar{E}_{3+}+\bar{E}_{3-})>0$. This shows that
$\bar{E}_{2}+\bar{E}_{3+}+\bar{E}_{3-}$ is nef. Since $\bar{S}_{i}$ is smooth,
Proposition 1.3.7 implies that $\bar{E}_{2}+\bar{E}_{3+}+\bar{E}_{3-}$ is big.
This proves (a).

Part (b) follows immediately from Proposition 4.2.6 (b) and the fact that
$\ddot{\gamma}_{i}$ is the contraction of one of the curves $\breve{F}_{i}$.

Part (c) follows immediately from Proposition 4.2.6 (a) and (c). ∎

#### Exclusion of anti-canonical polar cylinders.

In this section we will establish, again by contradiction, that the surface
$S=\operatorname{{\rm Proj}}B_{2,3,4,12}$ does not contain an anti-canonical
polar cylinder.

######  Assumption 4.2.9.

Assume the surface $S=\operatorname{{\rm Proj}}B_{2,3,4,12}$ contains an anti-
canonical polar cylinder $U\cong\mathbb{A}^{1}\times
Z=S\setminus\operatorname{{\rm Supp}}(D)$. Given $D$, define

$\tilde{D}=\sigma^{*}(D)+\frac{1}{3}\tilde{E}_{3+}+\frac{1}{3}\tilde{E}_{3-}\in\operatorname{{\rm
Div}}(\tilde{S},\mathbb{Q}).$

######  Proposition 4.2.10.

Assume 4.2.9 holds. Then,

  1. (a)

$\operatorname{{\rm Supp}}(D)$ contains $\operatorname{{\rm Sing}}(S)$;

  2. (b)

$\tilde{E}_{2},\tilde{E}_{3+},\tilde{E}_{3-}$ are contained in
$\operatorname{{\rm Supp}}(\tilde{D})$;

  3. (c)

$\tilde{D}$ is an effective anti-canonical $\mathbb{Q}$-divisor on
$\tilde{S}$;

  4. (d)

the set $\sigma^{-1}(U)=\tilde{S}\setminus\operatorname{{\rm
Supp}}(\sigma^{*}(D))=\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$
is an anti-canonical polar cylinder in $\tilde{S}$.

###### Proof.

Since $S$ is a normal surface and $U$ is a cylinder, $U$ must be smooth and so
$\operatorname{{\rm Sing}}(S)\subset\operatorname{{\rm Supp}}(D)$. This proves
(a). Part (b) follows from the definition of $\tilde{D}$ together with the
fact that $P_{2},P_{3+},P_{3-}\in\operatorname{{\rm Supp}}(D)$. Part (c)
follows from the definition of $\tilde{D}$ together with (17). We prove (d).
Since $P_{2},P_{3+},P_{3-}\in\operatorname{{\rm Supp}}(D)$,
$\sigma|_{\sigma^{-1}(U)}$ is an isomorphism onto its image. In particular,
$\sigma^{-1}(U)=\tilde{S}\setminus\operatorname{{\rm
Supp}}(\sigma^{*}(D))=\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$
is a cylinder which by (d) is an anti-canonical polar cylinder.

∎

######  Notation 4.2.11.

Assuming 4.2.9, the discussion in 1.6.12 implies that we have the following
commutative diagram:

---
$\textstyle{U\cong\mathbb{A}^{1}\times
Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{pr}_{Z}}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\tilde{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{\tilde{\rho}}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\phi}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota}$$\textstyle{\bar{Z}\cong\mathbb{P}^{1}}$

where $\tilde{\rho}$ denotes the rational map associated to the cylinder
$\sigma^{-1}(U)\subset\tilde{S}$, $\tilde{\mathcal{L}}$ is the linear system
associated to $\tilde{\rho}$, and $\pi$ is the resolution of indeterminacy of
$\tilde{\rho}$ so that $\phi$ is a $\mathbb{P}^{1}$-fibration. In what
follows, we define $\tilde{U}=\sigma^{-1}(U)$.

######  Lemma 4.2.12.

Assume 4.2.9 holds. Then $\tilde{\Delta}$ is not an irreducible component of
$\tilde{D}$.

###### Proof.

Assume that $\tilde{\Delta}$ is an irreducible component of $\tilde{D}$. Since
$\tilde{\Delta}\subset\operatorname{{\rm Supp}}{\tilde{D}}$ it follows from
Proposition 4.2.10 (b) that $\operatorname{{\rm Supp}}(\tilde{D})$ contains
$\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3+}\cup\tilde{E}_{3-}$. Let
$\tau=\tau_{2}\circ\tau_{1}$. The contraction $\tau:\tilde{S}\to\breve{S}$
restricts to an isomorphism between
$\tilde{U}=\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$ and its
image $\tau(\tilde{U})=\breve{S}\setminus\operatorname{{\rm Supp}}(\breve{D})$
where $\breve{D}=\tau_{*}(\tilde{D})$. Since $\breve{D}$ is an effective anti-
canonical $\mathbb{Q}$-divisor on $\breve{S}$, it follows that $\breve{S}$
contains an anti-canonical polar cylinder. Since by Proposition 4.2.6 (b),
$\breve{S}$ is a smooth del Pezzo surface of degree 2, this contradicts
Theorem 1.6.15. ∎

######  Lemma 4.2.13.

Assume 4.2.9 holds. Then the linear system $\tilde{\mathcal{L}}$ has a unique
proper base point.

###### Proof.

Since by 1.6.12 $\mathcal{L}$ has at most one base point,
$\tilde{\mathcal{L}}$ has at most one base point. So we need only show that
$\tilde{\mathcal{L}}$ is not base point free.

Assume for the sake of a contradiction that $\tilde{\mathcal{L}}$ is base-
point free. Then $\tilde{D}$ contains a section $\Gamma$ of the
$\mathbb{P}^{1}$-fibration $\tilde{\rho}$. In particular, for a general fiber
$L$ of $\tilde{\rho}$, the adjunction formula implies

$2=-K_{\tilde{S}}\cdot L=2\tilde{\Delta}\cdot L+\tilde{E}_{2}\cdot
L+\tilde{E}_{3+}\cdot L+\tilde{E}_{3-}\cdot L.$

Let $I=\\{2,3+,3-\\}$. If $\Gamma=\tilde{E}_{i}$ for some $i\in I$, then
$\tilde{\Delta}\cdot L=0$ and by Remark 1.6.11, $\tilde{E}_{j}$ is a section
for some $j\in I\setminus\\{i\\}$. This is impossible since by 1.6.12,
$\operatorname{{\rm Supp}}(\tilde{D})$ contains exactly one section of
$\tilde{\rho}$ in its support. It follows from 1.6.12 that for each $i\in I$,
$\tilde{E}_{i}$ is contained in a closed fiber of $\tilde{\rho}$.
Consequently, $\tilde{\Delta}\cdot L=1$ and so $\tilde{\Delta}$ is a section
by Remark 1.6.11.

Since the three curves $\tilde{E}_{i}$ intersect $\tilde{\Delta}$ at different
points, and at most one irreducible curve in each fiber can intersect a
section of a $\mathbb{P}^{1}$-fibration, we obtain that

each curve $\tilde{E}_{i}$ is contained in a different fiber of
$\tilde{\rho}$. (19)

Arguing as in the proof of (9) in Lemma 4.1.14, we obtain

every singular fiber of $\tilde{\rho}$ contains some curve $\tilde{E}_{i}$.
(20)

By combining (19) and (20), there are exactly $3$ singular fibers. Let us
denote these $3$ fibers by $\tilde{F}_{2},\tilde{F}_{3+},\tilde{F}_{3-}$ where
$\tilde{F}_{i}$ is the fiber containing $\tilde{E}_{i}$. We claim:

for each $i\in\\{2,3+,3-\\}$, the singular fibers $\tilde{F}_{i}$ have form
$\tilde{F}_{i}=\tilde{E}_{i}+\tilde{R}_{i}$ where $\tilde{R}_{2}$ consists of
$2$ disjoint $(-1)$-curves intersecting $\tilde{E}_{2}$ at distinct points of
$\tilde{E}_{2}\setminus\tilde{\Delta}$ and $\tilde{R}_{3+}$ (resp.
$\tilde{R}_{3-}$) consists of $3$ disjoint $(-1)$-curves intersecting
$\tilde{E}_{3+}$ (resp. $\tilde{E}_{3-}$) at distinct points of
$\tilde{E}_{3+}\setminus\tilde{\Delta}$ (resp.
$\tilde{E}_{3-}\setminus\tilde{\Delta}$). (21)

Consider the fiber $\tilde{F}_{i}$, and write
$\tilde{F}_{i}=\tilde{E}_{i}+\tilde{R}_{i}$ where
$\tilde{R}_{i}=\sum_{\ell=1}^{q_{i}}a_{i,\ell}\tilde{G}_{i,\ell}$, where
$q_{i}$ depends on $i$. Note that $\tilde{E}_{i}$ is the only component of
$\tilde{F}_{i}$ that intersects $\tilde{\Delta}$. Since $\Delta$ is ample,
every component of $\tilde{F}_{i}$ must intersect $\tilde{E}_{i}$ and can
intersect no other exceptional curve $\tilde{E}_{j}$ where $j\in
I\setminus\\{i\\}$. Since (by Lemma 1.6.17) $\tilde{F}_{i}$ is a tree and all
components $\tilde{G}_{i,\ell}$ intersect $\tilde{E}_{i}$, the components
$\tilde{G}_{i,\ell}$ must be disjoint and must intersect $\tilde{E}_{i}$ at
distinct points of $\tilde{E}_{i}\setminus\tilde{\Delta}$. Lemma 1.6.18
implies

  * •

$q_{2}=2$ and $q_{3+}=q_{3-}=3$

  * •

$\tilde{G}_{i,\ell}$ is a $(-1)$-curve and $a_{i,\ell}=1$ for all $i$ and all
$\ell=1,\dots,q_{i}$.

This proves (21).

Denote by
$\psi:\tilde{S}\to\mathbb{F}_{1}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1))$
the contraction of all 8 $(-1)$-curves $\tilde{G}_{i,\ell}$ in the $3$
singular fibers of $\tilde{\rho}$. Letting
$\rho_{1}:\mathbb{F}_{1}\to\mathbb{P}^{1}$ be the $\mathbb{P}^{1}$-bundle
structure, we have $\tilde{\rho}=\rho_{1}\circ\psi$ by Lemma 1.6.17 (2).

Since $\operatorname{{\rm Supp}}(\tilde{D})$ is connected and contains
$\tilde{E}_{2},\tilde{E}_{3+}$ and $\tilde{E}_{3-}$ and since $\tilde{\rho}$
restricts to a trivial $\mathbb{A}^{1}$-bundle on
$\tilde{S}\setminus\operatorname{{\rm Supp}}(\tilde{D})$, it follows that
$\operatorname{{\rm Supp}}(\tilde{D})$ contains at least two irreducible
components of both $\tilde{R}_{3+}$ and $\tilde{R}_{3-}$.

Choose one $(-1)$-curve $\tilde{G}_{3+,1}\subset\tilde{R}_{3+}$ which
intersects $\tilde{E}_{3+}$ and one $(-1)$-curve
$\tilde{G}_{3-,1}\subset\tilde{R}_{3-}$ which intersects $\tilde{E}_{3-}$. Let
$\check{\psi}:\tilde{S}\to\check{S}$ denote the contraction of the two curves
$\tilde{G}_{3+,1},\tilde{G}_{3-,1}$. We claim

$\text{ $\check{S}$ is a weak del Pezzo surface of degree 2}.$ (22)

First, let $\check{\Delta}=\check{\psi}_{*}(\tilde{\Delta})$, and let
$\check{E}_{i}=\check{\psi}_{*}(\tilde{E}_{i})$ for each $i\in I$. Note that
$\check{E}_{i}$ is a $(-2)$-curve for each $i\in I$. Then
$(-K_{\check{S}})^{2}=(2\check{\Delta}+\check{E}_{2}+\check{E}_{3+}+\check{E}_{3-})^{2}=2$,
so it suffices to show that $-K_{\check{S}}$ is nef and big. Let $\check{C}$
be a curve in $\check{S}$. If $\check{C}$ is one of
$\check{\Delta},\check{E}_{2},\check{E}_{3+},\check{E}_{3-}$ then
$\check{C}\cdot(2\check{\Delta}+\check{E}_{2}+\check{E}_{3+}+\check{E}_{3-})\geq
0$. Otherwise, since $\Delta$ is an ample divisor on $S$, the proper transform
of $\check{C}$, denoted $\tilde{C}$ must intersect one of
$\tilde{\Delta},\tilde{E}_{2},\tilde{E}_{3+},\tilde{E}_{3-}$, and consequently
$\check{C}\cdot(2\check{\Delta}+\check{E}_{2}+\check{E}_{3+}+\check{E}_{3-})>0$.
This shows that $-K_{\check{S}}$ is nef. Since $-K_{\check{S}}$ is nef and
$(-K_{\check{S}})^{2}=2$, Proposition 1.3.7 implies that $-K_{\check{S}}$ is
big. This proves (22).

