UPR-1319-T, CERN-TH-2022-073

Brill-Noether-general Limit Root Bundles:
Absence of vector-like Exotics in F-theory Standard Models

Martin Bies1,2, Mirjam Cvetič1,2,3,4, Ron Donagi2,1, Marielle Ong2

1Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104-6396, USA

2Department of Mathematics, University of Pennsylvania,
Philadelphia, PA 19104-6396, USA

3Center for Applied Mathematics and Theoretical Physics, University of
Maribor,
Maribor, Slovenia

4CERN Theory Department, CH-1211 Geneva, Switzerland

Root bundles appear prominently in studies of vector-like spectra of 4d
F-theory compactifications. Of particular importance to phenomenology are the
Quadrillion F-theory Standard Models (F-theory QSMs). In this work, we analyze
a superset of the physical root bundles whose cohomologies encode the vector-
like spectra for the matter representations $(\mathbf{3},\mathbf{2})_{1/6}$,
$(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$ and $(\mathbf{1},\mathbf{1})_{1}$.
For the family $B_{3}(\Delta_{4}^{\circ})$ consisting of
$\mathcal{O}(10^{11})$ F-theory QSM geometries, we argue that more than
$99.995\%$ of the roots in this superset have no vector-like exotics. This
indicates that absence of vector-like exotics in those representations is a
very likely scenario.

The QSM geometries come in families of toric 3-folds $B_{3}(\Delta^{\circ})$
obtained from triangulations of certain 3-dimensional polytopes
$\Delta^{\circ}$. The matter curves in $X_{\Sigma}\in B_{3}(\Delta^{\circ})$
can be deformed to nodal curves which are the same for all spaces in
$B_{3}(\Delta^{\circ})$. Therefore, one can probe the vector-like spectra on
the entire family $B_{3}(\Delta^{\circ})$ from studies of a few nodal curves.
We compute the cohomologies of all limit roots on these nodal curves.

In our applications, for the majority of limit roots the cohomologies are
determined by line bundle cohomology on rational tree-like curves. For this,
we present a computer algorithm. The remaining limit roots, corresponding to
circuit-like graphs, are handled by hand. The cohomologies are independent of
the relative position of the nodes, except for a few circuits. On these
_jumping circuits_ , line bundle cohomologies can jump if nodes are specially
aligned. This mirrors classical Brill-Noether jumps.
$B_{3}(\Delta_{4}^{\circ})$ admits a jumping circuit, but the root bundle
constraints pick the canonical bundle and no jump happens.

###### Contents

  1. 1 Introduction
  2. 2 Partial blowup limit root line bundles
    1. 2.1 Vector-like spectra in F-theory
    2. 2.2 Limit root bundles in the F-theory QSMs
    3. 2.3 Triangulation independence in the QSM geometries
    4. 2.4 The need for partial blowup limit root bundles
    5. 2.5 Beyond full blowup limit roots
  3. 3 Line bundle cohomology on tree-like, rational curves
    1. 3.1 Simplifying the tree
    2. 3.2 Global sections on terminal trees
    3. 3.3 Final algorithm
  4. 4 Towards Brill-Noether theory of limit root line bundles
    1. 4.1 The geometric multiplicity
    2. 4.2 Errata, corrections and immediate extensions of earlier work
    3. 4.3 Statistics of partial blowup limit roots in QSM geometries
  5. 5 $\mathcal{O}(10^{11})$ F-theory QSM without vector-like quark-doublets
    1. 5.1 Stationary circuits
    2. 5.2 Jumping circuit
    3. 5.3 Towards absence of vector-like exotics
  6. 6 Conclusion and Outlook
  7. A Speciality
  8. B Counts of partial blowup limit roots
    1. B.1 Spaces with $\overline{K}_{B_{3}}^{3}=6$
    2. B.2 Spaces with $\overline{K}_{B_{3}}^{3}=10$
  9. C Stationary circuits in $\mathcal{O}(10^{11})$ F-theory QSMs
  10. D Jumping circuit in $\mathcal{O}(10^{11})$ F-theory QSMs
    1. D.1 Ray generators
    2. D.2 Vanishing intersections and approximation of Stanley-Reisner ideal
    3. D.3 The nodal curve
    4. D.4 The location of the nodes

##  1 Introduction

String Theory is a leading candidate for a unified theory of quantum gravity
since it elegantly couples gauge dynamics to gravity. In particular, all
aspects of our physical reality, including the observed low energy particle
physics, must be accounted for by String Theory. Above all, an explicit
demonstration of the Standard Model from String Theory is desired.

Enormous efforts have been undertaken in this regard with perturbative String
Theory such as the $E_{8}\times E_{8}$ heterotic string [1, 2, 3, 4, 5, 6, 7,
8] or intersecting branes models in type II [9, 10, 11, 12, 13, 14, 15] (see
also [16] and references therein). These String Theory compactifications were
among the first from which the Standard Model gauge sector emerged with its
chiral or, in the case of [4, 5], vector-like spectrum. Typically, these
perturbative models suffer from chiral and vector-like exotic matter. The
first efforts to overcome this hurdle and provide a globally consistent MSSM
construction from String Theory are [4, 5] (see [17, 18] for more details on
the subtle global conditions for slope-stability of vector bundles).

A non-perturbative extension of type IIB String Theory must describe the gauge
dynamics on 7-branes _including_ their back-reactions (to all orders in the
string coupling) onto the compactification geometry $B_{n}$. This is achieved
elegantly in F-theory [19, 20, 21]. Namely, the back-reactions in question are
encoded in the geometry of an elliptically fibered Calabi-Yau space $\pi\colon
Y_{n+1}\twoheadrightarrow B_{n}$. Such a space $Y_{n+1}$ can be investigated
with well-established tools of algebraic geometry. This allows to study and
satisfy the global consistency conditions of the physics in $10-2n$ non-
compact dimension. For a modern overview of the recent developments towards
connecting F-theory and the Standard Model, we point the interested reader to
[22].

An important signature of the observed particle physics is the chiral
fermionic spectrum. A realistic 4d $\mathcal{N}=1$ F-theory compactifications
(i.e., $n=3$) must therefore reproduce this spectrum. From the F-theory
perspective, it is uniquely determined by a background gauge flux. The latter
can be conveniently specified by the internal $C_{3}$ profile in the dual
M-theory geometry. Importantly, the chiral spectrum of the F-theory
compactification only depends on $G_{4}=dC_{3}\in H^{(2,2)}(Y_{4})$. In recent
years, extensive toolkits have been designed to create and enumerate the
primary vertical subspace of $G_{4}$ configurations [23, 24, 25, 26, 27, 28,
29]. Applications of these tools to globally consistent chiral F-theory models
have been conducted in [30, 27, 29, 31]. Relatively recently, these efforts
culminated in the largest currently-known class of explicit string vacua that
realize the Standard Model gauge group with their exact chiral spectrum and
gauge coupling unification [32] \-- the _Quadrillion F-theory Standard Models_
(QSMs).

A 4d $\mathcal{N}=1$ F-theory compactification contains chiral (super)fields
in a representation $\mathbf{R}$ and chiral (super)fields in the charge
conjugate representation $\mathbf{\overline{R}}$. The difference between their
numbers is the chiral spectrum. Accounting for these fields individually is
achieved by the vector-like spectrum. This poses additional challenges as
those zero modes depend on the flat directions of the potential $C_{3}$ and
not only on $G_{4}=dC_{3}$. An F-theory ‘‘gauge field’’ that uniquely
specifies the vector-like spectrum is given by an element of the so-called
_Deligne cohomology_. The program initiated in [33, 34, 35] rests on the fact
that (a subset of) the Deligne cohomology can be parameterized by Chow
classes. This parametrization allows one to extract line bundles
$L_{\mathbf{R}}$ defined on curves $C_{\mathbf{R}}$ in $B_{3}$. In the dual
IIB picture, this can be interpreted as the localization of gauge flux on
matter curves $C_{\mathbf{R}}$, which can lift vector-like pairs on these
curves. Explicitly, the zero modes are counted by the sheaf cohomology groups
of the line bundle $L_{\mathbf{R}}$. There are
$h^{0}(C_{\mathbf{R}},\,{L}_{\mathbf{R}})$ massless chiral superfields in
representation $\mathbf{R}$. Similarly, there are
$h^{1}(C_{\mathbf{R}},\,{L}_{\mathbf{R}})$ massless chiral superfields in the
charge conjugate representation $\mathbf{\overline{R}}$ on $C_{\mathbf{R}}$.

In theory, this procedure always works. In practice however, technical
limitations arise since the line bundle cohomologies depend intricately on the
complex structure of $Y_{4}$. As a consequence, even state-of-the-art
algorithms, such as [36, 37, 38] (see also [34, 35]), on supercomputers
specifically designed for such computations (such as Plesken at Siegen
University) can oftentimes not perform the necessary operations in realistic
compactification geometries. This is exactly the reason why the works in [33,
34, 35] focused on computationally simple geometries. Even then, these models
have unrealistically large numbers of chiral fermions. Inspired by the
advances in machine learning applications to String Theory [39, 40, 41] (see
also [42] and references therein), the complex structure dependency of line
bundle cohomologies was investigated more systematically in [43]. Inspired by
the F-theory GUT models in [35] and the help of the algorithms in [36], it was
possible to generate a large data set [44] of line bundle cohomologies for
different complex structure moduli. This data was analyzed with data science
techniques. A theoretical understanding was achieved by supplementing these
data science insights by Brill-Noether theory [45] (see [46] for a modern
exposition and [47] for an earlier application of Brill-Noether theory in
F-theory). These insights led to a quantitative study of jumps of charged
matter vector pairs as a function of the complex structure moduli of the
matter curve $C_{\mathbf{R}}$.