Next we show that

$\text{ the three curves $\check{E}_{2},\check{E}_{3},\check{E}_{5}$ are
disjoint and are the only $(-2)$-curves on $\check{S}$}.$ (23)

It is easy to check that $\check{E}_{2},\check{E}_{3},\check{E}_{5}$ are
disjoint. Let $\check{E}$ be a $(-2)$-curve on $\check{S}$ other than
$\check{E}_{2},\check{E}_{3+},\check{E}_{3-}$. By adjunction, we have

$\check{E}\cdot(2\check{\Delta}+\check{E}_{2}+\check{E}_{3+}+\check{E}_{3-})=0$

and consequently, since $\check{E}$ is none of $\check{E}_{2}$,
$\check{E}_{3+}$, $\check{E}_{3-}$, $\check{\Delta}$, it follows that
$\check{E}$ is disjoint from $\operatorname{{\rm
Supp}}(2\check{\Delta}+\check{E}_{2}+\check{E}_{3+}+\check{E}_{3-})$.
Consequently, the proper transform of $\check{E}$ in $\tilde{S}$, denoted
$\tilde{E}$, is a $(-2)$-curve on $\tilde{S}$ that is disjoint from
$\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3+}\cup\tilde{E}_{3-}$ and so
$\hat{E}=\tau_{*}(\tilde{E})$ is a $(-2)$-curve on $\hat{S}$ that is neither
$\hat{E}_{3+}$ nor $\hat{E}_{3-}$. This contradicts Proposition 4.2.6 (c), so
we conclude no such $\check{E}$ exists. This proves (23). It follows
immediately from (23) that the contraction of
$\check{E}_{2},\check{E}_{3+},\check{E}_{3-}$ yields a morphism
$\gamma:\check{S}\to T$ where $T$ is a del Pezzo surface with exactly 3
$A_{1}$ singularities.

Since the two curves $\tilde{G}_{3+,1},\tilde{G}_{3-,1}$ are contained in
$\operatorname{{\rm Supp}}(\tilde{D})$, the divisor
$\check{D}=\check{\psi}_{*}(\tilde{D})$ is an anti-canonical divisor on
$\check{S}$. So $\check{S}\setminus\check{D}$ is an anti-canonical polar
cylinder on $\check{S}$ and $\check{D}$ contains
$\check{E}_{2},\check{E}_{3+},\check{E}_{3-}$ in its support. This contradicts
Corollary 1.6.16 and completes the proof. ∎

######  Lemma 4.2.14.

Assume 4.2.9 holds and let $\tilde{s}$ be the unique base point of
$\tilde{\mathcal{L}}$. Then
$\tilde{s}\notin(\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3+}\cup\tilde{E}_{3-})$.

###### Proof.

The proof consists of four parts. We will show

  1. (i)

$\tilde{s}\notin\tilde{\Delta}$,

  2. (ii)

$\tilde{s}\notin\tilde{E}_{2}$,

  3. (iii)

$\tilde{s}\notin\tilde{E}_{3+}$,

  4. (iv)

$\tilde{s}\notin\tilde{E}_{3-}$.

First we recall that by Lemma 1.6.14 we have

the log pair $(\tilde{S},\tilde{D})$ is not log-canonical at $\tilde{s}$. (24)

Assume $\tilde{s}\in\tilde{\Delta}$. By Lemma 4.2.12,
$\tilde{\Delta}\not\subseteq\operatorname{{\rm Supp}}(\tilde{D})$. Since
$\tilde{D}$ is an anti-canonical $\mathbb{Q}$-divisor on $\tilde{S}$ and
$\tilde{\Delta}$ is a $(-1)$-curve, we have

$\tilde{\Delta}\cdot\tilde{D}=\tilde{\Delta}\cdot(2\tilde{\Delta}+\tilde{E}_{2}+\tilde{E}_{3+}+\tilde{E}_{3-})=1$

and so

$1=\tilde{\Delta}\cdot\tilde{D}\geq\operatorname{{\rm
mult}}_{\tilde{s}}(\tilde{\Delta})\operatorname{{\rm
mult}}_{\tilde{s}}(\tilde{D})\geq\operatorname{{\rm
mult}}_{\tilde{s}}(\tilde{D})$ (25)

the first inequality because $\tilde{\Delta}\not\subseteq\operatorname{{\rm
Supp}}(\tilde{D})$, the second because $\tilde{s}\in\tilde{\Delta}$. Since by
(24), $(\tilde{S},\tilde{D})$ is not log-canonical at $\tilde{s}$, Lemma 1.5.3
implies $\operatorname{{\rm mult}}_{\tilde{s}}(\tilde{D})>1$, a contradiction
to (25). This proves (i).

For (ii), assume that $\tilde{s}\in\tilde{E}_{2}$. Let
$\hat{s}=\tau_{1}(\tilde{s})$. Consider the composition
$\eta\circ\tau_{1}:\tilde{S}\to\hat{S}\to\dot{S}$ as in 4.2.7. Since by (i)
$\tilde{s}\notin\tilde{\Delta}$, $\eta(\hat{s})$ is a smooth point of
$\dot{S}$ and $\eta_{*}\hat{E}_{2}$ is an integral anti-canonical divisor on
$\dot{S}$ and $\eta(\hat{s})$ is a regular point of $\eta_{*}\hat{E}_{2}$. By
Lemma 1.5.3, we obtain

the log pair $(\dot{S},\eta_{*}\hat{E}_{2})$ is log-canonical at
$\eta(\hat{s})$. (26)

On the other hand, $\dot{D}=\eta_{*}\hat{D}$ is an effective anti-canonical
$\mathbb{Q}$-divisor on $\dot{S}$ for which the log pair $(\dot{S},\dot{D})$
is not log-canonical at the smooth point $\eta(\hat{s})$ of $\dot{S}$. By
Lemma 1.5.7 (d), $\eta_{*}\hat{E}_{2}$ is not log-canonical at
$\eta(\hat{s})$, contradicting (26). This proves (ii).

For (iii), assume $\tilde{s}\in\tilde{E}_{3+}$. Consider the contraction
$\tau_{2}\circ\tau_{1}:\tilde{S}\to\hat{S}\to\breve{S}$ of
$\tilde{\Delta}\cup\tilde{E}_{2}$ as defined in 4.2.5. Then
$\breve{D}=(\tau_{2}\circ\tau_{1})_{*}\tilde{D}$ is an effective anti-
canonical $\mathbb{Q}$-divisor on $\breve{S}$, which by Proposition 4.2.6 (b)
is a smooth del Pezzo surface of degree $2$. By (24), $(\tilde{S},\tilde{D})$
is not log-canonical at $\tilde{s}$. It follows that, the log pair
$(\breve{S},\breve{D})$ is not log-canonical at the point
$\breve{s}=(\tau_{2}\circ\tau_{1})(\tilde{s})$. Recalling the notation from
4.2.5, we have that the divisor $\breve{E}_{3+}+\breve{E}_{3-}$ is an integral
anti-canonical divisor on $\breve{S}$ supported on the two $(-1)$-curves
$\breve{E}_{3+}$ and $\breve{E}_{3-}$ which intersect with multiplicity two at
$\breve{x}_{0}$. Also, $\operatorname{{\rm
Supp}}(\breve{E}_{3+}+\breve{E}_{3-})\subseteq\operatorname{{\rm
Supp}}(\breve{D})$. By (i) and (ii) $\breve{s}\neq\breve{x}_{0}$, so the log
pair $(\breve{S},\breve{E}_{3+}+\breve{E}_{3-})$ is log-canonical at
$\breve{s}$. Since the pair $(\breve{S},\breve{D})$ is not log-canonical at
$\breve{s}$, Lemma 1.5.9 (a) implies that there exists a unique divisor
$\breve{C}\in|-K_{\breve{S}}|$ such that $(\breve{S},\breve{C})$ is not log-
canonical at $\breve{s}$ and Lemma 1.5.9 (b) implies that

$\breve{C}$ is either an irreducible rational curve with a cusp at $\breve{s}$
or is the union of two smooth $(-1)$-curves meeting tangentially at
$\breve{s}$. (27)

Since $\breve{E}_{3+}$ is a $(-1)$-curve, we have
$\breve{C}\cdot\breve{E}_{3+}=(\breve{E}_{3+}+\breve{E}_{3-})\cdot\breve{E}_{3+}=1$.
If $\breve{C}$ does not contain $\breve{E}_{3+}$ in its support, we obtain
$1=\breve{C}\cdot\breve{E}_{3+}\geq\operatorname{{\rm
mult}}_{\breve{s}}(\breve{C})\operatorname{{\rm
mult}}_{\tilde{s}}(\breve{E}_{3+})\geq 2$, which is impossible. Consequently,
$\breve{E}_{3+}\subseteq\operatorname{{\rm Supp}}(\breve{C})$ and so
$\breve{C}$ is not an irreducible rational curve. It follows from (27) that
$\operatorname{{\rm Supp}}(\breve{C})=\breve{E}_{3+}\cup\breve{C}^{\prime}$
where $\breve{C}^{\prime}$ is a $(-1)$-curve intersecting $\breve{E}_{3+}$
tangentially at $\breve{s}$. Since $\breve{s}\neq\breve{x}_{0}$, we have
$\breve{C}^{\prime}\neq\breve{E}_{3-}$ and then, since $\breve{E}_{3-}$ is a
$(-1)$-curve, we obtain that
$1=\breve{C}\cdot\breve{E}_{3-}=(\breve{E}_{3+}+\breve{C}^{\prime})\cdot\breve{E}_{3-}=2+\breve{C}^{\prime}\cdot\breve{E}_{3-}$
which is impossible since $\breve{C}^{\prime}\cdot\breve{E}_{3-}\geq 0$. This
proves (iii). To prove (iv), simply permute the the indices $\\{3+,3-\\}$ and
repeat the proof of (iii). ∎

######  Remark 4.2.15.

It follows from Lemma 4.2.14 that any base point of $\mathcal{L}$ is not
supported on $\Delta\in\operatorname{{\rm Div}}(S)$.

######  Lemma 4.2.16.

Assume 4.2.9 holds. Let $\hat{D}={\tau_{1}}_{*}(\tilde{D})$ and let
$\hat{s}=\tau_{1}(\tilde{s})$. Then,

  1. (a)

the log pair $(\hat{S},\hat{D})$ is not log-canonical at $\hat{s}$;

  2. (b)

$(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})\cdot\hat{C^{\prime}}>0$ for every
irreducible curve $\hat{C^{\prime}}\in\operatorname{{\rm Div}}(\hat{S})$ other
than $\hat{E}_{3+}$ and $\hat{E}_{3-}$;

  3. (c)

there exists a unique divisor $\hat{C}\in|-K_{\hat{S}}|$ which passes through
$\hat{s}$, and $\hat{C}$ is an irreducible reduced curve.

###### Proof.

By Lemma 4.2.14, $\tilde{\mathcal{L}}$ has a proper base point
$\tilde{s}\in\tilde{S}\setminus(\tilde{\Delta}\cup\tilde{E}_{2}\cup\tilde{E}_{3+}\cup\tilde{E}_{3-})$.
Recall that by Proposition 4.2.6 (a), $\hat{S}$ is a weak del Pezzo surface of
degree $1$, and $\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ is an anti-canonical
divisor of $\hat{S}$ passing through $\hat{x}_{0}\in\hat{S}$. It follows from
Proposition 4.2.10 (b) and the definition of $\tau_{1}$ that $\hat{D}$ is an
effective anti-canonical $\mathbb{Q}$-divisor containing
$\hat{E}_{2}\cup\hat{E}_{3+}\cup\hat{E}_{3-}$ in its support. Since by Lemma
1.6.14, $(\tilde{S},\tilde{D})$ is not log-canonical at $\tilde{s}$, the log
pair $(\hat{S},\hat{D})$ is not log-canonical at $\hat{s}$. This proves (a).

Part (b) follows from the fact that $\Delta\in\operatorname{{\rm Div}}(S)$ is
ample and $\tilde{\Delta}$ is contracted by $\tau_{1}$, and is left to the
reader.

For (c), since $\hat{S}$ is a weak del Pezzo surface of degree 1, the anti-
canonical linear system $|-K_{\hat{S}}|$ has a unique base point which we
denote by $o$. Since $o\in\hat{E}_{2}\cup\hat{E}_{3+}\cup\hat{E}_{3-}$ and
$\hat{s}\in\hat{S}\setminus(\hat{E}_{2}\cup\hat{E}_{3+}\cup\hat{E}_{3-})$, it
follows that $\hat{s}\neq o$. By Proposition 4 in [Dem77], there exists an
irreducible divisor $\hat{C}\in|-K_{\hat{S}}|$ which passes through $\hat{s}$.
Since $\hat{s}\neq o$, the divisor $\hat{C}$ is unique. Furthermore, since
$(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})\cdot\hat{C}=1$, $\hat{C}$ is reduced.
This proves (c). ∎

######  Proposition 4.2.17.

Assume 4.2.9 holds. Let $\hat{s}=\tau_{1}(\tilde{s})$. Then, there exists an
effective anti-canonical $\mathbb{Q}$-divisor $\hat{D}_{0}$ on $\hat{S}$ such
that the following hold:

  1. (i)

the log pair $(\hat{S},\hat{D}_{0})$ is not log-canonical at the point
$\hat{s}$;

  2. (ii)

the support of $\hat{D}_{0}$ does not contain the unique member $\hat{C}$ of
$|-K_{\hat{S}}|$ passing through $\hat{s}$.

###### Proof.

Let $\hat{D}=\tau_{1*}(\tilde{D})$. If the support of $\hat{D}$ does not
contain the support of $\hat{C}$, set $\hat{D}_{0}=\hat{D}$, and Lemma 4.2.16
(a) gives the result. Assume from now on that $\operatorname{{\rm
Supp}}(\hat{D})$ contains $\operatorname{{\rm Supp}}(\hat{C})$. Let
$\tilde{C}\in\operatorname{{\rm Div}}(\tilde{S})$ denote the proper transform
of the curve $\hat{C}$. Then, the support of $\tilde{D}$ contains $\tilde{C}$
and we can write

$\tilde{D}=(\sum_{i\in
I}a_{i}\tilde{D}_{i})+a_{\tilde{C}}\tilde{C}+a_{2}\tilde{E}_{2}+a_{3+}\tilde{E}_{3+}+a_{3-}\tilde{E}_{3-}$
(28)

where $\sum_{i\in I}a_{i}\tilde{D}_{i}$ is an effective $\mathbb{Q}$-divisor
whose support does not contain $\tilde{C}$, $\tilde{E}_{2}$, $\tilde{E}_{3+}$,
$\tilde{E}_{3-}$ and all the coefficients
$a_{i},a_{\tilde{C}},a_{2},a_{3+},a_{3-}$ are positive rational numbers. We
claim:

  1. (a)

$\tilde{C}$ does not intersect $\tilde{\Delta}$;

  2. (b)

$\tilde{C}$ intersects $\tilde{E}_{2}$ transversely in a single point;

  3. (c)

the curve $\tilde{C}$ is the unique irreducible component of $\tilde{D}$ which
intersects $\tilde{E}_{2}$;

Since $1=\hat{C}\cdot(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})$, the curve
$\hat{C}$ does not pass through $\hat{x}_{0}$ and so, $\tilde{C}$ does not
intersect $\tilde{\Delta}$, proving (a). Since $\hat{E}_{2}$ is a $(-1)$-curve
on $\hat{S}$ and $\hat{C}$ is an anti-canonical divisor on $\hat{S}$, we have
by adjunction that $\hat{E}_{2}\cdot\hat{C}=1$, so $\hat{C}$ intersects
$\hat{E}_{2}$ transversely at a unique point. Let $y=\hat{E}_{2}\cap\hat{C}$
denote the point of intersection. Since $\hat{x}_{0}\not\in\hat{C}$,
$\hat{x}_{0}\neq y$ and so $\tilde{C}$ intersects $\tilde{E}_{2}$ transversely
at a single point belonging to $\tilde{E}_{2}\setminus\tilde{\Delta}$. This
proves (b). For (c), assume there exists an irreducible component
$\tilde{D}_{i}$ of $\tilde{D}$ which intersects $\tilde{E}_{2}$, noting that
$\tilde{D}_{i}\neq\tilde{\Delta}$ by Lemma 4.2.12. Since
$\tilde{D}_{i}\notin\\{\tilde{E}_{3+},\tilde{E}_{3-}\\}$, Lemma 4.2.16 (b)
implies that
$\tau_{*}(\tilde{D}_{i})\cdot\hat{C}=\tau_{*}(\tilde{D}_{i})\cdot(\hat{E}_{2}+\hat{E}_{3}+\hat{E}_{5})>0$.
Since $\hat{x}_{0}\not\in\hat{C}$, it follows that $\tilde{D}_{i}$ intersects
both $\tilde{C}$ and $\tilde{E}_{2}$. Let $W$ be as in Notation 4.2.11. If
$\tilde{D}_{i}$ intersects $\tilde{C}$ and $\tilde{E}_{2}$ at $y$ then the
total transform of $\operatorname{{\rm Supp}}(\tilde{D})$ on $W$ would contain
the proper transforms of $\tilde{D}_{i}$, $\tilde{C}$ and $\tilde{E}_{2}$
meeting at the same point (because $\tilde{s}\neq y$ and $W$ is obtained from
$\tilde{S}$ by blowing up points over $\tilde{s}$). This is impossible since
the total transform of $\operatorname{{\rm Supp}}(\tilde{D})$ on $W$ (the
union of the section of the $\mathbb{P}^{1}$-fibration and finitely many
fibers) is a tree. If $\tilde{D}_{i}$ intersects $\tilde{C}$ and
$\tilde{E}_{2}$ at a point other than $y$, then the total transform of
$\operatorname{{\rm Supp}}(\tilde{D})$ on $W$ contains a cycle, which again
contradicts the fact that the total transform of $\operatorname{{\rm
Supp}}(\tilde{D})$ on $W$ is a tree. This proves (c).

Next we claim:

the coefficients of $\tilde{D}$ satisfy the inequalities
$a_{3+},a_{3-}>\frac{1}{3}$ and $a_{\tilde{C}}=2a_{2}<\frac{2}{3}$. (29)

Since $\operatorname{{\rm Supp}}(\tilde{D})$ is connected (by Remark 1.6.13)
and contains $\tilde{E}_{3+}$ and $\tilde{E}_{3-}$ in its support (by
Proposition 4.2.10 (b)), it follows that there exists an irreducible component
$\tilde{\delta}$ of
$\tilde{D}=\sigma^{*}D+\frac{1}{3}(\tilde{E}_{3+}+\tilde{E}_{3-})$ which
intersects $\tilde{E}_{3+}$. We have
$\sigma^{*}\sigma_{*}(\tilde{\delta})=\tilde{\delta}+\frac{1}{3}(\operatorname{{\rm
mult}}_{P_{3+}}\sigma_{*}(\tilde{\delta}))\tilde{E}_{3+}$ and since
$\operatorname{{\rm mult}}_{P_{3+}}\sigma_{*}(\tilde{\delta})>0$,
$a_{3+}\geq\frac{1}{3}+\frac{1}{3}\operatorname{{\rm
mult}}_{P_{3+}}\sigma_{*}(\tilde{\delta})>\frac{1}{3}$. The same argument
applies to $\tilde{E}_{3-}$, showing that $a_{3-}>\frac{1}{3}$. By (c),
$\tilde{C}$ is the unique irreducible component of $\tilde{D}$ intersecting
$\tilde{E}_{2}$, so $\sigma_{*}(\tilde{C})$ is the unique irreducible
component of $D$ passing through $P_{2}\in S$ (again since $\tilde{\Delta}$ is
not an irreducible component of $\tilde{D}$ by Lemma 4.2.12). Furthermore,
since $\tilde{C}\cdot\tilde{E}_{2}=1$, we have
$\sigma^{*}\sigma_{*}(\tilde{C})=\tilde{C}+\frac{1}{2}\tilde{E}_{2}$ and hence
we obtain that
$a_{\tilde{C}}\tilde{C}+a_{2}\tilde{E}_{2}=a_{\tilde{C}}(\tilde{C}+\frac{1}{2}E_{2})$,
showing that $a_{2}=\frac{1}{2}a_{\tilde{C}}$. Since
$\tilde{D}\sim(-K_{\tilde{S}})$ and $\tilde{\Delta}$ is a $(-1)$-curve on
$\tilde{S}$, the adjunction formula implies

$1=\tilde{D}\cdot\tilde{\Delta}=\left(\sum_{i\in
I}a_{i}(\tilde{D}_{i}\cdot\tilde{\Delta})\right)+a_{\tilde{C}}(\tilde{C}\cdot\tilde{\Delta})+a_{2}+a_{3+}+a_{3-}.$

Since $\tilde{C}\cdot\tilde{\Delta}=0$ by (a),
$\tilde{D}_{i}\cdot\tilde{\Delta}\geq 0$, $a_{3+}>\frac{1}{3}$ and
$a_{3-}>\frac{1}{3}$, we obtain that $a_{2}<\frac{1}{3}$ and that
$a_{\tilde{C}}=2a_{2}<\frac{2}{3}$. This proves (29).

Now, let $\epsilon=\frac{a_{\tilde{C}}}{2-3a_{\tilde{C}}}$ and observe that by
(29), $\epsilon>0$. Consider the $\mathbb{Q}$-divisor

$\begin{split}\hat{D}_{0}&=(1+3\epsilon)\hat{D}-2\epsilon\hat{C}-\epsilon(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})\\\
&=\left(\sum_{i\in
I}(1+3\epsilon)a_{i}\hat{D}_{i}\right)+((1+3\epsilon)a_{\tilde{C}}-2\epsilon)\hat{C}\\\
&\quad+((1+3\epsilon)a_{2}-\epsilon)\hat{E}_{2}+((1+3\epsilon)a_{3+}-\epsilon)\hat{E}_{3+}+((1+3\epsilon)a_{3-}-\epsilon)\hat{E}_{3-}.\end{split}$
(30)

We claim

the divisor $\hat{D}_{0}$ is an effective anti-canonical $\mathbb{Q}$-divisor
on $\hat{S}$. (31)

Since $\hat{D}$, $\hat{C}$ and $\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-}$ are
anti-canonical $\mathbb{Q}$-divisors on $\hat{S}$, the first equality in (30)
gives that $\hat{D}_{0}$ too is an anti-canonical $\mathbb{Q}$-divisor. To
prove (31), it suffices to show that all coefficients in (30) are non-
negative. Since $1+3\epsilon>0$, $(1+3\epsilon)a_{i}>0$ for all $i\in I$. Next
we have

$\displaystyle(1+3\epsilon)a_{\tilde{C}}-2\epsilon=0$
$\displaystyle(1+3\epsilon)a_{2}-\epsilon=\frac{2\epsilon}{a_{\tilde{C}}}a_{2}-\epsilon=0$
$\displaystyle(1+3\epsilon)a_{3+}-\epsilon=\frac{2\epsilon}{a_{\tilde{C}}}a_{3+}-\epsilon=\frac{\epsilon(2a_{3+}-a_{\tilde{C}})}{a_{\tilde{C}}}>0$
$\displaystyle(1+3\epsilon)a_{3-}-\epsilon=\frac{2\epsilon}{a_{\tilde{C}}}a_{3-}-\epsilon=\frac{\epsilon(2a_{3-}-a_{\tilde{C}})}{a_{\tilde{C}}}>0$

where the last two inequalities are because $\epsilon,a_{\tilde{C}}>0$,
$a_{3+},a_{3-}>\frac{1}{3}$ and $a_{\tilde{C}}<\frac{2}{3}$. This shows that
$\hat{D}_{0}$ is effective, proving (31).

Finally, we show

the log pair $(\hat{S},\hat{D}_{0})$ is not log-canonical at $\hat{s}$. (32)

Recall that $\pi:W\to\tilde{S}$ is the minimal resolution of indeterminacy of
the rational map determined by $\tilde{\rho}$ (as in Notation 4.2.11) and
consider the composition $\beta=\tau_{1}\circ\pi:W\to\hat{S}$. Since
$K_{\hat{S}}+\hat{D}_{0}\sim_{\mathbb{Q}}0$, the ramification formula for
$\beta$ reads

$K_{W}+D_{0,W}\sim_{\mathbb{Q}}\beta^{*}(K_{\hat{S}}+\hat{D}_{0})+\sum_{i=1}^{r}c_{i}E_{i}+(1-\operatorname{{\rm
mult}}_{x_{0}}\hat{D}_{0})\Delta_{W}$ (33)

where

  * •

$K_{W}$ is a canonical divisor of $W$,

  * •

$D_{0,W}$, $\Delta_{W}$ are, respectively, the proper transforms of
$\hat{D}_{0}$ and $\tilde{\Delta}$ on $W$,

  * •

the curves $E_{i}$ are the irreducible components of the exceptional locus of
$\pi$.