A $G_{4}$-flux is subject to the quantization condition
$G_{4}+\frac{1}{2}c_{2}(Y_{4})\in H^{4}_{\mathbb{Z}}(Y_{4})$. Unfortunately,
this condition can often not be checked in a computationally feasible manner.
The interested reader is referred to [48] for recent advances in this
direction. _Quadrillion F-theory Standard Models_[32] evaluated necessary
conditions for a specific $G_{4}$-flux to satisfy the quantization condition.
In this work, we proceed under the assumption that the quantization condition
is satisfied. It is worth noting that the QSM $G_{4}$-flux candidate also
satisfies the D3-tadpole cancelation and masslessness of the $U(1)$-gauge
boson [32].

Therefore, the vector-like spectra in the QSMs beg to be investigated. Those
modes localize on five matter curves and were first studied in [49]. It was
shown that on three of these matter curves, the line bundle ${L}_{\mathbf{R}}$
is necessarily a fractional power $P_{\mathbf{R}}$ of the canonical bundle. On
the remaining two matter curves, the line bundles are modified by
contributions from the Yukawa points. Such fractional powers of line bundles
are known as root bundles, which may be thought of as generalizations of spin
bundles. Inspired by Ρίζα, which is ‘‘root" in Greek, $P$ refers to root
bundles throughout this article. Similar to spin bundles, root bundles are far
from unique. The mathematics of root bundles indicates that we should think of
the different root bundles as being induced from inequivalent gauge potentials
for a given $G_{4}$-flux.

In general, we cannot expect that all physically relevant root bundles on the
matter curve ${C}_{\mathbf{R}}$ are induced from F-theory gauge potentials in
the Deligne cohomology. This mirrors the expectation that only some of the
spin bundles on the matter curves are consistent with the F-theory geometry
$Y_{4}$. Rather, an F-theory gauge potential induces a collection
$\\{P_{\mathbf{R}}\\}$ consisting of one root bundle on each matter curve
$C_{\mathbf{R}}$. By repeating this for all physically relevant F-theory gauge
potentials, we should expect that in general only a proper subset of all root
bundles on $C_{\mathbf{R}}$ will be reached. Conversely, if we are given a
collection of root bundles $\\{P_{\mathbf{R}}\\}$ \-- one for each matter
curve -- then this collection can fail to stem from an F-theory gauge
potential. Namely, it is possible for one of the root bundles to not be
induced at all, or not in combination with one of the other roots.

While it is of great importance for the physics to determine which roots are
physical (i.e. induced from an F-theory gauge potential), this question is
also extremely challenging. For this reason, this question was not
investigated by [49] in favor of conducting a systematic study of all root and
spin bundles on the individual matter curves. This local bottom-up analysis
allowed us to name a root bundle and a spin bundle on all matter curves,
except for the Higgs curve in one particular QSM geometry [32], such that
their tensor product is a line bundle with the physically desired
cohomologies. On a technical level, this is achieved by considering a
deformation of the smooth, physical matter curves $C_{\mathbf{R}}$ into a
nodal curve $C^{\bullet}_{\mathbf{R}}$. On the latter, root bundles are
described in a diagrammatic way by _limit roots_ [50]. For limit root bundles
on the full blowup of $C^{\bullet}_{\mathbf{R}}$, the global sections can be
counted by the techniques put forward in [49].

This study was subsequently extended in [51] by a computer algorithm which
enumerates all root bundles with exactly three global sections on the full
blowup of $C^{\bullet}_{\mathbf{R}}$. This in turn was repeated for
$\mathcal{O}(10^{15})$ different QSM geometries. The latter rests on the fact
that the QSM geometries are defined by families of toric spaces
$B_{3}(\Delta^{\circ})$ associated to the (full, regular, star) triangulations
of 708 (3-dimensional, reflexive) polytopes $\Delta^{\circ}$ in the Kreuzer-
Skarke list [52] (see also [53]). One may think of these spaces as different
desingularizations of toric K3-surfaces [54, 55, 56, 57, 58]. These well-known
results allowed to argue that the nodal curve $C^{\bullet}_{\mathbf{R}}$ \--
and thus, the number of limit root bundles -- only depends on the polytope
$\Delta^{\circ}$. Hence, by repeating a somewhat involved computer scan for 23
different polytopes, one finds results that apply to a large fraction of the
QSM geometries. This allowed, in a first approximation and statistical
meaning, to find base spaces for which absence of vector-like exotics in the
representation $(\mathbf{3},\mathbf{2})_{1/6}$ is very likely.

##### Results

In this work, we analyze a superset of all physical root bundles for the
matter representations $(\mathbf{3},\mathbf{2})_{1/6}$,
$(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$ and $(\mathbf{1},\mathbf{1})_{1}$
in $\mathcal{O}(10^{11})$ F-theory QSM geometries $B_{3}(\Delta_{4}^{\circ})$.
We find that more than $99.995\%$ of the roots in this superset have no
vector-like exotics. We arrived at this result from a systematic study of a
large fraction of QSM geometries. While our results are strongest for the
family $B_{3}(\Delta_{4}^{\circ})$, this study provides statistical evidence
that the absence of the vector-like exotics in
$(\mathbf{3},\mathbf{2})_{1/6}$, $(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$
and $(\mathbf{1},\mathbf{1})_{1}$ is a very likely scenario for a large
fraction of the QSM geometries.

As mentioned above, the vector-like spectra in the QSM geometries are
necessarily counted by cohomologies of root bundles[49]. Such root bundles
$P_{\mathbf{R}}$ on the matter curve $C_{\mathbf{R}}$ are by no means unique.
Even more, it is not clear exactly which ones are induced top-down from
F-theory gauge potentials in the Deligne cohomolgy. First, on one matter
curve, only a subset of all mathematically allowed root bundles could be
induced from all physically allowed F-theory gauge potentials. Secondly, it is
conceivable that fluxes which induce a specific root bundle on matter curve
$C_{1}$, only induce a few selected root bundles on another matter curve
$C_{2}$. In this work, we have not addressed this involved question, but
rather conducted a local and bottom-up analysis. Our study is bottom-up in
that we study _all_ root bundles that could possibly be induced from the
$G_{4}$-flux and all spin bundles on the matter curve in question. Our study
is local in that we focus on one matter curve at a time. Correlations among
the vector-like spectra of different matter curves are not taken into account.

A crucial step in our study is triangulation independence. Specifically, each
family $B_{3}(\Delta^{\circ})$ of toric 3-fold QSM base spaces -- obtained
from the (fine, regular, star) triangulations of 708 (3-dimensional,
reflexive) polytopes $\Delta^{\circ}$ in the Kreuzer-Skarke list [52] admits a
triangulation-independent nodal limit [51]. This observation allows one to
probe the vector-like spectra on the entire family $B_{3}(\Delta^{\circ})$
from the vector-like spectra on just 5 nodal curves. For the family
$B_{3}(\Delta_{4}^{\circ})$ \-- associated to the 4-th polytope in the
Kreuzer-Skarke list of 3-dimensional polytopes -- which consists of
$\mathcal{O}(10^{11})$ different toric spaces, we argue that all limit root
bundles on the nodal curves $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$,
$C_{(\mathbf{\overline{3}},\mathbf{1})_{-2/3}}^{\bullet}$ and
$C_{(\mathbf{1},\mathbf{1})_{1}}^{\bullet}$ have exactly three global
sections. By upper semi-continuity, this must hold true on the corresponding
smooth, irreducible matter curves in each space of
$B_{3}(\Delta_{4}^{\circ})$. This rules out vector-like exotics.

Apparently, this argument relies on the ability to compute the cohomologies of
all limit root bundles. This is exactly the domain of Brill-Noether theory. By
following the original work in [50], we find that for some limit roots, this
is equivalent to line bundle cohomology on tree-like curves. The remaining
limit roots require an understanding of line bundle cohomology on circuit-like
curves. In our applications to the F-theory QSMs, the majority of limit roots
arise from tree-like curves with rational irreducible components. We present a
computer algorithm which counts line bundle cohomology on such curves and
handle the circuit-like cases by hand. On tree-like, rational curves, the
cohomologies are independent of the relative position of the nodes. While true
for most circuit-like cases, special circuits admit jumps provided that the
nodes are specially aligned. Such _jumping circuits_ mirror classical Brill-
Noether jumps. For $B_{3}(\Delta_{4}^{\circ})$, one jumping circuit exists.
The root bundle constraints select the canonical bundle on this jumping
circuit so that we find four global sections. This leads to $99.995\%$ instead
of $100\%$.

In contrast to earlier works [49, 51], we take the _geometric multiplicity_
into account [50]. Under a deformation of a smooth to a nodal curve, distinct
root bundles on the smooth curve coalesce as we approach the nodal limit. The
exact number of distinct roots which coalesce is the geometric multiplicity.
Alternatively, this multiplicity ensures that we count as many limit root
bundles on nodal curves as root bundles on a smooth curve. This ensures
comparability of the limit root counts for different QSM geometries and
slightly modifies the conclusions in [51].