Recall that by 1.6.12 there exists a unique section of the
$\mathbb{P}^{1}$-fibration $\phi:W\to\mathbb{P}^{1}$ among the curves
$E_{1},\dots,E_{r}$ and the proper transforms of the components of
$\hat{D}_{0}$ are contained in closed fibers of $\phi$. We assume that $E_{r}$
is this section. Let $L$ be a general fiber of $\phi:W\to\mathbb{P}^{1}$.
Since $L$ is a $(0)$-curve which does not intersect
$D_{0,W}\cup\bigcup_{i=1}^{r-1}E_{i}$ and intersects the section $E_{r}$ of
$\phi$ transversely at a unique point, we obtain by adjunction and (33), that

$-2=c_{r}+(1-\mathrm{mult}_{x_{0}}\hat{D}_{0})\Delta_{W}\cdot L.$

Since $\tilde{\Delta}$ is a $(-1)$-curve on $\tilde{S}$, we have by adjunction
that $\tilde{D}\cdot\tilde{\Delta}=(-K_{\tilde{S}})\cdot\tilde{\Delta}=1$.
This implies that $\operatorname{{\rm mult}}_{\hat{x}_{0}}(\hat{D})=1$. Since
$\hat{C}$ does not pass through $\hat{x}_{0}$ and $\operatorname{{\rm
mult}}_{\hat{x}_{0}}(\hat{E}_{2}+\hat{E}_{3+}+\hat{E}_{3-})=3$, it follows
from (30) that $\operatorname{{\rm
mult}}_{\hat{x}_{0}}\hat{D}_{0}=(1+3\epsilon)-3\epsilon=1$. This implies that
$c_{r}=-2$, which shows that the log pair $(\hat{S},\hat{D}_{0})$ is not log-
canonical at $\hat{s}$.

∎

######  Proposition 4.2.18.

The surface $S=\operatorname{{\rm Proj}}B_{2,3,4,12}$ does not contain an
anti-canonical polar cylinder.

###### Proof.

Assume that $S$ contains an anti-canonical polar cylinder. Then, preserving
the notation of Proposition 4.2.17, it follows that there exists an effective
anti-canonical $\mathbb{Q}$-divisor $\hat{D}_{0}$ such that

  1. (i)

the log pair $(\hat{S},\hat{D}_{0})$ is not log-canonical at the point
$\hat{s}$;

  2. (ii)

the support of $\hat{D}_{0}$ does not contain the unique member $\hat{C}$ of
$|-K_{\hat{S}}|$ passing through $\hat{s}$.

Recall that by Proposition 4.2.6 (c), $\hat{E}_{3+}$ and $\hat{E}_{3-}$ are
the only $(-2)$-curves on $\hat{S}$, and it follows from Lemma 4.2.14 (iii)
and (iv), that $\hat{s}$ is not contained in the support of a $(-2)$-curve on
$\hat{S}$. Since $(\hat{S},\hat{D}_{0})$ is not log-canonical at $\hat{s}$,
Corollary 1.5.8 implies that $\operatorname{{\rm
Supp}}(\hat{C})\subseteq\operatorname{{\rm Supp}}(\hat{D}_{0})$, a
contradiction. ∎

######  Corollary 4.2.19.

Let $B=B_{a_{0},a_{1},a_{2},a_{3}}$ where
$(a_{0},a_{1},a_{2},a_{3})\in\\{(2,3,5,30),(2,3,4,12)\\}$. Then $B^{(d)}$ is
rigid for all $d\in\mathbb{N}^{+}$.

###### Proof.

Let $X=\operatorname{{\rm Proj}}B$. By Propositions 4.1.17 and 4.2.18, $X$
does not contain an anti-canonical polar cylinder, so by Theorem 3.2.1,
$B^{(d)}$ is rigid for all $d\in\mathbb{N}^{+}$. ∎

We can finally prove the Main Theorem.

######  Main Theorem.

Let $n\geq 2$, and let $B_{a_{0},a_{1},a_{2},a_{3}}={\rm\bf
k}[X_{0},X_{1},X_{2},X_{3}]/\langle
X_{0}^{a_{0}}+X_{1}^{a_{1}}+X_{2}^{a_{2}}+X_{3}^{a_{3}}\rangle$ be a Pham-
Brieskorn ring. If $\min\\{a_{0},a_{1},a_{2},a_{3}\\}\geq 2$ and at most one
element $i$ of $\\{0,1,2,3\\}$ satisfies $a_{i}=2$, then
$(B_{a_{0},a_{1},a_{2},a_{3}})^{(d)}$ is rigid for all $d\in\mathbb{N}^{+}$.

###### Proof.

Recall by the reduction made previously, it suffices to prove that
$B_{a_{0},a_{1},a_{2},a_{3}}$ is rigid for all
$(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{+}\cup\Gamma_{3}^{-}$, see Notation
3.0.1). For $(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{+}$, the result follows
from Corollary 3.1.2. For $(a_{0},a_{1},a_{2},a_{3})\in\Gamma_{3}^{-}$, the
result follows immediately from Lemma 3.2.2 together with Propositions 3.2.3,
3.2.7 and Corollary 4.2.19. ∎

##  5 Some Remarks for Higher Dimensions

######  5.0.1.

We conclude by offering some remarks on the Main Conjecture in the higher
dimensional cases (where $n>3$). First, we recall that because of Theorem
2.1.13, it suffices to prove the conjecture for the cases
$B_{a_{0},\dots,a_{n}}$ where $\operatorname{{\rm
cotype}}(a_{0},\dots,a_{n})=0$. Again, by Proposition 2.1.4, in exactly these
cases $\operatorname{{\rm Proj}}B_{a_{0},\dots,a_{n}}$ is a well-formed
quasismooth weighted complete intersection. We also note that canonical sheaf
and canonical divisor of $X=\operatorname{{\rm Proj}}B_{a_{0},\dots,a_{n}}$
can still be computed using Theorems 1.4.9 and 2.2.5.

######  Lemma 5.0.2.

Let $n\geq 4$ and let $X=\operatorname{{\rm Proj}}B_{a_{0},\dots,a_{n}}$. Then
$\operatorname{{\rm CaCl}}(X)\cong\mathbb{Z}$.

###### Proof.

For every $(b_{0},\dots,b_{n})$, there exists some $(a_{0},\dots,a_{n})$ such
that $\operatorname{{\rm Proj}}(B_{b_{0},\dots,b_{n}})\cong\operatorname{{\rm
Proj}}(B_{a_{0},\dots,a_{n}})$ where $\operatorname{{\rm
cotype}}(a_{0},\dots,a_{n})=0$. (See Lemma 3.1.14 (b) in [Chi23b].) So, we may
assume that $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$. Since $X$ is
integral, we have $\operatorname{{\rm CaCl}}(X)\cong\operatorname{{\rm
Pic}}(X)\cong\mathbb{Z}$, the second isomorphism by Theorem 3.2.4 of [Dol82]
(using that $\dim(X)\geq 3$). ∎

Recall that morphism of varieties $f:X\to Y$ is affine if the inverse image of
every affine open subvariety of $Y$ is affine. Also, if $i:U\hookrightarrow Y$
is an open immersion of varieties where $U$ is affine, then $i$ is an affine
morphism.

######  Proposition 5.0.3.

Let $X$ be a projective normal $\mathbb{Q}$-factorial variety and suppose that
$\operatorname{{\rm CaCl}}(X)\cong\mathbb{Z}$. Let $H$ be any ample
$\mathbb{Q}$-divisor on $X$. Then every cylinder $U$ in $X$ is an $H$-polar
cylinder.

###### Proof.

Let $U$ be a cylinder in $X$ and recall that $U$ is affine (by definition).
The inclusion $i:U\to X$ is an affine morphism so we can apply Corollary
21.12.7 of [Gro67] which (since $U$ is dense in $X$) implies that
$U=X\setminus\operatorname{{\rm Supp}}(D)$ for some effective divisor
$D\in\operatorname{{\rm Div}}(X)$. Since $X$ is $\mathbb{Q}$-factorial, we may
assume that $D$ is Cartier. Let $H$ be any ample $\mathbb{Q}$-divisor of $X$.
We must show that $U$ is $H$-polar.

If $D=0$, then $\operatorname{{\rm Supp}}(D)=\emptyset$ and so $X=U$ is both
affine and projective. Since $X$ is a projective variety over ${\rm\bf k}$,
$\operatorname{\mathcal{O}}_{X}(X)={\rm\bf k}$, and since $X$ is affine,
$X=\operatorname{{\rm
Spec}}\operatorname{\mathcal{O}}_{X}(X)=\operatorname{{\rm Spec}}{\rm\bf k}$
is a point. This contradicts the assumption that $\operatorname{{\rm
CaCl}}(X)\cong\mathbb{Z}$; it follows that $D\neq 0$. Since $D$ is a nonzero
effective divisor and $\operatorname{{\rm Pic}}(X)\cong\operatorname{{\rm
CaCl}}(X)\cong\mathbb{Z}$, Example 1.2.4 of [Laz04] implies that $D$ is ample.

Choose $s\in\mathbb{N}^{+}$ such that $sH$ is Cartier. Then both $sH$ and $D$
belong to $\operatorname{{\rm CaDiv}}(X)$. Since $\operatorname{{\rm
CaCl}}(X)\cong\mathbb{Z}$, there exist $m,n\in\mathbb{Z}$ such that $m>0$ and
$m(sH)\sim nD$. Since $m(sH)$ is ample, so is $nD$, and it is not too hard to
show that $n>0$. Since $m(sH)\sim nD$ where $m,s,n\in\mathbb{N}^{+}$, Remark
1.6.3 implies that every $D$-polar cylinder is $H$-polar. Since $U$ is
$D$-polar, it is therefore $H$-polar. ∎

######  Corollary 5.0.4.

Let $n\geq 4$, and let $B=B_{a_{0},\dots,a_{n}}$ be such that
$\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$. Then $B$ is not rigid if
and only if $\operatorname{{\rm Proj}}B$ contains a cylinder.

###### Proof.

Let $\alpha$ denote the amplitude of $B$ and let $X=\operatorname{{\rm
Proj}}B$. If $\alpha=0$, then Theorem 2.2.5 (c) implies that $B$ is rigid and
that $\operatorname{{\rm Proj}}B$ does not contain a cylinder so the result
holds in this special case.

Assume now that $\alpha>0$ (resp. $\alpha<0$). Then $K_{X}$ is ample (resp.
$-K_{X}$ is ample) by Theorem 2.2.5 (d). Then $B$ is not rigid if and only if
$X$ contains a $K_{X}$-polar cylinder (resp. $-K_{X}$ polar cylinder) if and
only if $X$ contains a cylinder. (The first equivalence is by Theorem 2.2.5
(d), the second is by Proposition 5.0.3.) ∎

The following proposition is well-known. See Proposition 4.9.7 of [Chi23b] for
one proof.

######  Proposition 5.0.5.

Let $n\geq 4$. Then $B_{a_{0},\dots,a_{n}}$ is a unique factorization domain.

######  Remark 5.0.6.

Proposition 5.0.5 fails for $n=2,3$. See Exercise II.6.5 in [Har77] as well as
[Sto84].)

######  5.0.7.

Let $n\geq 4$, suppose $\operatorname{{\rm cotype}}(a_{0},\dots,a_{n})=0$ and
let $X=\operatorname{{\rm Proj}}(B_{a_{0},\dots,a_{n}})$. Similar to the case
when $n=3$, the problem splits quite naturally into the cases where the
amplitude $\alpha$ is either negative, zero or positive.

If $\alpha<0$, then Theorem 2.2.5 implies that $-K_{X}$ is an ample
$\mathbb{Q}$-Cartier divisor, so $X$ is a Fano variety. By Corollary 5.0.4, we
are motivated to investigate the more general question of existence of
cylinders in singular Fano varieties, a question which is generally open in
dimension 2 or larger.

If $\alpha=0$, we know that $B_{a_{0},\dots,a_{n}}$ is rigid by Theorem 2.2.5
(c).

If $\alpha>0$, then the cases satisfying $(B_{a_{0},\dots,a_{n}})_{\alpha}\neq
0$ are rigid by Theorem 2.2.5 (c). For the cases where $B_{\alpha}=0$ (such as
$B_{5,5,5,6,6}$ for example), noting that $K_{X}$ is an ample
$\mathbb{Q}$-Cartier divisor, we are led to investigate the question of which
varieties with ample canonical divisors can contain cylinders (again by
Corollary 5.0.4). In particular, it is worthwhile investigating to what extent
Lemma 3.1.1 can be generalized to higher dimensions.

## References

  * [Arc93] Andrew Archibald.  Intersection theory on surfaces.  https://escholarship.mcgill.ca/downloads/t722h9520.PDF, 1993.
  * [BR86] Mauro Beltrametti and Lorenzo Robbiano.  Introduction to the theory of weighted projective spaces.  Expositiones Mathematicae, 4, 01 1986.
  * [CD20] Michael Chitayat and Daniel Daigle.  On the rigidity of certain Pham-Brieskorn rings.  Journal of Algebra, 550:290–308, 2020.
  * [CD22] Michael Chitayat and Daniel Daigle.  Locally nilpotent derivations of graded integral domains and cylindricity.  Transformation Groups, 2022.
  * [Chi23a] Michael Chitayat.  Rationality of weighted hypersurfaces of special degree, 2023.
  * [Chi23b] Michael Chitayat.  Rigidity of Pham-Brieskorn Threefolds.  University of Ottawa, Theses and Dissertations, 2023.
  * [CPPZ21] Ivan Cheltsov, Jihun Park, Yuri Prokhorov, and Mikhail Zaidenberg.  Cylinders in Fano varieties.  EMS Surveys in Mathematical Sciences, 8(1):39–105, August 2021\.
  * [CPS10] Ivan Cheltsov, Jihun Park, and Constantin Shramov.  Exceptional del Pezzo hypersurfaces.  Journal of Geometric Analysis, 20(4):787–816, 2010.
  * [CPW16a] Ivan Cheltsov, Jihun Park, and Joonyeong Won.  Cylinders in del Pezzo Surfaces.  International Mathematics Research Notices, 2017(4):1179–1230, 05 2016.
  * [CPW16b] Ivan Cheltsov, Jihun Park, and Joonyeong Won.  Cylinders in singular del Pezzo surfaces.  Compositio Mathematica, 152(6):1198–1224, 2016.
  * [CPW16c] Ivan Cheltsov, Jihun Park, and Joonyeong Wong.  Affine cones over smooth cubic surfaces.  Journal of the European Mathematical Society, 18(7):1537–1564, 2016\.
  * [Dai12] Daniel Daigle.  Tame and wild degree functions.  Osaka Journal of Mathematics, 49:53–80, 2012.
  * [Dai23] Daniel Daigle.  Rigidity of graded integral domains and of their Veronese subrings, 2023\.
  * [Dem77] Michel Demazure.  Surfaces de Del Pezzo : IV - Systèmes anticanoniques.  Séminaire sur les singularités des surfaces, 1976-1977.  talk:6.
  * [Dem88] Michel Demazure.  Anneaux gradués normaux.  In Introduction à la théorie des singularités. II: Méthodes algébriques et géométriques. (Introduction to the theory of singularities. II: Algebraic and geometric methods), pages 35–68. Paris: Hermann, 1988.
  * [DFMJ17] Daniel Daigle, Gene Freudenburg, and Lucy Moser-Jauslin.  Locally nilpotent derivations of rings graded by an abelian group.  In Algebraic Varieties and Automorphism Groups, pages 29–48, Tokyo, Japan, 2017. Mathematical Society of Japan.
  * [Dim86] Alexandru Dimca.  Singularities and coverings of weighted complete intersections.  Journal für die reine und angewandte Mathematik, 366:184–193, 1986\.
  * [Dol82] Igor Dolgachev.  Weighted projective varieties.  In James B. Carrell, editor, Group Actions and Vector Fields, pages 34–71, Berlin, Heidelberg, 1982. Springer Berlin Heidelberg.
  * [FMJ13] Gene Freudenburg and Lucy Moser-Jauslin.  Locally nilpotent derivations of rings with roots adjoined.  Michigan Mathematical Journal, 62(2):227–258, 2013.
  * [Fre17] Gene Freudenburg.  Algebraic Theory of Locally Nilpotent Derivations.  Encyclopaedia of Mathematical Sciences. Springer Berlin Heidelberg, 2017\.
  * [FZ03] Hubert Flenner and Mikhail Zaidenberg.  Rational curves and rational singularities.  Mathematische Zeitschrift, 244(3):549–575, 2003.
  * [Gro67] Alexander Grothendieck.  éléments de géométrie algébrique : IV. étude locale des schémas et des morphismes de schémas, quatrième partie.  Publications Mathématiques de l’IHÉS, 32:5–361, 1967.
  * [Har72] Robin Hartshorne.  Algebraic de Rham cohomology.  manuscripta mathematica, 7(2):125–140, 1972.
  * [Har77] Robin Hartshorne.  Algebraic geometry.  Springer-Verlag, New York, 1977.  Graduate Texts in Mathematics, No. 52.
  * [HTU17] Paul Hacking, Jenia Tevelev, and Giancarlo Urzúa.  Flipping surfaces.  Journal of Algebraic Geometry, 26(2):279–345, 2017.
  * [IF00] A. R. Iano-Fletcher.  Working with weighted complete intersections, page 101–174.  London Mathematical Society Lecture Note Series. Cambridge University Press, 2000.
  * [Ish14] S. Ishii.  Introduction to Singularities.  Springer, 2014.
  * [KM98] Janos Kollar and Shigefumi Mori.  Birational Geometry of Algebraic Varieties.  Cambridge Tracts in Mathematics. Cambridge University Press, 1998.
  * [Kov12] Sandor J Kovacs.  Singularities of stable varieties.  In Handbook of Moduli, Vol. II, pages 171–216. Advanced Lectures in Mathematics, 2012.
  * [KPZ13] Takashi Kishimoto, Yuri Prokhorov, and Mikhail Zaidenberg.  ${G}_{a}$-actions on affine cones.  Transformation Groups, 18:1137–1153, 2013.
  * [KZ00a] Shulim Kaliman and Mikhail Zaidenberg.  Miyanishi’s characterization of the affine 3-space does not hold in higher dimensions.  Annales de l’Institut Fourier (Grenoble), 50(6):1649–1669, 2000\.
  * [KZ00b] Shulim Kaliman and Mikhail Zaidenberg.  Miyanishi’s characterization of the affine 3-space does not hold in higher dimensions.  In Annales de l’institut Fourier, volume 50, pages 1649–1669, 2000\.
  * [Laz04] R.K. Lazarsfeld.  Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series.  Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics. Springer, 2004.
  * [LS22] Dayan Liu and Xiaosong Sun.  On the Rigidity of Some Extensions of Domains.  Michigan Mathematical Journal, pages 1 – 16, 2022.
  * [Miy00] Masayoshi Miyanishi.  Open Algebraic Surfaces.  American Mathematical Soc., 2000.
  * [Par03] Jihun Park.  A note on del pezzo fibrations of degree 1.  Communications in Algebra, 31(12):5755–5768, 2003.
  * [Sto84] Uwe Storch.  Die Picard-zahlen der singularitäten $t_{1}^{r_{1}}+t_{2}^{r_{2}}+t_{3}^{r_{3}}+t_{4}^{r_{4}}=0$.  Journal für die reine und angewandte Mathematik, 350:188–202, 1984.


∎

e1e-mail<EMAIL_ADDRESS>
11institutetext: GSI Helmholtzzentrum für Schwerionenforschung GmbH Darmstadt,
Germany 22institutetext: Institute of Modern Physics, Chinese Academy of
Sciences Lanzhou, China 33institutetext: Helmholtz Forschungsakademie Hessen
für FAIR (HFHF), GSI Helmholtzzentrum für Schwerionenforschung, Campus
Frankfurt, Germany 44institutetext: Ruhr-Universität Bochum, Germany
55institutetext: Goethe-Universität Frankfurt, Germany

# Deep Machine Learning for the PANDA Software Trigger

P. Jiange1,GSI,IMP,HFHF  K. GötzenGSI  R. KliemtGSI,RUB  F.
NerlingGSI,HFHF,GUF and K. PetersGSI,HFHF,GUF
on behalf of the PANDA Collaboration

(Received: date / Accepted: date)

###### Abstract

Deep machine learning methods have been studied for the software trigger of
the future PANDA experiment at FAIR, using Monte Carlo simulated data from the
GEANT-based detector simulation framework PandaRoot. Ten physics channels that
cover the main physics topics, including electromagnetic, exotic, charmonium,
open charm, and baryonic reaction channels, have been investigated at four
different anti-proton beam momenta. Binary and multi-class classification
together with seven different network architectures have been studied. Finally
a residual convolutional neural network with four residual blocks in a binary
classification scheme has been chosen due to its extendability, performance
and stability. The presented study represents a feasibility study of a
completely software-based trigger system. Compared to a conventional selection
method, the deep machine learning approach achieved a significant efficiency
gain of up to 200%, while keeping the background reduction factor at the
required level of 1/1000. Furthermore, it is shown that the use of additional
input variables can improve the data quality for subsequent analysis. This
study shows that the PANDA software trigger can benefit greatly from the deep
machine learning methods.

###### Keywords:

Software Trigger Deep learning High efficiency

††journal: Eur. Phys. J. C

##  1 Introduction

###  1.1 Motivation

The antiProton ANnihilation at DArmstadt experiment (PANDA) at the Facility
for Antiproton and Ion Research (FAIR) bib_FAIR  aims to explore hadron
physics by means of antiproton induced reactions bib_PANDA ; bib_PANDA2020 . A
wide array of physics topics and open questions is in the scope of PANDA,
including strangeness physics, charm and exotics, nucleon structure and
hadrons in nuclei PANDAPhysicsBook ; PANDAPhaseOne . Some examples are the
investigation of the charmonium spectrum, hyperon spectroscopy,
electromagnetic form factors as well as the search for exotic states, e.g.
hybrids, glueballs and hypernuclei. At centre-of-mass energies between 2 and
5.5 GeV the total $\bar{p}p$ cross section ranges between 50 and 100 mb
Workman:2022ynf , while the signals of interest have cross sections between a
couple of microbarns and a few nanobarns or even picobarns. Thus, with signal
cross sections being many orders of magnitude smaller than the total proton-
antiproton cross section, extracting the physics of interest is challenging in
terms of background suppression.

With an expected average reaction rate of about 20 MHz, and with the average
event size of 10-20 kB, the full data rate will be roughly 200 GB/s or more.
Because there is only a tiny fraction of events containing physics processes
of interest, it would be inefficient to store all the data. In order to
identify the interesting events in an online environment, a sophisticated
trigger system is necessary. The PANDA triggering system is foreseen to reduce
the rate of events to be stored by an approximate factor of 1000 to 20 kHz,
reducing the stored data to 200 MB/s, which still leads to about 1 PB stored
data per year.

Other experiments such as E835 E760 ; E835 , ATLAS ATLAS , BaBar Babar , or
BESIII BESIII  have different approaches to trigger and perform event
selection. Detector coincidence combined with clean background conditions,
e.g. in $e^{+}e^{-}$ collisions, allow for more or less straightforward
triggering decisions. In PANDA, the kinematic similarity of signal and
background reactions combined with a wide physics interest and the high
interaction rate puts the key challenge on the trigger system and consequently
the selections have to be made on full event candidates with high-level
information. Similar to experiments like LHCb LHCb ; LHCb3 , ALICE
ALICE_Trigger  or CMS CMS , PANDA will use a Software Trigger system based on
fully reconstructed event information. To cope with the challenges, high-level
properties need to be obtained from the reaction data for proper signal-
background separation, and deep machine learning methods are studied for the
PANDA Software Trigger to improve performance compared to a conventional cut-
and-count method.

###  1.2 Experiment Environment

Figure 1: Overview of the FAIR facility (top). The PANDA experimental setup
(bottom) with the initial detectors (in black) and staged upgrades (in red)
bib_PANDA2020 (and references therein).

FAIR is an international accelerator facility being under construction in
Darmstadt, Germany bib_FAIR ; bib_PANDA2020 . It expands the existing
accelerator complex in a large scale, shown in Figure 1. An antiproton beam
will be prepared in a cascade of accelerators, making use of almost the whole
facility. The synchrotron High-Energy Storage Ring (HESR) bib_HESR  will
store, cool and accelerate the antiprotons, which will have momenta in the
range between 1.5 GeV/$c$ and 15 GeV/$c$, corresponding to centre-of-mass
energies in the antiproton-proton reactions at PANDA between about 2 and 5.5
GeV. With a luminosity of up to $L=2\cdot 10^{32}\>\rm{cm^{-2}s^{-1}}$ in the
“high luminosity mode” and with a beam momentum resolution of $dp/p<2\cdot
10^{-5}$ in the “high resolution mode” HESR is an essential component for
PANDA that allows for precise measurements.

The PANDA experiment, shown in Figure 1, consists of two spectrometers, one
barrel-shaped surrounding the interaction point, the other covering the
forward direction, both providing measurements for precise charged particle
tracking and particle identification as well as electromagnetic calorimetry.

###  1.3 Software Trigger in PANDA

Figure 2: Schematic overview of the complete PANDA online trigger system.

The PANDA online trigger system will consist of online reconstruction, event
building, and the Software Trigger, as illustrated in the schematic in Figure
2. First, the online reconstruction of neutral particles and charged particles
will be performed as completely as possible using fast algorithms, and the
Particle IDentification (PID) information will be assigned to the
corresponding tracks. This will be combined with the event reconstruction
process, possibly in an iterative manner, to provide the online event
candidate with reconstructed final state particle information to the Software
Trigger module. The event candidate will then be processed in the Software
Trigger module by the selection algorithms. Then the event candidates will be
tagged when being consistent with a signal signature. Finally, an event
candidate will be written to the data storage if any of the trigger algorithms
accepts it to be a signal event.

##  2 Physics channels

The PANDA experiment has a rich physics program, so that the composition of
active trigger signatures will be adapted depending on the current physics aim
PANDAPhysicsBook ; PANDAPhaseOne . Therefore, the PANDA Software Trigger
system needs to identify many types of physics reactions and offer a high
flexibility in configuration.