##### Outline

In Section 2, we briefly recall the motivation for root bundles in F-theory
vacua [49]. Subsequently, we outline the steps necessary to extend the
techniques of [49, 51] to limit roots on partial blowup curves, which are
compactly summarized in Figure 3, Figure 4 and Figure 5. This approach makes
it natural to think about line bundle cohomology on tree-like nodal curves. We
put forward an algorithm that determines these line bundle cohomologies on
rational curves in Section 3. This algorithm can be optimized further to
detect speciality, i.e. if a line bundle has more sections than on a smooth
$\mathbb{P}^{1}$. While this optimization currently seems not relevant for
studying Brill-Noether theory of limit root bundles or for arguing the absence
of vector-like exotics in $\mathcal{O}(10^{11})$ QSM geometries, we outline
these steps in Appendix A. With these techniques, we return to the QSM
geometries and re-analyze the setups that were originally discussed in [51].
In Section 4, we first discuss the geometric multiplicity and compare our
results to [51]. Indeed, we obtain more refined results, which also identify
two errors in the earlier work. We summarize our findings in Table 3 and Table
4. We propose to view these tables as a first approximation of the Brill-
Noether theory of the root bundles in question. More details can be found in
Appendix B. We also mention that the necessary computer implementations are
available in the gap-4-package _QSMExplorer_ as part of the
ToricVarieties$\\_$project [36]. Many, but definitely not all, of these
computations can be conducted on a simple personal computer. Finally in
Section 5, we focus on the family of QSM geometries
$B_{3}(\Delta_{4}^{\circ})$ associated to full regular star triangulations of
the 4-th polytope $\Delta_{4}^{\circ}$ in the Kreuzer-Skarke database [52]. By
hand, we analyze the global sections on a couple of circuit-like nodal curves.
For most of them, the line bundle cohomologies are independent of the relative
positions of the nodes (cf. Appendix C). However, there is one circuit-like
nodal curve for which four nodes are located on two irreducible components. By
studying the embedding of this nodal curve into each space in
$B_{3}(\Delta_{4}^{\circ})$, we can show that the location of all nodes is
determined in terms of the input parameters to the QSMs. In particular, we
argue that the root bundle constraints select the canonical bundle on this
jump circuit, which leads to exactly four (and not three) global sections. The
detailed study of this embedding is conducted in Appendix D. We summarize and
discuss our findings in Section 6.

##  2 Partial blowup limit root line bundles

###  2.1 Vector-like spectra in F-theory

The story begins with an F-theory compactification to four dimensions given by
a singular, elliptically fibered 4-fold $\pi:Y_{4}\twoheadrightarrow B_{3}$.
Suppose that $Y_{4}$ admits a smooth, crepant resolution which is a flat
elliptic fibration $\widehat{\pi}:\widehat{Y}_{4}\twoheadrightarrow B_{3}$. A
$G_{4}$-flux in $H^{(2,2)}(\widehat{Y}_{4})$ satisfies the quantization
condition precisely if [59]

$\displaystyle G_{4}+\frac{1}{2}c_{2}(T_{\widehat{Y}_{4}})\in
H^{(2,2)}_{\mathbb{Z}}(\widehat{Y}_{4}):=H^{(2,2)}(\widehat{Y}_{4})\cap
H^{4}(\widehat{Y}_{4},\mathbb{Z})\,.$ (1)

For F-theory compactifications on an elliptically fibered smooth Calabi-Yau
4-fold with globally defined Weierstrass model, the class
$c_{2}(T_{\widehat{Y}_{4}})$ is even [60]. Consequently in such geometries, we
demand $G_{4}\in H^{(2,2)}_{\mathbb{Z}}(\widehat{Y}_{4})$. Unfortunately, this
condition can often not be checked in a computationally feasible manner. The
interested reader is referred to [48] for recent advances in this direction.
_Quadrillion F-theory Standard Models_[32] evaluated necessary conditions for
a specific $G_{4}$-flux to satisfy the quantization condition. In this work,
we proceed under the assumption that the quantization condition is satisfied.
It is worth noting that the QSM $G_{4}$-flux candidate also satisfies the
D3-tadpole cancelation and masslessness of the $U(1)$-gauge boson [32].

There is a surjection from the Deligne cohomology group

$\displaystyle\widehat{c}:H^{4}_{D}(\widehat{Y}_{4},\mathbb{Z}(2))\twoheadrightarrow
H^{(2,2)}_{\mathbb{Z}}(\widehat{Y}_{4})\,,$ (2)

and a $G_{4}$-flux lifts to an F-theory gauge potential $A$ in the Deligne
cohomology group $H^{4}_{D}(\widehat{Y}_{4},\mathbb{Z}(2))$. Although this
group is intractable when it comes to explicit computations, we can
parametrize a subset of it via the map

$\displaystyle\widehat{\gamma}:\mathrm{CH}^{2}(\widehat{Y}_{4},\mathbb{Z})\rightarrow
H^{4}_{D}(\widehat{Y}_{4},\mathbb{Z}(2))\,.$ (3)

Thus, we turn our attention to studying
$\mathcal{A}\in\mathrm{CH}^{2}(\widehat{Y}_{4},\mathbb{Z})$ satisfying
$(\widehat{c}\circ\widehat{\gamma})(\mathcal{A})=G_{4}$.

For some representation R of the gauge group $G$ of the F-theory
compactification in question, let $C_{\textbf{R}}\subseteq B_{3}$ be the
matter curve upon which the massless matter states localize and transform in
R. The corresponding matter surface $S_{\mathbf{R}}\subseteq\widehat{Y}_{4}$
\-- the mathematicians will notice that $S_{\mathbf{R}}$ is actually a 2-cycle
and not a surface -- is a linear combination of $\mathbb{P}^{1}$-fibration
over $C_{\textbf{R}}$ and the coefficients correspond to the weight vector
$\mathbf{w}$ of the representation $\mathbf{R}$ (see [61] and references
therein). By restricting to $S_{\mathbf{R}}$ and integrating over the fibers
over $C_{\textbf{R}}$, we can associate to each
$\mathcal{A}\in\mathrm{CH}^{2}(\widehat{Y}_{4},\mathbb{Z})$ a divisor via the
cylinder map

$\displaystyle
D_{\textbf{R}}:\mathrm{CH}^{2}(\widehat{Y}_{4},\mathbb{Z})\rightarrow\mathrm{CH}^{1}(C_{\textbf{R}},\mathbb{Z})\cong\operatorname{Pic}(C_{\textbf{R}})\,.$
(4)

Specifically, we consider $\iota_{S_{\mathbf{R}}}\colon
S_{\mathbf{R}}\hookrightarrow\widehat{Y}_{4}$ and $\pi_{S_{\mathbf{R}}}\colon
S_{\mathbf{R}}\twoheadrightarrow C_{\mathbf{R}}$. Then the cylinder map is
given by

$\displaystyle
D_{\mathbf{R}}\left(\mathcal{A}\right)=\pi_{S_{\mathbf{R}}\ast}\left(\iota_{S_{\mathbf{R}}}^{\ast}\left(\mathcal{A}\right)\right)\in\mathrm{Pic}\left(C_{\mathbf{R}}\right)\,.$
(5)

As explained in [33, 34, 35], the divisor
$D_{\mathbf{R}}\left(\mathcal{A}\right)$ gives rise to the following line
bundle

$\displaystyle
P_{\mathbf{R}}\left(\mathcal{A}\right)=\mathcal{O}_{C_{\mathbf{R}}}\left(D_{\mathbf{R}}\left(\mathcal{A}\right)\right)\otimes_{\mathcal{O}_{C_{\mathbf{R}}}}\mathcal{O}_{C_{\mathbf{R}}}^{\text{spin}}\,,$
(6)

where $\mathcal{O}_{C_{\mathbf{R}}}^{\text{spin}}$ is a spin bundle on
$C_{\mathbf{R}}$ compatible with the global structure of the F-theory
compactification on $\widehat{Y}_{4}$, whose sheaf cohomologies encode the
vector-like spectrum on $C_{\mathbf{R}}$, i.e.

$\displaystyle\begin{split}h^{0}\left(C_{\mathbf{R}},P_{\mathbf{R}}\left(\mathcal{A}\right)\right)&\quad\leftrightarrow\quad\text{chiral
zero modes in representation $\mathbf{R}$}\,,\\\
h^{1}\left(C_{\mathbf{R}},P_{\mathbf{R}}\left(\mathcal{A}\right)\right)&\quad\leftrightarrow\quad\text{chiral
zero modes in charge conjugate representation
$\overline{\mathbf{R}}$}\,.\end{split}$ (7)

The generic case
$h^{0}\left(C_{\mathbf{R}},P_{\mathbf{R}}\left(\mathcal{A}\right)\right)=\chi(P_{\mathbf{R}}\left(\mathcal{A}\right))$
corresponds to the absence of exotic vector-like pairs. The situation
$h^{0}\left(C_{\mathbf{R}},P_{\mathbf{R}}\left(\mathcal{A}\right)\right)=\chi(P_{\mathbf{R}}\left(\mathcal{A}\right))+1$,
which implies the existence of exactly one Higgs pair, features prominently as
the desired vector-like spectrum on the Higgs curve of an F-theory MSSM
construction.