Table 1: List of ten physics channels for the Software Trigger study together with the corresponding codes used in this document and the trigger signatures. Physics topic |  Reaction channel |  Code |  Trigger
---|---|---|---
Electromagnetic |  $\bar{p}p\rightarrow e^{+}e^{-}$ |  $ee$ |  $\bar{p}p\rightarrow e^{+}e^{-}$
Exotics |  $\bar{p}p\rightarrow\phi_{(1)}\phi_{(2)}$ |  $Phi$ |  $\phi\rightarrow K^{+}K^{-}$
Charmonium |  $\bar{p}p\rightarrow\eta_{c}\pi^{+}\pi^{-}$ |  $Etac$ |  $\eta_{c}\rightarrow K_{S}^{0}K^{-}\pi^{+}$
|  $\bar{p}p\rightarrow J/\psi\,\pi^{+}\pi^{-}$ |  $J2e$ |  $J/\psi\rightarrow e^{+}e^{-}$
|  $\bar{p}p\rightarrow J/\psi\,\pi^{+}\pi^{-}$ |  $J2mu$ |  $J/\psi\rightarrow\mu^{+}\mu^{-}$
Open charm |  $\bar{p}p\rightarrow D^{0}\bar{D^{0}}$ |  $D0$ |  $D^{0}\rightarrow K^{-}\pi^{+}$
|  $\bar{p}p\rightarrow D^{+}D^{-}$ |  $Dch$ |  $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$
|  $\bar{p}p\rightarrow D_{s}^{+}D_{s}^{-}$ |  $Ds$ |  $D_{s}^{+}\rightarrow K^{+}K^{-}\pi^{+}$
Baryons |  $\bar{p}p\rightarrow\Lambda\bar{\Lambda}$ |  $Lam$ |  $\Lambda\rightarrow p\pi^{-}$
|  $\bar{p}p\rightarrow\Lambda_{c}\bar{\Lambda_{c}}$ |  $Lamc$ |  $\Lambda_{c}\rightarrow pK^{-}\pi^{+}$
Background |  $\bar{p}p\hskip 3.61371ptgeneric$ |  DPM |  -

For this study a total of ten channels are considered to verify the
feasibility of the Software Trigger inspired by the physics motivation given
in the PANDA Physics Book PANDAPhysicsBook . These ten physics channels have
simple and clear decay modes in their respective field, spanning a number of
event topologies which may occur in the experimental runs, while covering the
main physics topics, namely exotic hadrons, charmonium, open charm, and baryon
states, including the resonances $\phi$, $\eta_{c}$, $J/\psi$, $D^{0}$,
$D^{+}$, $D_{s}^{+}$, $\Lambda$, and $\Lambda_{c}$.

Table 2: Summarised list of input observables. Category |  Description
---|---
|  \- momentum $p$ of reconstructed candidate (lab/cms)
Candidate kinematics |  \- transverse momentum $p_{t}$ of reconstructed candidate
|  \- polar angle $\theta$ of reconstructed candidate (lab/cms)
Daughter kinematics |  \- kinematic variables from each daughter of the candidate
|  \- Electron/Muon/Kaon/Pion/Proton PID probability of each daughter
|  \- maximum (transverse) particle momentum in event
Event info & multiplicities |  \- sum of (transverse) momenta of charged particles in event (cms)
|  \- sum of cluster energies in EMC
|  \- maximum of cluster energies in EMC
|  \- Particle multiplicities
|  \- Minimum/maximum/sum of momenta
|  \- Number of loose ($P_{\rm PID}>~{}0.25$) candidates
|  \- Aplanarity of the event
Event shape |  \- Sphericity of the event
|  \- Magnitude of the event thrust vector
|  \- Reduced Fox-Wolfram Moments 1-4

The physics topics, the reaction physics channels together with the
corresponding codes and the trigger signatures are listed in Table 1. Signal
events are generated with EvtGen EvtGen  using a simple phase space decay
model (PHSP) as well as the VSS (decay of vector to two scalar particles) and
VLL (decay of vector to two charged leptons) models for the decays of $\phi$
and $J/\psi$, respectively. More complex decay patterns e.g. in the Dalitz
decay channels are omitted in order to populate the phase space more evenly
for data quality studies. Generic inelastic background reactions are generated
by the Dual Parton Model (DPM) DPM , which models the various production cross
sections in antiproton-proton reactions. These background events have to be
rejected effectively by the triggering algorithm.

For each reaction type, a specific selection procedure, the trigger line, is
defined. To determine the performance, the individual trigger lines are under
investigation as well as the complete trigger system for the ten reaction
types to be tagged simultaneously.

A trigger line includes the reconstruction of a certain composite resonance /
particle candidate and the classification whether this composite candidate
originates from signal or background events. A comprehensive list of input
quantities (Table 2) is computed based on reconstructed properties, momenta,
angles, particle identification probabilities of the composite candidates as
well as final state particles involved. Furthermore, event specific variables
such as minimal and maximal momenta, momentum sums, event planarity and
sphericity, thrust magnitude, Fox-Wolfram moments EventShape  and
multiplicities are deduced, to further support the trigger decision.

One trigger line consists of four parts:

  * •

Reconstruction of composite candidates by final state combinatorics

  * •

Preselection with a broad cut on the invariant mass of the candidate of $\pm
10\sigma$ around its peak position

  * •

Calculation of the necessary observables, Table 2

  * •

Selection by algorithm (focus of this work)

The selection target is determined by achieving the required background
suppression factor. In this study, the overall required suppression factor is
$s=1/1000$, i.e. the number of background events is suppressed by 99.9% with
all active trigger lines acting simultaneously. Applying $n_{\mathrm{trig}}$
trigger lines simultaneously, the background suppression for each trigger line
is required to satisfy $s_{\mathrm{i}}=s/n_{\mathrm{trig}}$. In turn, due to
the possibility of different trigger lines accepting the same background
event, the actual total background suppression factor $s_{\mathrm{all}}$ can
be a little bit better, such that $s_{\mathrm{all}}\leq s=\sum
s_{\mathrm{i}}$.

##  3 Data preparation

The data for the physics channels and background are obtained by Monte-Carlo
simulations with the PandaRoot software PandaRoot . This simulation and
reconstruction software framework is based on the publicly available software
package FairRoot FairRoot  v18.2.0 and the the external package collection
FairSoft in version “jun19p1”.

EvtGen EvtGen  provides the particles for propagation through the detector
volumes with the GEANT 3 transport modeller GEANT3 , which is followed by
detailed detector simulations, digitisation and reconstruction. Per particle
candidate, tracking, calorimetry and PID information, such as particle energy
loss, time of flight, Cherenkov angle, EMC energy deposit, is being provided.
All events are self-contained and no event time relevant effects, such as
event mixing or incomplete events, are considered. In Table 3, the number of
simulated events is shown for the chosen channels where the centre-of-mass
energy permits the reaction. Also the Monte-Carlo truth information, matched
to the reconstructed particle candidates, is provided to distinguish
combinatorial background from signal events for the training stage.

Table 3: Numbers of simulated events of each physics channel at each energy, given in million events. The dashes mark the cases, where the reaction is energetically not possible. $\sqrt{s}[GeV]$ | $ee$ | $Phi$ | $Etac$ | $J2e$ | $J2mu$ | $D0$ | $Dch$ | $Ds$ | $Lam$ | $Lamc$ | DPM
---|---|---|---|---|---|---|---|---|---|---|---
2.4 | 3.1 | 1.3 | - | - | - | - | - | - | 2.4 | - | 20.0
3.8 | 2.4 | 1.2 | 6.0 | 2.7 | 1.2 | 1.2 | 1.4 | - | 1.3 | - | 20.0
4.5 | 2.2 | 1.2 | 6.6 | 2.3 | 1.2 | 1.2 | 1.3 | 1.6 | 1.2 | - | 20.0
5.5 | 2.2 | 1.2 | 7.7 | 2.1 | 1.2 | 1.2 | 1.2 | 1.3 | 1.2 | 1.7 | 20.0

##  4 Methods

Our figure of merit for comparing different approaches to the triggering
process will be the triggering efficiency, defined as the ratio of the number
of triggered events ($N_{\rm{trig}}$) to the number of events passing the
detector reconstruction, combinatorics and a 10$\,\sigma$ preselection mass
window ($N_{\rm{rec}}$),

$\epsilon=\frac{N_{\rm{trig}}}{N_{\rm{rec}}}$

with $\sigma$ being the width/resolution of the individual reconstructed
resonance. The triggering efficiency can be defined for two cases.
Individually the efficiency focuses on the selection performance of each
trigger line, serving one particular physics channel. This is a useful
quantity to optimise the triggering algorithms in question. For the complete
trigger setup, the total efficiencies will be affected by cross-tagging. Here
events can be rejected by their intended trigger line, but are accepted
accidentally by one or more of the other trigger lines. This is an
unintentional but fortunate effect. The total triggering efficiency is the
practically more relevant measure for the experiment.

Since we require a fixed background reduction for each neural network (trigger
line) individually, the resultant signal trigger efficiencies already
represent a comparable measure for network performance. As an additional
quality measure we provide the integral (AUC = area-under-curve) of the
receiver-operator-characteristics (ROC) being more common in Machine Learning
context, while not being of large practical value for this study.

###  4.1 Cut-and-count method

The conventional benchmark trigger scheme is following a cut-and-count
approach employing an optimised set of trigger line specific one-dimensional
cuts on the measured observable distributions.

In order to identify this set of criteria all input observables are evaluated
individually for each trigger line. We integrate the distributions of both
signal and background events for each of the $n$ observables in both
directions up to a threshold value retaining 90% of the signal events. This
threshold defines the cut to be applied. From these $2n$ possible cuts, we
select and apply the one leading to the largest suppression of background
events. This procedure is iterated until the total background suppression
matches the required factor. The last criterion in the set is adapted in the
way, that the background requirement is exactly met, so that the corresponding
signal efficiency can be larger than 90%.

The complete trigger configuration is given by 30 sets of one-dimensional
selection criteria, one for each trigger line at each accessible centre-of-
mass energy.

###  4.2 Deep learning methods

In order to let a neural network decide on which event belongs to one of the
signal channels or to background, the data has to be presented in a sensible
choice of observables. The same set of input variables (Table 2) used for the
cut-and-count approach is being used for direct comparison. Eventually this
set of variables is extended by additional event shape observables to
determine the optimal performance achievable using all available quantities.
The neural network forms its answer in a single number or a set of numbers in
case of a multiple classification. Performing a selection cut on this output
will determine the rejection rate of background events as well as the
efficiency of signal event acceptance.

The PyTorch framework pytorch  is chosen to provide the underlying
functionality to build the neural networks, which are trained and evaluated
with the prepared simulation data, roughly in a 1:1 ratio between the training
and testing data sets.

##  5 Neural network optimisation

Neural network setups are quite diverse and mostly tailored for the problem at
hand. For the Software Trigger, several choices are made based on the
performance, especially the signal efficiency and background reduction.
Individual networks are optimised by:

  * •

Network depth DNN

  * •

Width or conventional kernel size of each layer

  * •

Choice of the optimiser, learning rate and related items, e.g. momentum
overviewGD ,

  * •

Choice of the activation function Sigmoid ; ReLU ; LeakyReLU ; GeLU ; SeLU ;
SoftPlus ,

  * •

Weight decay generalisation L2

  * •

Weight initialisation Xavier .

###  5.1 Multi-class and binary approaches

The first question investigated is if a single network serving all trigger
lines simultaneously performs better than a single network for each physics
channel. Allowing multi-class classification would mean fewer but bigger
networks to be trained. Binary classification in contrast allows for easier
adaption and modification of the list of channels. The multi-class approach
reduces the number of required training experiments for each trigger setup to
find the optimal network size at the cost of longer training times due to the
increased complexity.

A set of Dense NNs DNN  (DNN) is used to study this issue exemplarily for one
setting at $\sqrt{s}=4.5\,\mathrm{GeV}$ by the means of a Bayesian approach
for the best triggering efficiency Bayesian . The results are shown in Table
4, where the individual trigger efficiencies serve as figure of merit. Here,
only signal candidates that match the Monte-Carlo truth are under
investigation, eliminating combinatorial effects. In those channels where the
triggering efficiency is not close to 100%, the binary classification
outperforms the multi-class approach by up to a factor of two.

Therefore, we choose the binary classification approach with one neural
network per trigger line and energy point.