###  2.2 Limit root bundles in the F-theory QSMs

In [51], we followed this logic to study the vector-like spectra in the
largest currently-known class of globally-consistent F-theory Standard Model
constructions without chiral exotics and gauge coupling unification [32]. The
geometry of the compactification space of those Quadrillion F-theory Standard
Models (F-theory QSMs) determines a class
$\mathcal{A}^{\prime}\in\mathrm{CH}^{2}(\widehat{Y}_{4},\mathbb{Z})$ and an
integer $\xi\in\mathbb{Z}_{>0}$ such that
$D_{\textbf{R}}(\mathcal{A}^{\prime})$ is -- possibly with additional
contributions from the Yukawa points -- expressible in terms of the canonical
divisor on $C_{\textbf{R}}$ , the integer $\xi$ divides the degree of
$D_{\textbf{R}}(\mathcal{A}^{\prime})$ and $\mathcal{A}$ is subject to two
constraints

$\displaystyle(\widehat{c}\circ\widehat{\gamma})(\mathcal{A})=G_{4},\quad\xi\cdot
D_{\textbf{R}}(\mathcal{A})\sim D_{\textbf{R}}(\mathcal{A}^{\prime})$ (8)

The condition on the right of Equation 8 says that
$D_{\textbf{R}}(\mathcal{A})$ is a $\xi$-th root divisor of
$D_{\textbf{R}}(\mathcal{A}^{\prime})$ and the associated line bundle
$P_{\mathbf{R}}\left(\mathcal{A}\right)$ is a $\xi$-th root bundle on
$C_{\textbf{R}}$. To count the zero modes in the F-theory QSMs, we are
therefore led to study the sheaf cohomologies of root bundles.

An important task is to find the root bundles that are induced from $F$-theory
gauge potentials. First, on one matter curve, only a subset of all
mathematically allowed root bundles could be induced from all physically
allowed F-theory gauge potentials. Secondly, it is conceivable that fluxes
which induce a specific root bundle on matter curve $C_{1}$, only induce a few
selected root bundles on another matter curve $C_{2}$. The study of these
questions is involved and currently beyond our abilities. While we hope to
return to these questions in future work, in the current work we opt for a
local and bottom-up analysis instead. In this sense, we follow the spirit of
the earlier works [49, 51]. Our study is bottom-up in that we study _all_
mathematically allowed root bundles. Our study covers all root bundles that
could possibly be induced from the $G_{4}$-flux and all spin bundles on the
matter curve in question. Our study is local in that we focus on one matter
curve at a time. Correlations among the vector-like spectra of different
matter curves are therefore not taken into account.

Consequently, we study all root bundles $P_{\mathbf{R}}$ on $C_{\textbf{R}}$
\-- including all spin bundles on $C_{\textbf{R}}$. Since $C_{\textbf{R}}$ is
smooth and the integer $\xi$ divides the degree of
$D_{\textbf{R}}(\mathcal{A}^{\prime})$, there are $\xi^{2g}$ root bundles
where $g$ is the genus of $C_{\textbf{R}}$. Unfortunately, computing their
sheaf cohomology is difficult. In particular, it is not easy to distinguish
between trivial and non-trivial root bundles in general. To overcome this
challenge, we consider deformations of $C_{\textbf{R}}$ to a reducible, nodal
curve $C^{\bullet}_{\textbf{R}}$ by modifying the defining polynomials of
$C_{\textbf{R}}$ in a concrete base geometry $B_{3}$. This comes at a double
cost.

First, the deformation $C^{\bullet}_{\textbf{R}}\to C_{\textbf{R}}$ is, per se
only subject to upper semicontinuity of global sections. This means, that we
may lose global sections that exist on $C^{\bullet}_{\textbf{R}}$ as we deform
back to the smooth, irreducible $C_{\textbf{R}}$. However, if the number of
global sections match the chiral index, then it is protected by topology and
cannot change. We exploited this fact in [49] and will mimic this strategy in
later part of this work.

Second, the $\xi$-th root bundles of a line bundle $L$ on a reducible nodal
curve $C$ do not exist if $\xi$ does not divide the degree of the restriction
of $L$ to the irreducible components of $C$. Thanks to the work of [50], this
is elegantly circumvented by passing to limit root bundles
$P^{\circ}_{\textbf{R}}$ on (partial) blow-ups $C^{\circ}_{\textbf{R}}$, whose
existence only requires for $\xi$ to divide $\deg(L_{\textbf{R}})$. By
properly keeping track of multiplicities, there exist $\xi^{2g}$ limit roots,
each determined by the combinatorial data associated to the dual graph of
$C^{\bullet}_{\textbf{R}}$. Fortunately, it turns out that there are less than
$2\times 708$ different dual graphs that arise in the context of the QSMs.
This observation allows us to gain insights into a large fraction of these
geometries.

###  2.3 Triangulation independence in the QSM geometries

To understand why there are only $2\times 708$ different dual graphs to
considered, we briefly recall the genesis of the F-theory QSM geometries,
specifically the base spaces in question. These geometries are obtained from
desingularizations of toric K3-surfaces. Such desingularizations were first
studied in [54]. It is well-established by now that they correspond to three-
dimensional, reflexive lattice polytopes $\Delta\subset M_{\mathbb{R}}$ and
their polar duals $\Delta^{\circ}\subset N_{\mathbb{R}}$ defined by
$\left\langle\Delta,\Delta^{\circ}\right\rangle\geq-1$. In [52], Kreuzer and
Skarke listed all possible 3-dimensional polytopes [52]. Just as in [51], we
consider the i-th polytope in the Kreuzer-Skarke list as subset of
$N_{\mathbb{R}}$ and denote it by $\Delta_{i}^{\circ}$.

Recall that to each polytope $\Delta\subset M_{\mathbb{R}}$, one can associate
a normal fan $\Sigma_{\Delta}$. The ray generators of this fan are the facet
normals of $\Delta$ and the maximal cones correspond to the vertices of
$\Delta$. Even though the toric variety $X_{\Delta}\equiv X_{\Sigma_{\Delta}}$
or the CY-hypersurface may not be smooth, we can resolve these hypersurfaces.
Such resolutions were introduced in [54] as _maximal projective crepant
partial_ (MPCP) desingularizations. Equivalently, [56] refers to them as
maximal projective subdivisions of the normal fan. MPCPs are associated to
fine, regular, star triangulation (FRST) of the lattice polytope
$\Delta^{\circ}$. We recall that _star_ means that every simplex in the
triangulation contains the origin, _fine_ ensures that every lattice point of
$\Delta^{\circ}$ is used as ray generator111This property is also referred to
as _full_[62]. and that _regular_ implies that $X_{\Sigma(T)}$ is projective.
Together, this implies that $\Sigma(T)$ defines a _maximal_ projective
refinement of $\Sigma_{\Delta}$. Note that in applications to toric
K3-surfaces, $X_{\Sigma(T)}$ is guaranteed to be smooth. Namely, a maximal
projective subdivision of $\Sigma_{\Delta}$ then constructs a 3-dimensional
Gorenstein orbifold with terminal singularities [56] which must be smooth by
proposition 11.4.19 in [63]. For more details, we refer the interested reader
to [54].

708 of the 4319 polytopes in [52] satisfy
$\overline{K}_{X_{\Sigma}(T)}^{3}\in\\{6,10,18,30\\}$. These are exactly the
polytopes whose geometries define the base spaces for the Quadrillion F-theory
Standard Models (QSMs) [32]. Just as in [51], we use $B_{3}(\Delta^{\circ})$
to denote the toric 3-folds obtained from FRSTs of the polytope
$\Delta^{\circ}$. An instructive example is $\Delta_{52}^{\circ}$, which we
display in Figure 1.

Figure 1: $\Delta_{52}^{\circ}\subset N_{\mathbb{R}}$ on the left and
$\Delta_{52}\subseteq M_{\mathbb{R}}$ on the right [52]. The magenta point is
the origin. The generic K3-surface meets trivially with the gray divisors, in
an irreducible curve with the pink ones, and in finite families of
$\mathbb{P}^{1}$s with the cyans. This picture is taken from [51].

For each QSM geometry, we introduced a nodal quark-doublet curve in [49]. This
curve is canonically associated with the family $B_{3}(\Delta_{4}^{\circ})$.
We can write this curve as

$\displaystyle
C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}=\bigcup\limits_{a\in
A}{V(x_{a},s_{9})}\cup\bigcup\limits_{b\in
B}{V(y_{b},s_{9})}\cup\bigcup\limits_{c\in C}{V(z_{c},s_{9})}\,.$ (9)

We explained the rationale behind this notation in [51]. Namely, one can show
that $V(x_{a},s_{9})$ is irreducible, $V(y_{b},s_{9})$ a finite collection of
$\mathbb{P}^{1}$s and $V(z_{c},s_{9})=\emptyset$. This rests on arguments
originally presented in [54, 55, 56, 57, 64].

This classification is directly related to the geometry of the polytope
$\Delta^{\circ}$. Namely, the homogeneous coordinates are one-to-one with the
lattice points of $\Delta^{\circ}$ (minus the origin) since we use these as
rays of the refined fan. A homogeneous coordinate associated to a facet
interior point of $\Delta^{\circ}$ is denoted by $z_{c}$. For a lattice point
in the interior of an edge $\Theta^{\circ}\subset\Delta^{\circ}$, two facets
$F_{1}$, $F_{2}$ of $\Delta^{\circ}$ meet at $\Theta^{\circ}$. In the dual
polytope $\Delta$, they correspond to vertices $m_{1},m_{2}$ and the dual edge
$\Theta$ is the edge connecting $m_{1}$ and $m_{2}$. If $\Theta$ has interior
points, we denote the homogeneous coordinate by $y_{b}$. All remaining
coordinates are indicated as $x_{a}$. In Figure 1, distinct types of lattice
points are marked in different colors.