Table 4: The comparison on the individual trigger efficiencies of truth matched events between multi-class classification and binary classification based on a DNN. Channels | Multi-class[%] | Binary[%]
---|---|---
$ee$ | 100.0 | -
$Phi$ | 87.9 | -
$Etac$ | 48.1 | 76.2
$J2e$ | 98.6 | -
$J2mu$ | 99.9 | -
$D0$ | 69.9 | 93.4
$Dch$ | 48.4 | 78.6
$Ds$ | 32.7 | 73.7
$Lam$ | 48.1 | 94.7

###  5.2 NN type selection

In order to identify the optimal network architecture and meta-configuration,
seven different types of networks are studied, listed in Table 5: A dense
neural network (DNN), a convolutional neural network (CNN) CNN , both with and
without residual blocks ResNet ; ResNeXt ; shatteringGradient , a CNN with
bottleneck residual blocks 1x1  as well as 1D and a 2D Long-Short-Term-Memory
network LSTM .

\begin{overpic}[width=433.62pt,tics=10,trim=0.0pt 0.0pt 0.0pt
20.075pt,clip]{effiMedianDNN.png}
\put(15.0,60.0){\fcolorbox{black}{white}{DNN}}
\end{overpic}\begin{overpic}[width=433.62pt,tics=10,trim=0.0pt 0.0pt 0.0pt
20.075pt,clip]{effiMedianCNNRes.png}
\put(15.0,60.0){\fcolorbox{black}{white}{CNNRes}} \end{overpic}

\begin{overpic}[width=433.62pt,tics=10,trim=0.0pt 0.0pt 0.0pt
20.075pt,clip]{effiMedianCNN.png}
\put(15.0,60.0){\fcolorbox{black}{white}{CNN}}
\end{overpic}\begin{overpic}[width=433.62pt,tics=10,trim=0.0pt 0.0pt 0.0pt
20.075pt,clip]{effiMedianLSTM1D.png}
\put(15.0,60.0){\fcolorbox{black}{white}{LSTM1D}} \end{overpic}

Figure 3: Individual trigger efficiencies (blue dots) for the channel $Dch$ at
$5.5$ GeV/$c$ for four of the seven NN types with varying depth parameters as
well as the median with standard deviation (red makers and bars).

Figure 4: Normalised triggering performance $\hat{\epsilon}$ as function of
the used layers/blocks in the seven network types, combined for the three
channels $Etac$, $Dch$ and $Lamc$ at $5.5$ GeV/$c$.

Some of the trigger lines perform reasonably well and show stable results
under all kind of network configurations due to particularities of the decay
kinematics. For example a $J/\psi$ decaying into two leptons will leave two
strongly correlated high-momentum tracks in the detector, which is
significantly different to the average multi-pion background event and in
consequence always leads to high selection quality.

For a meaningful optimisation, these “simple” cases are ignored, and the
networks are optimised in depth and layer size for a high triggering
efficiency for the three more challenging channels $Etac$, $Dch$ and $Lamc$.
For instance, the depth is optimised by obtaining the trigger performance as
function of used layers/blocks, and ten runs are carried out for each NN
framework of a certain depth. The performance is evaluated by the individual
efficiencies as well as a combined efficiency (“normalised trigger
performance” $\hat{\epsilon}$), defined as the geometric mean

$\hat{\epsilon}=(\epsilon_{Etac}\cdot\epsilon_{Dch}\cdot\epsilon_{Lamc})^{1/3}$

of the efficiencies of the channels $Etac$, $Dch$ and $Lamc$.

Table 5: The list and the description of the NNs investigated for the PANDA Software Trigger in this note. Abbreviation | Description
---|---
DNN | Dense NN
DNNRes | Dense NN with residual blocks
CNN | Convolutional NN
CNNRes | CNN with residual blocks
CNNResBN | CNN with Bottleneck residual blocks
LSTM1D | Long-Short-Term Memory with 1D input
LSTM2D | Long-Short-Term Memory with 2D input

The absolute value of $\hat{\epsilon}$ is the primary measure how well a
network type performs. The standard deviation of the individual efficiencies
from a series of training runs with varying random initialisation of the
internal weights is considered as a measure for the training stability.
Furthermore the networks are investigated for their robustness under changing
the network size, in terms of number of layers or blocks, which may be
required in a future scenario for more complex triggering setups.

In Figure 4, we present exemplarily four network types (DNN, CNN, CNNRes and
LSTM1D) and their individual efficiencies for just the channel $Dch$. For each
network size ten networks have been trained and evaluated, providing a median
and standard deviation from the individual results. The straight forward
approach would be a DNN (Figure 4, top left), however, the triggering
efficiency and stability for training is inferior as it is visible from the
large spread of the results. For a NN with increasing depth, in general
significant degradation of performance is expected because of shattered
gradients shatteringGradient . This can be most clearly seen in case of the
CNN with decreasing performance using a larger number of layers (Figure 4, top
right). Using a CNN with residual blocks (Figure 4, bottom left) mitigates
this behaviour. The LSTM model also looks stable but shows a worse
generalisation than CNNRes, with regard to the different physics channels
(Figure 4, bottom right)

For the selection of the best architecture the combined efficiencies
$\hat{\epsilon}$, shown in Figure 4, are compared. Here all considered network
types are presented, which contains the information of a total of $\approx$
2200 network training and evaluation runs, three channels times ten networks
per marker. We identify the CNNRes type network (orange) as optimal for the
given purpose based on its performance in triggering efficiency as well as
training stability and robustness concerning varying size.

###  5.3 Chosen Network Architecture

The final choice to evaluate the performance of a neural network approach for
the PANDA Software Trigger is the CNNRes, a Convolutional Neural Network with
four residual blocks. It is “deep” enough to ensure that the NN has enough
capacity, while performing relatively stable when adding more blocks, which
may be considered within a production triggering environment, e.g. featuring
many more trigger lines.

Figure 5: Schematic view of the chosen network architecture.

In Figure 5, the details of the network are presented. In the first stage the
127 input channels are mapped to a 11$\times$11 matrix with a fully connected
121 large linear layer in between. Here, the network has the possibility to
sort any linear combination of observables next to each other, which will
enhance the performance of the image recognition type architecture to follow.
Then two convolutional layers extend the dimensionality to
16$\times$11$\times$11, allowing for a detailed feature extraction performed
by four residual blocks - pairs of convolutional layers with a residual path.
Three convolutional and two linear layers then reduce the dimensions and
perform the classification for the output. Between each layer batch
normalisation BatchNorm  is being performed.

##  6 Results and discussion

###  6.1 Individual trigger performance

Measuring the performance of the neural network approach is done by the
individual trigger efficiency determined per channel with that corresponding
single trigger line active. The cuts on the network output are tuned to
achieve a background suppression of 1/1000 in total, with an equal fraction of
background contribution by each trigger channel for a certain centre-of-mass
energy. For example at the energy of $\sqrt{s}=4.5\,\mathrm{GeV}$ with nine of
the ten channels being energetically accessible, the targeted background
suppression per channel is 1/9000. Another commonly established measure of the
network capabilities are the ROC curves and, more condensed, the corresponding
AUCs. Both, the individual triggering efficiencies and the network AUCs are
summarised in Table 6.

Table 6: Collection of individual trigger efficiency results for a target 1/1000 background suppression, the AUC values of the ROC curves as well as the simultaneous trigger efficiency gains for all accessible channels. eff[%] | $ee$ | $Phi$ | $Etac$ | $J2e$ | $J2mu$ | $D0$ | $Dch$ | $Ds$ | $Lam$ | $Lamc$
---|---|---|---|---|---|---|---|---|---|---
Individual trigger efficiencies [%] |
Conventional trigger |
2.4 GeV | 94.79 | 38.09 | - | - | - | - | - | - | 21.50 | -
3.8 GeV | 88.80 | 54.64 | 9.71 | 91.34 | 92.52 | 39.50 | 12.50 | - | 40.25 | -
4.5 GeV | 98.88 | 56.24 | 11.92 | 85.85 | 84.62 | 31.42 | 10.19 | 6.38 | 49.21 | -
5.5 GeV | 95.94 | 61.47 | 17.87 | 76.31 | 75.09 | 36.79 | 17.71 | 11.87 | 56.27 | 6.59
NN trigger |
2.4 GeV | 99.98 | 63.67 | - | - | - | - | - | - | 29.14 | -
3.8 GeV | 99.98 | 78.34 | 22.45 | 99.53 | 99.75 | 39.54 | 23.89 | - | 56.67 | -
4.5 GeV | 99.99 | 78.71 | 21.77 | 98.46 | 99.40 | 54.77 | 24.77 | 15.78 | 63.81 | -
5.5 GeV | 99.99 | 80.94 | 28.53 | 96.44 | 98.35 | 62.74 | 35.43 | 22.74 | 70.55 | 11.85
NN trigger with additional event shape observables |
2.4 GeV | 100.00 | 73.72 | - | - | - | - | - | - | 30.33 | -
3.8 GeV | 100.00 | 85.05 | 26.56 | 99.76 | 99.88 | 69.30 | 30.48 | - | 68.52 | -
4.5 GeV | 100.00 | 84.13 | 25.21 | 99.21 | 99.64 | 58.01 | 27.32 | 18.92 | 73.47 | -
5.5 GeV | 100.00 | 84.95 | 38.10 | 98.86 | 98.90 | 65.02 | 40.16 | 27.26 | 78.19 | 14.75
Individual NN performance |
AUC values for the NN trigger |
2.4 GeV | 1.000 | 0.948 | - | - | - | - | - | - | 0.978 | -
3.8 GeV | 1.000 | 0.987 | 0.869 | 0.999 | 0.999 | 0.960 | 0.854 | - | 0.987 | -
4.5 GeV | 1.000 | 0.988 | 0.912 | 0.998 | 0.999 | 0.951 | 0.866 | 0.865 | 0.990 | -
5.5 GeV | 1.000 | 0.986 | 0.929 | 0.996 | 0.998 | 0.936 | 0.890 | 0.852 | 0.992 | 0.821
AUC values for the NN trigger with the additional event shape observables |
2.4 GeV | 1.000 | 0.959 | - | - | - | - | - | - | 0.977 | -
3.8 GeV | 1.000 | 0.988 | 0.887 | 0.999 | 1.000 | 0.963 | 0.838 | - | 0.990 | -
4.5 GeV | 1.000 | 0.990 | 0.913 | 0.998 | 0.999 | 0.956 | 0.877 | 0.890 | 0.994 | -
5.5 GeV | 1.000 | 0.989 | 0.941 | 0.999 | 0.998 | 0.955 | 0.900 | 0.873 | 0.994 | 0.834
Cross-tagging efficiency gains as effect of simultaneous triggering [%] |
Conventional trigger |
2.4 GeV | 0.01 | 0.43 | - | - | - | - | - | - | 1.96 | -
3.8 GeV | 22.74 | 4.52 | 21.93 | 0.83 | 1.45 | 4.12 | 5.36 | - | 3.74 | -
4.5 GeV | 25.53 | 5.68 | 48.65 | 3.68 | 1.00 | 6.94 | 31.82 | 18.51 | 4.12 | -
5.5 GeV | 19.68 | 19.37 | 93.05 | 5.01 | 3.53 | 12.05 | 50.72 | 40.75 | 2.18 | 15.50
NN trigger |
2.4 GeV | 0.00 | 0.00 | - | - | - | - | - | - | 1.26 | -
3.8 GeV | 26.39 | 4.75 | 12.17 | 2.28 | 0.47 | 9.36 | 5.02 | - | 3.57 | -
4.5 GeV | 34.92 | 6.51 | 42.59 | 3.42 | 1.75 | 6.12 | 18.13 | 16.22 | 4.58 | -
5.5 GeV | 26.37 | 15.80 | 132.98 | 6.22 | 5.11 | 9.08 | 27.23 | 30.85 | 4.71 | 10.99
NN trigger with additional event shape observables |
2.4 GeV | 0.01 | 0.00 | - | - | - | - | - | - | 0.98 | -
3.8 GeV | 26.93 | 6.72 | 14.57 | 3.92 | 1.23 | 4.13 | 6.48 | - | 4.10 | -
4.5 GeV | 35.89 | 7.22 | 61.43 | 4.71 | 3.22 | 6.54 | 22.28 | 16.78 | 4.08 | -
5.5 GeV | 29.18 | 21.76 | 118.00 | 12.93 | 8.09 | 10.37 | 31.27 | 32.09 | 4.15 | 11.46

Figure 6: Individual triggering efficiency gains of the neural networks
compared to the conventional approach, based on the same set of input
observables (light colours), and on the extended set of observables (dark
colours).

Figure 7: Cross-tagging efficiency gain as effect of simultaneous triggering
of multiple reactions, for the conventional benchmark approach (light
colours), the NN approach with the same set of observables (medium colours),
and the NN approach with the extended set of observables (dark colours).

In all cases the triggering with the aid of the neural networks improves the
efficiencies compared to the cut-and-count approach. This improvement is of
course expected, mainly because possible correlations between observables are
exploited by the new algorithm. Figure 7 shows the relative efficiency gain,
where the bars in light colours represent the increase based on the same set
of input variables, the dark coloured bars the performance gain using the
extended set of variables. It reaches almost up to 200% for some of the
channels, corresponding to about a factor three in performance. The actual
benefit strongly depends on the channel in question. For example the channel
$ee$ has already a very good triggering efficiency from the cut-and-count
approach, leaving no room for improvements. In the case of the two $J/\psi$
channels $J2e$ and $J2mu$, the triggering efficiency is dropping with
increasing beam momentum (see Table 6), which could be almost completely
recovered by the new approach. For the open charm and $Etac$ and $Lamc$
channels the triggering efficiency has about doubled.

Adding more observables, describing the overall event topology, leads in many
cases to a substantial gain in triggering efficiency. Hence, in a future
software trigger setup it would be desirable to provide these event shape
observables, even if computationally expensive in an online environment. Based
on the AUC values all networks show good (0.8 - 0.9) to excellent ($>0.9$)
classification performance.