Consequently, all irreducible components of
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ are smooth, irreducible curves
embedded into a smooth and projective K3-surface. This alone implies, and
equivalently follows from [54, 55, 56, 57, 64], that the pairwise topological
intersection numbers are non-negative. Furthermore, we can restrict to open
subsets near intersection points. Since an open subset of an irreducible
topological space is irreducible, it follows that the multiplicity of
intersection at each intersection point is non-negative. Consequently, for
generic $s_{9}$, the number of pairwise intersection points among the
irreducible components of $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$
matches the topological intersection number. This is how we concluded in [51]
that the dual graph of $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ is the
same for all FRSTs of $\Delta^{\circ}$. This leads to at most 708 distinct
dual graphs. In fact, one finds that distinct polytopes in the QSM database
can lead to identically the same dual graphs. So, there are strictly less than
708 distinct dual graphs for the canonical nodal quark-doublet curve
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ in the QSMs. There are two
matter curves with distinct topology, namely the Higgs curve
$C_{(\mathbf{1},\mathbf{2})_{-1/2}}^{\bullet}$ and a curve supporting right-
handed quarks $C_{(\mathbf{\overline{3}},\mathbf{1})_{1/3}}^{\bullet}$.
However, their topologies are identical, contributing to at most 708 more
distinct dual graphs. In this work, we will ignore those two curves and focus
on the curve with dual graph identical to that of
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$.

In Appendix D, we refine this result for the polytope $\Delta_{4}^{\circ}$. We
will explain in Section 2.5, that the counting of limit root bundles on
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ only depends on some irreducible
components and the nodes on these components. For $\Delta_{4}^{\circ}$, we
therefore focus on four components $C_{0}$, $C_{1}$, $C_{2}$, $C_{3}$. Our
arguments show that there are homogeneous coordinates for these four
irreducible components such that, in these coordinates, the location of the
nodes is the same for all FRSTs of $\Delta_{4}^{\circ}$. This in turn is
critical for our analysis in Section 5.2, where we study the one jumping
circuit associated to $\Delta_{4}^{\circ}$. To the best of our knowledge, this
FRSTs independence of the location of the nodes does not follow from earlier
works [54, 55, 56, 57, 64]. At this point, our argument rests on a detailed
computational analysis which can, in principle, be extended to all QSM spaces.
One can speculate that such an extension would lead to the very same
conclusion, namely triangulation independent positions of the relevant nodes.
This extension is beyond the scope of the current work and we leave this study
for future investigation.

###  2.4 The need for partial blowup limit root bundles

In order to approximate physical root bundle on the smooth matter curve
$C_{(\mathbf{3},\mathbf{2})_{1/6}}$, we thus employ limit root bundles on the
nodal curve $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$. As we explained
before, the dual graph of $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ is
independent of the FRSTs. This extends to certain topological intersection
numbers and implies that the operations performed on basis of [50], which
construct and count limit root line bundles $P^{\bullet}_{\textbf{R}}$ on
(partial) blow-ups of $C^{\bullet}_{\textbf{R}}$ from combinatorial data, are
independent of the FRSTs.

In [49], we compute the zeroth sheaf cohomology of all limit root bundles that
arise from the full blow-up of $C^{\bullet}_{\textbf{R}}$. We refer to this
work for additional background for the application of limit root bundles
towards F-theory QSMs. In particular, we explained that the sheaf cohomology
of such limit root bundles is given by the following formula,

$\displaystyle
h^{0}(C^{\circ}_{\textbf{R}},P^{\circ}_{\textbf{R}})=\sum_{Z}h^{0}(Z,P^{\circ}_{\textbf{R}}|_{Z}),$
(10)

where $Z$ is an irreducible component of $C^{\bullet}_{\textbf{R}}$. Since the
pushforward of the limit roots $P^{\circ}_{\textbf{R}}$ along the blow-up map
$\pi:C^{\circ}_{\textbf{R}}\rightarrow C^{\bullet}_{\mathbf{R}}$ preserves the
number of global sections, we also have that

$\displaystyle
h^{0}(C^{\circ}_{\textbf{R}},P^{\circ}_{\textbf{R}})=h^{0}(C^{\bullet}_{\textbf{R}},\pi_{*}P^{\circ}_{\textbf{R}}).$
(11)

Along the deformation of $C^{\bullet}_{\textbf{R}}$ back to the original curve
$C_{\textbf{R}}$, the number of sections either decreases or remains constant
as the dimension of the zeroth cohomology is an upper semi-continuous
function, i.e.

$\displaystyle h^{0}(C_{\textbf{R}},P_{\textbf{R}})\leq
h^{0}(C^{\bullet}_{\textbf{R}},\pi_{*}P^{\circ}_{\textbf{R}}).$ (12)

In this work, we aim to establish the absence of vector-like exotics for
_physical_ roots. Since we cannot yet tell which roots are physical, we cover
all (mathematical) roots and identify their sheaf cohomologies. With an eye
towards the nodal quark-doublet curve
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$, we would ideally find that all
of its roots have exactly three global sections. Then, by upper semicontinuity
this remains true as we deform back to the smooth, irreducible (and physical)
curve $C_{(\mathbf{3},\mathbf{2})_{1/6}}$. But for this argument to work, we
need to extend to all roots, including those on partial blowups. This requires
an extension of the techniques in [51]. As a side effect of these extensions,
we shall present -- what we believe are the very first -- results on Brill-
Noether theory of limit root bundles.

###  2.5 Beyond full blowup limit roots

We show the generalizations necessary to go from full to partial blowups in an
example that will be relevant in later part of this work. Namely, we focus on
the family of F-theory QSMs defined in terms of the fourth polytope
$\Delta_{4}^{\circ}$ in the Kreuzer-Skarke database [52]. There are
$\mathcal{O}(10^{11})$ such geometries [53] and by following [49, 51], each
contains a canonical nodal curve $C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$
with dual graph displayed in Figure 2(a). The edges in this graph represent
nodal singularities and the vertices are irreducible components of
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$. The four curves $C_{i}$ each
are smooth $\mathbb{P}^{1}$, which are marked in pink in the graph. In
addition, $L_{i_{1},...,i_{k}}$ represents a chain of $k$ smooth genus zero
curves $C_{i_{j}}$ such that $C_{i_{j}}$ and $C_{i_{j+1}}$ intersect each
other at a single node. Hence, as dual graph, $L_{i_{1},...,i_{k}}$
corresponds to a chain or link of $k$ vertices. It is ordered so that
$C_{i_{1}}$ intersects the $\mathbb{P}^{1}$ on the left and $C_{i_{k}}$
intersects the $\mathbb{P}^{1}$ on the right. For $L_{21,27}$, we have that
$C_{21}$ intersects $C_{3}$ while $C_{27}$ intersects $C_{1}$.

##### Simplifying the dual graph

On this curve, we are looking for $12$-th roots $P^{\bullet}$ of the $12$-th
power of the canonical bundle $K^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$.
To this end, we can replace the above dual graph by another graph, for which
the combinatorics task of enumerating the limit root bundles is simpler. To
see this, we consider the following:

$C_{0}$$C_{1}$$\dots$$C_{n-1}$$C_{n}$ (13)

In [51], we argued that when constructing limit roots on the full blow-up of a
chain of curves, the weights along the chain in the weighted subgraph are
uniquely fixed. Thus, we were able to replace these chains by a single edge
and we would still count the same number of weighted subgraphs and hence,
limit root bundles. Indeed, this is the only weighted subgraph which satisfies
the rules (C1) and (C2) in [50]. Namely, a curve $C$ with dual graph
$\Gamma_{C}$ has $r^{b_{1}(\Gamma_{C})}$ weighted subgraphs satisfying (C1)
and (C2). If $C$ is a chain of rational curves, then

$\displaystyle
b_{1}(\Gamma_{C})=\\#\left(\text{edges}\right)+\\#\left(\text{connected
components}\right)-\\#\left(\text{vertices}\right)=0\,.$ (14)

Hence, there is only $1=r^{0}$ weighted subgraph, which corresponds to the
full blow-up. Consequently, we can transition to the simpler dual graph
displayed in Figure 2(b).

$L_{16,13}$$L^{0}_{24,28}$$L^{1}_{24,28}$$L_{8,7,6,5,4}$$L_{25,17}$$L_{21,27}$$L_{14,9}$$h^{0}=2$$C_{0}$$h^{0}=3$$C_{1}$$h^{0}=3$$C_{2}$$h^{0}=2$$C_{3}$
(a) Dual graph of nodal curve.

$C_{0}$$C_{1}$$C_{2}$$C_{3}$ (b) Dual graph of simplified nodal curve.

Figure 2: Dual graph of (simplified) nodal curve
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$ in the QSM spaces
$B_{3}(\Delta_{4}^{\circ})$.

##### Global section counting of partial blowup limit roots

$C_{0}$$C_{1}$$C_{2}$$C_{3}$$d=1$$d=1$$d=1$$d=1$$d=1$$d=1$$d=1$ (a) Full
blowup of nodal curve.

$C_{0}$$C_{1}$$C_{2}$$C_{3}$ (b) Disconnected curve from removing nodes.

Figure 3: For full blowup limit roots, each node is replaced by an exceptional
$\mathbb{P}^{1}$. The global sections are isomorphic to those of a line bundle
on the disconnected curve.

In order to find $12$-th roots $P^{\bullet}$ of the $12$-th power of the
canonical bundle $K^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$ and count their
global sections, we blew up every node in our earlier works [49, 51]. In
following [50, 49, 51] we first focus on limit roots $P^{\bullet}$ that
correspond to a certain line bundle on the full blowup curve of
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$. This full blowup curve is a
curve $C^{\circ}_{(\mathbf{3},\mathbf{2})_{1/6}}$ in which each node is
replaced by an exceptional $\mathbb{P}^{1}$. Its dual graph is therefore of
the form displayed in Figure 3(a). In this diagram, we have marked the
exceptional $\mathbb{P}^{1}$s in blue color. Furthermore, it holds
$\left.P\right|_{E_{i}}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)$ for each
exceptional $\mathbb{P}^{1}$ by [50]. The (local) sections on the exceptional
$\mathbb{P}^{1}$s are uniquely fixed by the demand to bridge between the
sections on the components $C_{i}$ of
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$. Therefore, we called these
sections _bridging sections_ in [49]. The number of global sections of
$P^{\bullet}$ is therefore identical to those of the restriction of
$P^{\bullet}$ to the completely disconnected curve in Figure 3(b). In
particular, it holds

$\displaystyle
h^{0}(C^{\bullet},P^{\bullet})=\sum_{i=0}^{3}{h^{0}(C_{i},\left.P^{\bullet}\right|_{C_{i}})}\,.$
(15)

$C_{0}$$C_{1}$$C_{2}$$C_{3}$ (a) No blowup on red edge.

$C_{0}$$C_{1}$$C_{2}$$C_{3}$$d=1$$d=1$$d=1$$d=1$$d=1$$d=1$ (b) Partial blowup.

$C_{0}$$C_{1}$$C_{2}$$C_{3}$ (c) Tree-like nodal curve.

Figure 4: For partial blowup limit roots, some nodes are not replaced by an
exceptional $\mathbb{P}^{1}$. In some cases, global sections are then counted
on tree-like nodal curves.

##### Tree-like and circuit-like graphs

In order to generalize this procedure to partial blowups, let us first look at
the situation in which we decide not to blow up the edge marked in red in
Figure 4(a). It is in general not guaranteed that such a partial blowup admits
limit roots. However, if limit roots $P^{\bullet}$ exist, then they correspond
to line bundles on the nodal curve in Figure 4(b). Again, the sections on the
exceptional $\mathbb{P}^{1}$s bridge between the sections on neighboring
components $C_{i}$. However, since one node -- namely the one corresponding to
the red edge -- has not been blown up, the number of global sections of
$P^{\bullet}$ match that of the restriction of $P^{\bullet}$ to the nodal
curve with dual graph displayed in Figure 4(c). There, $C_{2}\cup C_{3}$ is a
nodal, tree-like curve. If we are able to compute its line bundle cohomology,
then we have

$\displaystyle
h^{0}(C^{\bullet},P^{\bullet})=h^{0}\left(C_{0},\left.P^{\bullet}\right|_{C_{0}}\right)+h^{0}\left(C_{1},\left.P^{\bullet}\right|_{C_{1}}\right)+h^{0}\left(C_{2}\cup
C_{3},\left.P^{\bullet}\right|_{C_{2}\cup C_{3}}\right)\,.$ (16)

Apparently, this generalizes Equation 15.

In our applications, we find that for the majority of limit roots, the number
of global sections is the same as the number of global sections of a line
bundle on a rational, tree-like nodal curve. For example, 99% of the limit
root bundles on $\Delta_{4}^{\circ}$ arise in this way. Consequently, in order
to classify the limit root bundles with regard to their number of global
sections, it becomes rather important for us to compute line bundle cohomology
of line bundles on tree-like nodal curves. We will therefore discuss this at
length in Section 3.

All remaining root bundles stem from circuit-like graphs. For example, suppose
that we do not blow-up the two nodes marked in red in Figure 5(a), then this
leads to the computation of line bundle cohomology on the nodal curve with
dual graph displayed in Figure 5(b). This curve has a circuit-like component
$C_{1}\cup C_{2}$, which we cannot yet treat systematically.

Fortunately, in our current applications, these circuits account for very few
limit root bundles. The reason why seems related to the fact that we are
looking for $12$-th and even $20$-th roots of the canonical bundle. If
instead, we were looking at second roots, one should expect equal
contributions from tree-like and circuit-like graphs. It would be even
conceivable that the circuits would dominate. However, in our applications,
this is not the case. Therefore, we can conduct a case-by-case study for those
few circuit-like curves that we encounter. We will come back to this in
Section 5.1 and Section 5.2 and distinguish two types of circuits. First,
there are circuits for which the number of global sections is the same
irrespective of the position of the nodes. We call them _stationary circuits_.
On the other hand, circuits for which the number of global sections depends on
the (relative) location of the nodes are called _jumping circuits_. For these,
it is possible to achieve more than the minimal number of global sections upon
rearranging the position of the nodes.

$C_{0}$$C_{1}$$C_{2}$$C_{3}$ (a) No blowup on red edges.

$C_{0}$$C_{1}$$C_{2}$$C_{3}$ (b) Circuit-like nodal curve.

Figure 5: Some choices of partial blowups lead to limit roots whose
cohomologies are counted by line bundles on circuit-like nodal curves.

##  3 Line bundle cohomology on tree-like, rational curves

###  3.1 Simplifying the tree

Consider a connected rational nodal curve $C:=\bigcup_{i\in I}{C_{i}}$ and a
line bundle $L$ on it. Then, $I$ is a simply connected graph and all
components $C_{i}$ are $\mathbb{P}^{1}$’s.

######  Definition 3.1.

Let $d_{i}=\deg(L|_{C_{i}})$. Write $I=I_{+}\cup I_{-}$, where

$\displaystyle I_{-}:=\left\\{i\in I|d_{i}<0\right\\}\,,\qquad
I_{+}:=\left\\{i\in I|d_{i}\geq 0\right\\}\,.$ (17)

Let

$\displaystyle C_{+}:=\bigcup_{i\in I_{+}}{C_{i}}\,,\qquad
C_{-}:=\bigcup_{i\in I_{-}}{C_{i}}\,.$ (18)

Note that the curves $C_{+}$, $C_{-}$ could be disconnected. Let $e_{i}$ be
the intersection number of $C_{i}$ with $C_{-}$ and let $L_{+}$ be the line
bundle on $C_{+}$ with $\deg(L_{+}|_{C_{i}})=d_{i}-e_{i}$.

######  Proposition 3.2.

$h^{0}(C,L)=h^{0}(C_{+},L_{+})$.

###### Proof.

Consider a connected component $Y=\cup Y_{i}$ of $C_{+}$. Suppose that $e_{i}$
connected components $Z_{j}$ of $C_{-}$ intersect $Y_{i}$. Since $C$ is
connected and rational, $Y_{i}$ intersects each $Z_{j}$ at exactly one point
$n_{j}$. Hence, $e_{i}$ is exactly the number of connected components of
$C_{-}$ that meet $Y_{i}$. The connected components of $C_{-}$ support only
the trivial section, which automatically glues across the intersections
between the components of $Z_{j}$. Hence, after choosing local coordinates, we
can parametrize the local sections as follows:

(22)

Without loss of generality, we can assume that the nodes are at the following
positions:

(25)

Each intersection between $Y_{i}$ and $Z_{j}$ forces the local sections on
$Y_{i}$ to vanish at $n_{j}$. This amounts to exactly one equation of linear
dependence on $h^{0}(Y_{i},L|_{Y_{i}})$, namely

$\displaystyle\sum^{d_{i}}_{k=0}\alpha_{k}a_{j}^{k}b_{j}^{d_{i}-k}=0,$ (26)

If $d_{i}\geq e_{i}$, then

$\displaystyle
h^{0}\Bigl{(}Y_{i}\cup\Bigl{(}\bigcup^{e_{i}}_{j=1}Z_{j}\Bigr{)},L\Bigr{)}=h^{0}(Y_{i},L|_{Y_{i}})-e_{i}=h^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(d_{i}-e_{i}))=h^{0}(Y_{i},L_{+}|_{Y_{i}}).$
(27)

If $d_{i}<e_{i}$, then the number of equations of linear dependence imposed by
the $e_{i}$ intersections exceeds $h^{0}(Y_{i},L|_{Y_{i}})$. Hence,

$\displaystyle
h^{0}\Bigl{(}Y_{i}\cup\Bigl{(}\bigcup^{e_{i}}_{j=1}Z_{j}\Bigr{)},L\Bigr{)}=0,$
(28)

which coincides with
$h^{0}(Y_{i},L_{+}|_{Y_{i}})=h^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(d_{i}-e_{i}))$.
Counting the sections of $L$ over $C$ is the same as counting the sections
over $C_{+}$ and then, enforcing the gluing conditions at $C_{+}\cap C_{-}$.
Hence, $h^{0}(C,L)=h^{0}(C_{+},L_{+})$. ∎

######  Note.

By iterating this process we have

$\displaystyle
h^{0}\left(C,L\right)=h^{0}\left(C_{+},L_{+}\right)=\dots=h^{0}(C_{+}^{(n)},L_{+}^{(n)})\,,$
(29)

In fact, since $(C_{+},L_{+})$ has at most as many components as $C$ this
process must eventually become “constant”. Let us illustrate this with two
examples.