###  6.2 Simultaneous trigger performance

In a realistic setup, all active trigger lines have the ability to trigger
events simultaneously. Therefore it can happen, that a signal event of a
certain type missed by the dedicated trigger line is accidentally tagged by
another one. This cross tagging by simultaneous triggering has the potential
to further improve the triggering efficiency. Nevertheless, at the same time
the background level will not exceed the requirements.

Figure 7 quantifies this effect in the current scenario, showing the increase
in the triggering efficiencies, also presented in Table 6, which are
calculated as the fraction of triggered events from events that passed the
reconstruction in the trigger line designed for the channel. The light, medium
and dark coloured bars correspond to the conventional cut-and-count benchmark
algorithm, the neural network approach and the neural network approach based
on the extended variable set, respectively. In some cases this adds a
substantial increase in triggering efficiency. For example the channel $Etac$
at 5.5 GeV gains a factor of two in total triggering efficiency by cross
tagging in this particular setup. It is important to note that these
“accidental” gains are highly dependent on the actual composition of trigger
lines, the centre-of-mass energy, and the actual trained networks.

Figure 8: Candidate mass distributions of accepted and rejected signal (red
and green, respectively) and background (blue and black, respectively) for the
channel $Dch$ at 4.5 GeV/$c$. Left, all kinematic input observables, right
without the energy of the candidate and the invariant mass of the recoiling
system. The histograms are normalised to the same integral.

###  6.3 Computing performance

The bulk of training processes goes into finding the optimal solution for each
network architecture, which is time consuming and requires the intensive usage
of GPU and CPU resources. A dedicated server with one Intel® Xeon® W-2135 CPU
with 3.70 GHz and one NVIDIA® 3090 GPU is used, equivalent in performance to
179 cores of the AMD® EPYC® 7551 32-Core Processors on the computing cluster
at GSI. Processing times are in the order of two days for a single trigger
line on 12 cores of the computing cluster. Since the training can be performed
in parallel for the trigger lines this is quite promising. The model inference
is able to run in the order of 1.4 M candidates per second on the GPU server
with six trigger lines in parallel. The main bottleneck is the CPU based data
reading and preparation.

###  6.4 Data quality and choice of observables

The input features and observables have varying impact on the trigger
decision. Some features are correlated, some are not relevant for the
classification problem at hand, which differs between the channels and even
between the data sets of the same channel at different beam energies. The
ranking of the observable importance can guide the observable choice and thus
may reduce the required computations in the online processing before.

Furthermore, it is important to maintain a certain quality of the data during
the selection process, which in turn means that the underlying physics
constrains the choice of observables.

The $Dch$ channel ($D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$) for example has a
significant enhancement in the invariant mass distribution of background
events around the signal peak position (blue histogram in Figure 8, left) when
using both the energy of the $D^{+}$ candidate and the invariant mass of the
recoil system at the same time as input for the NN. This effect originates
from the specific kinematics of the two body reaction $\bar{p}p\rightarrow
D^{+}D^{-}$ considered here. However, it is the goal to trigger $D^{+}$
particles from many different reactions inclusively. Dropping these
observables from the NN input removes the undesirable peaking background
(Figure 8, right) at the cost of some signal efficiency. We observed now a
smooth distribution (blue histogram) in the signal region similar to the
rejected background (black histogram).

Figure 9: Trigger efficiency deviation from the mean value in Dalitz
coordinates for the channel $Etac$ at 3.8 GeV/$c$, without (left) and with
(right) the additional observables.

When analysing complex decay patterns and mixtures of resonances, it is
important to cover the phase space without steep drops or holes in the
triggering efficiency distributions introduced by the triggering algorithm.
Complex fitting algorithms, such as a partial-wave analysis PrimerPWA ,
benefit most from a flat triggering efficiency distribution if possible, but
they require at least a smooth dependency on any kinematic observable.

For example a three-body-decay, such as the channel $\eta_{c}\rightarrow
K_{s}K^{-}\pi^{+}$, could be studied in the Dalitz plot Dalitz
representation. We find that the relative efficiency of the neural network
trigger introduces a cut-off in one corner of the Dalitz plot (Figure 9, left)
with the observables that are used in the cut-and-count approach. It could be
that the network identified this corner to be particularly occupied with
background. Introducing the event shape observables (Table 2) as additional
input, the training is able to produce a significantly more flat triggering
efficiency (Figure 9, right), being beneficial for later Dalitz plot analysis.

##  7 Summary and outlook

Based on a set of ten channels with typical event topologies in the reach of
PANDA physics investigated at four anti-proton beam energies, it is
demonstrated, that PANDA will greatly benefit from a neural network supported
software trigger system.

As one result the use of binary classification networks outperforms an
approach with multi-class classification. As an additional advantage, it will
make the setup much easier to extend the system with a larger set of
simultaneous trigger lines and improves the computational scalability. From
seven network architectures, the convolutional network with four residual
blocks showed the best results for the three channels with the lowest
triggering efficiency, while producing stable results under different training
attempts and size changes.

In all cases the network approach performs better than the cut-and-count
method, which further improves when adding more observables. In the comparison
the triggering efficiencies show gains of up to 200% and all networks show
good up to excellent performance.

Data quality is an important topic which deserves to be studied in more
detail. It would be desirable to give a feedback about e.g. the triggering
efficiency flatness in the Dalitz plot coordinates to the neural networks
during the training process. This also holds for the background flatness in
critical distributions, such as the candidate invariant mass in order to avoid
peaking structures.

For the study presented here, the background suppression requirement is set
equally among the physics channels. However, this approach neglects the
differences in reconstruction and varying background levels in the different
channels. Finding an approach to achieve the best background suppression
requirements for each trigger line, while maintaining the desired overall
background reduction, is a complex challenge. Enhancing the triggering
efficiency of one channel will reduce the triggering efficiency of another
channel, which has to be carefully balanced in the future experiment.

###### Acknowledgements.

We acknowledge the internal review work of the accompanying analysis memo by
A. Belias, M. Papenbrock and T. Stockmanns, and we thank all members of the
PANDA Collaboration. This work has been supported by the Bundesministerium für
Bildung und Forschung (BMBF), Germany; the GSI Helmholtzzentrum fr̈
Schwerionenforschung GmbH, Germany; the Helmholtz Forschungsakademie Hessen
für FAIR (HFHF), Germany; the Helmholtz-Institut Mainz, Germany; the Institute
of Modern Physics, Chinese Academy of Sciences, China. And in particular, P.
Jiang has received financial support via the Helmholtz – OCPC Postdoc Program
and the Office of China Postdoc Council, Ministry of Human Resources and
Social Security, China.

## References

  * (1) P. Spiller and G. Franchetti, The FAIR accelerator project at GSI, NIMA, 556, 305-309 (2006)
  * (2) M. Destefanis for the PANDA Collaboration, The PANDA experiment at FAIR, Nuclear Physics B (Proc. Suppl.), 245, 199-206, (2013)
  * (3) A. Belias, FAIR status and the PANDA experiment, JINST 15 C10001 (2020)
  * (4) PANDA Collaboration, Physics Performance Report for PANDA: Strong Interaction Studies with anti-protons, FAIR/PANDA/Physics Book, 28-30 (2009)
  * (5) Barucca, G., Davì, F., Lancioni, G. et al. PANDA Phase One, Eur. Phys. J. A 57, 184 (2021)
  * (6) Particle Data Group, R L Workman, V D Burkert et al., Review of Particle Physics, PTEP 2022, 083C01 (2022)
  * (7) R. Ray, J. Rosen, M. Masuzawa et al., A trigger for the Fermilab E760 lead-glass calorimeter, NIM, A307, 254 (1991)
  * (8) The E835 Collaboration, High precision charmonium spectroscopy in anti-p p annihilations, Nucl. Phys., A655, 29-34 (1999)
  * (9) The ATLAS Collaboration, Performance of the ATLAS Trigger System in 2010, Eur. Phys. J. C, 72, 1-5 (2012)
  * (10) The BABAR Collaboration, The BABAR Detector, NIM, A479, 90-100 (2001)
  * (11) M. Ablikim, Z. H. An, J. Z. Bai et al., Design and Construction of the BESIII Detector. NIM, A614, 349 (2010)
  * (12) V. Khachatryan, A.M. Sirunyan, A. Tumasyan et al., The CMS trigger system, JINST, 12 P01020, 3-18 (2017)
  * (13) The LHCb Collaboration, The LHCb Detector at the LHC, JINST, 3 S08005, 151-166 (2008)
  * (14) C. Bozzi, S. Roiser on behalf of the LHCb Collaboration, The LHCb software and computing upgrade for Run 3: opportunities and challenges, J. Phys., Conf. Ser. 898 112002 (2017)
  * (15) S. Acharya et al., ALICE Collaboration, Comput. Phys. Commun. 242, 25-48 (2019)
  * (16) A. Lehrach, O. Boine-Frankenheim, F. Hinterberger et al., Beam performance and luminosity limitations in the high-energy storage ring (HESR), Nucl. Instrum. Meth. A 561, 289 (2006)
  * (17) D. J. Lange, The EvtGen particle decay simulation package, Nucl. Instrum. Meth. A 462, 152-155 (2001) doi:10.1016/S0168-9002(01)00089-4
  * (18) A. Capella, U. Sukhatme, C. -I. Tan et al., Dual Parton Model, Physics Reports, 236 225-329 (1994)
  * (19) A. J. Bevan et al., The Physics of the B Factories, Eur. Phys. J. C 74, 3026 (2014)
  * (20) S. Spataro for the PANDA Collaboration, The PandaRoot framework for simulation, reconstruction and analysis, J. Phys. Conf. Ser. 331 (2011).
  * (21) M. Al-Turany, D. Bertini, R. Karabowicz et al., The FairRoot framework, J. Phys.: Conf. Ser. 396 022001 (2012)
  * (22) CERN Program Library Long Writeup W5013, https://cds.cern.ch/record/1073159/files/cer-002728534.pdf, (1993, last retrieved Oct. 2022)
  * (23) A. Paszke, S. Gross, C. Soumith et al., Automatic differentiation in PyTorch, NIPS-W (2017)
  * (24) Y. LeCun, Y. Bengio and G. Hinton, Deep learning, Nature, 521, 436–444 (2015)
  * (25) S. Ruder, An overview of gradient descent optimizer, arXiv:1609.04747 [cs.LG] (2016)
  * (26) J. Han, C. Moraga, The influence of the sigmoid function parameters on the speed of backpropagation learning, IWANN, From Natural to Artificial Neural Computation, 195-201 (1995)
  * (27) V. Nair and G. Hinton, Rectified Linear Units Improve Restricted Boltzmann Machines, Proc. 27th ICML, 807-814 (2010)
  * (28) A. Maas, A. Hannun and A. Ng, Rectifier non-linearities improve NN acoustic Models, Proc. icml 30 (1), 3, (2013)
  * (29) D. Hendrycks and K. Gimpel, Gaussian Error Linear Units (GELUs), (2016)
  * (30) G. Klambauer, T. Unterthiner, A. Mayr et al., Self-normalizing neural networks, Proc. NIPS17, 972-981 (2017)
  * (31) X. Glorot, A. Bordes, Y. Bengio, Deep sparse rectifier neural networks, Proc. AISTATS, PMLR 15, 315-323 (2011)
  * (32) A. Krogh and J. Hertz, A Simple Weight Decay Can Improve Generalization, Advances in Neural Information Processing Systems, Morgan-Kaufmann, 4, (1991)
  * (33) X. Glorot, Y Bengio, Understanding the difficulty of training deep feedforward neural networks, Proc. ICAIS, PMLR 9, 249-256 (2010)
  * (34) J. Mockus, Bayesian approach to global optimisation: theory and applications, 79-116, Kluwer Academic, Dordrecht/Boston/London (2012)
  * (35) K. O’Shea and R. Nash, An Introduction to Convolutional Neural Networks, arXiv:1511.08458 [cs.NE] (2015)
  * (36) K. He, X. ßZhang, S. Ren et al., Deep Residual Learning for Image Recognition, CVPR, 770-778 (2016)
  * (37) S. Xie, R. Girshick, P. Dollar et al., Aggregated Residual Transformations for Deep Neural Networks, CVPR, 1492-1500 (2017)
  * (38) D. Balduzzi, M. Frean, L. Leary et al., The Shattered Gradients Problem: If resnets are the answer, then what is the question? Proc. ICML, PMLR 70, 342-350 (2017)
  * (39) M. Lin, Q. Chen, S. Yan, Network In Network, (2013)
  * (40) S. Hochreiter, J. Schmidhuber, Long short-term memory. Neural Computation, 9(8), 1735–1780 (1997)
  * (41) S. Ioffe, C. Szegedy, Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift, Proc. ICML, PMLR 37, 448-456 (2015)
  * (42) K. J. Peters, A Primer on partial wave analysis, Int. J. Mod. Phys. A21, 5618-5624 (2006)
  * (43) R. H. Dalitz, Decay of $\tau$ Mesons of Known Charge, Phys. Rev. 94, 1046-1051 (1954)