######  Example 3.3.

We first perform the following simplification:

$-2$$1$$2$$0$$2$ (30)

The latter cannot be simplified further.

######  Example 3.4.

We perform the following simplifications:

$1$$1$$1$$-2$$-1$$-1$$-1$$-1$$-1$$1$$0$ (31)

The latter cannot be simplified further.

######  Definition 3.5.

Let $C$ be a tree-like nodal (but not necessarily connected) curve and
$L\in\operatorname{Pic}(C)$. We call a pair $(C,L)$ _terminal_ iff $C_{+}\cap
C_{-}=\emptyset$.

######  Consequence.

Once we know how to compute global sections on a terminal tree-like pair
$(C,L)$, we can employ the following algorithm to compute the global sections
on a connected nodal pair $(C,L)$:

  1. 1.

By iterating the simplification process, we have:

$\displaystyle
h^{0}\left(C,L\right)=h^{0}(C_{+},L_{+})=\dots=h^{0}(C_{+}^{(n)},L_{+}^{(n)})\,,$
(32)

with terminal $(C_{+}^{(n)},L_{+}^{(n)})$.

  2. 2.

Compute $h^{0}(C_{+}^{(n)},L_{+}^{(n)})$.

###  3.2 Global sections on terminal trees

We will now turn to line bundle cohomology on terminal trees. Recall that such
terminal trees have $C_{+}\cap C_{-}=\emptyset$. Since $C_{-}$ does not
support global sections, we can focus on just $C_{+}$. We begin with the
following lemma.

######  Lemma 1.

Consider a curve $C=\mathbb{P}^{1}_{a}\cup\mathbb{P}^{1}_{b}$ with
$\mathbb{P}^{1}_{a}\cap\mathbb{P}^{1}_{b}=\left\\{n\right\\}$ a nodal
singularity:

$\mathbb{P}^{1}_{a}$$\mathbb{P}^{1}_{b}$ (33)

On this curve, consider the line bundle

$\displaystyle\mathcal{L}=\mathcal{O}_{C}(d_{a},d_{b})\in\mathrm{Pic}(C)\,.$
(34)

That is, we consider
$\mathcal{O}_{\mathbb{P}^{1}_{a}}(d_{a})\in\mathrm{Pic}(\mathbb{P}^{1}_{a})$
and
$\mathcal{O}_{\mathbb{P}^{1}_{b}}(d_{b})\in\mathrm{Pic}(\mathbb{P}^{1}_{b})$
subject to gluing at the node $n$. Assume $d_{a},d_{b}\geq 0$. Then

$\displaystyle
h^{0}(C,L)=h^{0}\left(\mathbb{P}^{1}_{a},\mathcal{O}_{\mathbb{P}^{1}_{a}}(d_{a})\right)+h^{0}\left(\mathbb{P}^{1}_{b},\mathcal{O}_{\mathbb{P}^{1}_{b}}(d_{b})\right)-1=d_{a}+d_{b}+1\,.$
(35)

###### Proof.

We pick the following local coordinates and parametrize the local sections by
$\alpha_{i},\beta_{i}\in\mathbb{C}$:

(39)

Without loss of generality, we can assume that the node is at the following
position:

(42)

Enforcing gluing at $n_{1}$ is equivalent to $\alpha_{d_{a}}=\beta_{d_{b}}$.
Hence it follows:

$\displaystyle
h^{0}(C,L)=h^{0}(\mathbb{P}^{1}_{a},\mathcal{O}_{\mathbb{P}^{1}_{a}}(d_{a}))+h^{0}(\mathbb{P}^{1}_{b},\mathcal{O}_{\mathbb{P}^{1}_{b}}(d_{b}))-1=d_{a}+d_{b}+1\,.$
(43)

This completes the argument. ∎

It is not too hard to generalize this result.

######  Lemma 2.

Consider a _connected_ rational nodal curve $C:=\bigcup_{i\in I}{C_{i}}$ and
$L$ a line bundle on it. (This means that $I$ is a simply connected graph and
the $C_{i}$ are $\mathbb{P}^{1}$’s.) Let $d_{i}$ be the degree of the
restriction of $L$ to $C_{i}$. Suppose that $d_{i}\geq 0$ for all $i\in I$.
Then

$\displaystyle h^{0}(C,L)=d+1\,,\qquad d=\sum_{i\in I}{d_{i}}\,.$ (44)

###### Proof.

We prove this statement by induction on $|I|$. If $|I|=1$, then the statement
is obviously true.

For the induction step, consider a connected, rational curve $C$ with $n$
components. Assume that our statement is true for $C$. By adding one component
$C_{n+1}$ to $C$, we obtain a new curve $C^{\prime}$. Assume that $C^{\prime}$
is tree-like, i.e. $C_{n+1}$ intersects exactly one component of $C$ in
exactly one point. Without loss of generality, let us denote this component as
$C_{n}$ and $p=C_{n}\cap C_{n+1}$.

The global sections on $C_{n}$ correspond to a subset $S\subseteq
H^{0}(C_{n},\mathcal{O}_{C_{n}}(d_{n}))$ on $C_{n}$. Let us glue these
sections to $H^{0}(C_{n+1},\mathcal{O}_{C_{n+1}}(d_{n+1}))$ at $p$. Without
loss of generality, we make the following choices:

(48)

Choose $p=V(x)$ on $C_{n}$ and $p=V(u)$ on $C_{n+1}$. Let us now assume that
no gluing condition in $p$ was required. This would mean that for any
$\varphi\in S$ and $\psi\in H^{0}(C_{n+1},\mathcal{O}_{C_{n+1}}(d_{n+1}))$,
they satisfy

$\displaystyle\varphi(p)=\psi(p)\,.$ (49)

However, since $u^{d_{n+1}},v^{d_{n+1}}\in
H^{0}(C_{n+1},\mathcal{O}_{C_{n+1}}(d_{n+1}))$, this would imply that any
$\varphi\in S$ satisfies

$\displaystyle\varphi(p)=u^{d_{n+1}}(p)=0\,,\qquad\varphi(p)=v^{d_{n+1}}(p)=1\,.$
(50)

This cannot be satisfied in $\mathbb{C}$. Hence, there is at least one
condition to be imposed at $p$. However, since $p$ imposes at most one
condition, it imposes exactly one condition. Recall that by assumption,

$\displaystyle h^{0}(C,L)=d+1\,,\qquad d=\sum_{i\in I}{d_{i}}\,.$ (51)

Consequently, we find for $C^{\prime}$ that:

$\displaystyle
h^{0}(C^{\prime},L)=h^{0}(C,L)+h^{0}(C_{n+1},L)-1=d+1+(d_{n+1}+1)-1=\left(d+d_{n+1}\right)+1\,.$
(52)

This completes the argument. ∎

######  Corollary 3.6.

Consider a rational nodal curve $C:=\bigcup_{i\in I}{C_{i}}$ and a line bundle
$L$ on it. Suppose that $C$ has $k$ connected components and the restrictions
of $L$ to $C_{i}$ have degrees $d_{i}\geq 0$. Then

$\displaystyle h^{0}(C,L)=\Bigl{(}\sum_{i\in I}{d_{i}}\Bigr{)}+k\,.$ (53)

We are now in the position to formulate the result which allows one to count
the global sections on any terminal tree.

######  Proposition 3.7.

Let $(C,L)$ be terminal and $C_{+}$ have $k$ connected components. Then,

$\displaystyle h^{0}(C,L)=\Bigl{(}\sum_{i\in I_{+}}{d_{i}}\Bigr{)}+k\,.$ (54)

###### Proof.

Since $(C,L)$ is terminal, $C_{+}\cap C_{-}=\emptyset$ and

$\displaystyle
h^{0}(C,L)=h^{0}(C_{+},L|_{C_{+}})+h^{0}(C_{-},L|_{C_{-}})=h^{0}(C_{+},L|_{C_{+}})\,.$
(55)

Now, let $e$ be the number of intersections between curves in $C_{+}$. Then,
$e=|I_{+}|-k$ and

$\displaystyle h^{0}(C,L)=\Bigl{(}\sum_{i\in
I_{+}}{(d_{i}+1)}\Bigr{)}-e=\Bigl{(}\sum_{i\in
I_{+}}{d_{i}}\Bigr{)}+|I_{+}|-e=\Bigl{(}\sum_{i\in I_{+}}{d_{i}}\Bigr{)}+k\,.$
(56)

This completes the argument. ∎

###  3.3 Final algorithm

######  Algorithm 1.

Consider a connected, rational nodal curve $C:=\bigcup_{i\in I}{C_{i}}$ and a
line bundle $L$ on it. Then, the following algorithm computes $h^{0}(C,L)$.

  1. 1.

Simplify $(C,L)\to(C_{+},L_{+})\to\dots(C_{+}^{(n)},L_{+}^{(n)})$ with
terminal $(C_{+}^{(n)},L_{+}^{(n)})$.

  2. 2.

Then $h^{0}(C,L)=d_{+}\left(L_{+}^{(n)}\right)+c_{+}\left(C_{+}^{(n)}\right)$,
where

     * •

$d_{+}(L_{+}^{(n)})=\sum_{i\in I_{+}(C_{+}^{(n)}}{d_{i}}$.

     * •

$c_{+}(C_{+}^{(n)})$ is the number of connected components of $C_{+}^{(n)}$.

######  Note.

Since $C_{+}$ has less or as many components as $C$, this algorithm must
terminate after a finite number of iterations. Let us illustrate this
algorithm in a couple of examples.

######  Example 3.8.

We first perform the following simplification:

$-2$$1$$2$$0$$2$ (57)

The latter is terminal and we have

$\displaystyle h^{0}(C,L)=(0+2)+1=3\,.$ (58)

######  Example 3.9.

We perform the following simplifications:

$1$$1$$1$$-2$$-1$$-1$$-1$$-1$$-1$$1$$0$ (59)

The latter is terminal and we have $h^{0}(C,L)=0+1=1$.

##  4 Towards Brill-Noether theory of limit root line bundles

We now turn back to the QSM geometries. Our task is to enumerate all
physically relevant limit root bundles on the nodal quark-doublet curve of the
QSM geometries. Subsequently, we want to identify their number of global
sections. To this end, we first briefly explain the geometric multiplicity
(see [50] for more details) and then turn towards the counts relevant for the
QSMs.

###  4.1 The geometric multiplicity

Recall that every line bundle $L$ with degree divisible by $r$ has $r^{2g}$
roots on a smooth curve of genus $g$. For a nodal curve $C^{\bullet}$, the
root bundle construction in [50] yields less than $r^{2g}$ limit roots.
Indeed, let $\Gamma_{C}$ be the dual graph of $C^{\bullet}$ with edge set
$E(\Delta)$ and vertex set $V(\Delta)$. Fix a weighted subgraph $\Delta^{w}$.
Its edges are the nodes at which we blow up and so, it determines a partial
normalization
$\widetilde{\pi}_{\Delta^{w}}:\widetilde{C}_{\Delta^{w}}\rightarrow
C^{\bullet}$ with dual graph $\Gamma_{C}\setminus E(\Delta^{w})$ and partial
blow-up $C^{\circ}_{\Delta^{w}}$. Its weights $u_{i},v_{i}$ determine a line
bundle $\widetilde{\pi}^{*}_{\Delta_{w}}(L)(-\sum_{i}(u_{i}p_{i}+v_{i}q_{i}))$
on $\widetilde{C}_{\Delta^{w}}$ of which we can take the $r$th root
$\widetilde{P}_{\Delta^{w}}\in\operatorname{Pic}(\widetilde{C}_{\Delta^{w}})$,
i.e.

$\displaystyle(\widetilde{P}_{\Delta^{w}})^{\otimes
r}\cong\widetilde{\pi}^{*}_{\Delta_{w}}(L)\Bigl{(}-\sum_{n_{i}\in
E(\Delta^{w})}(u_{i}p_{i}+v_{i}q_{i})\Bigr{)}.$ (60)

Degree 1 line bundles on the exceptional components are then glued to
$\widetilde{P}_{\Delta^{w}}$ to obtain a limit root $P^{\circ}_{\Delta^{w}}$
on $C_{\Delta^{w}}^{\circ}$. To count the limit roots associated to
$\Delta^{w}$, consider the tower of normalizations:

$\displaystyle\nu:C^{\nu}\xrightarrow{\nu_{\Delta^{w}}}\widetilde{C}_{\Delta^{w}}\xrightarrow{\widetilde{\pi}_{\Delta^{w}}}C^{\bullet},$
(61)

where $\nu_{\Delta^{w}}$ is the normalization of $\widetilde{C}_{\Delta^{w}}$
and $C^{\nu}$ is the full normalization of $C^{\bullet}$ with geometric genus
$g^{\nu}$ and arithmetic genus $g(C^{\nu})=g^{\nu}+1-|V(\Delta)|$. The other
arithmetic genera can be expressed in terms of $g^{\nu}$ as:

$\displaystyle g(C^{\bullet})$
$\displaystyle=g(C^{\nu})+|E(\Delta)|=g^{\nu}+b_{1}(\Gamma_{C}).$ (62)
$\displaystyle g(\widetilde{C}_{\Delta^{w}})$
$\displaystyle=g(C^{\nu})+|E(\Delta)\setminus
E(\Delta^{w})|=g^{\nu}+b_{1}(\Gamma_{C}\setminus E(\Delta^{w})).$ (63)

The two genera above are related to each other as

$\displaystyle g(C^{\bullet})$
$\displaystyle=g(\widetilde{C}_{\Delta^{w}})+|E(\Delta^{w})|=\sum_{i\in
I}g(X_{i})+b_{1}(\Sigma_{C}),$ (64)

where $X_{i}$ is a _connected_ component of $\widetilde{C}_{\Delta^{w}}$ and
$\Sigma_{C}$ is the dual graph whose vertices are $X_{i}$ and whose edges are
the exceptional components. Taking pullbacks induces the composition

$\displaystyle\nu^{*}:\operatorname{Pic}(C^{\bullet})\xrightarrow{\widetilde{\pi}^{*}_{\Delta^{w}}}\operatorname{Pic}(\widetilde{C}_{\Delta^{w}})\xrightarrow{\nu^{*}_{\Delta^{w}}}\operatorname{Pic}(C^{\nu}).$
(65)

Each root $\widetilde{P}_{\Delta^{w}}$ on $\widetilde{C}_{\Delta^{w}}$
corresponds to a root on the full normalization with a choice of gluings at
the nodes that were not blown up. Since there are $r^{2g^{\nu}}$ roots of
$\nu_{\Delta^{w}}^{*}\widetilde{\pi}^{*}(L)(-\sum_{i}(u_{i}p_{i}+v_{i}q_{i}))$
on $C^{\nu}$ and $r^{b_{1}(\Gamma_{C}\setminus E(\Delta^{w}))}$ choices of
gluings, every $\Delta^{w}$ is associated to
$r^{2g^{\nu}+b_{1}(\Gamma_{C}\setminus E(\Delta^{w}))}$ limit $r$th roots.
Summing over the $r^{b_{1}(\Gamma_{C})}$ weighted subgraphs gives

$\displaystyle\sum_{\Delta^{w}}r^{2g^{\nu}+b_{1}(\Gamma_{C}\setminus
E(\Delta^{w}))}$ (66)

limit $r$th roots. To get $r^{2g}$ limit $r$th roots, we need to multiply each
summand with a prefactor called the _geometric multiplicity_ in [50], namely

$\displaystyle\mu_{\Delta^{w}}=r^{b_{1}(\Gamma_{C})-b_{1}(\Gamma_{C}\setminus
E(\Delta^{w}))}=r^{|E(\Delta^{w})|}=r^{b_{1}(\Sigma_{C})}.$

One can interpret this geometrically by saying that a single limit root line
bundle corresponds to several root bundles on the corresponding smooth,
irreducible curve. In [50], $\mu_{\Delta^{w}}$ is the geometric multiplicity
of the connected component of the moduli space of limit roots supported at a
given limit root (Lemma 4.1.1). Taking the geometric multiplicity into
account, we do indeed get $r^{2g}$ limit roots because

$\displaystyle\sum_{\Delta^{w}}(r^{2g^{\nu}+b_{1}(\Gamma_{C}\setminus
E(\Delta^{w}))}\cdot\mu_{\Delta_{w}})=\sum_{\Delta^{w}}r^{2g^{\nu}+b_{1}(\Gamma_{C})}=r^{b_{1}(\Gamma_{C})}\cdot
r^{2g^{\nu}+b_{1}(\Gamma_{C})}=r^{2g}.$ (67)

In [51], only the blow-ups at the full set of nodes were considered, i.e.
$E(\Delta^{w})=E(\Delta)$. Hence, $\mu_{\Delta^{w}}=r^{b_{1}(\Gamma_{C})}$ in
that case. The first equality of Equation 67 shows that each $\Delta^{w}$ is
associated to $r^{2g^{\nu}+b_{1}(\Gamma_{C})}$ limit roots when the geometric
multiplicity is taken into account. Since this quantity does not depend on the
weighted subgraph, we will define this specially.

######  Definition 4.1.

The number of limit roots associated to each weighted subgraph $\Delta^{w}$ is

$\displaystyle\mu:=r^{2g^{\nu}+b_{1}(\Gamma_{C})}\,.$ (68)

We conclude this section by studying a simple example taken from [50]. To this
end, consider two curves $C_{1}$, $C_{2}$ with general genera $g_{1}$, $g_{2}$
(marked in dark pink) and assume that they touch in exactly 3 nodes. The goal
is to identify all 3rd limit roots of the structure sheaf. Consequently, we
look at the following dual graph:

00 (69)

This graph has $b_{1}(\Gamma_{C})=2$. We list all limit roots in Table 1. In
particular, we find a total number of roots:

$\displaystyle\begin{split}N_{\text{total}}&=1\cdot r^{0}\cdot
r^{2g_{\nu}+2}+6\cdot r^{2}\cdot r^{2g_{\nu}}+2\cdot r^{2}\cdot
r^{2g_{\nu}}=(1+6+2)\cdot\mu=3^{2\left(2+g_{\nu}\right)}\,.\end{split}$ (70)

Since $g(C)=2+g_{\nu}=2+g_{1}+g_{2}$, this is exactly the expected number.

$N$ Weighted diagram $\Delta^{w}$ $\mu_{\Delta^{w}}$ $N_{P}$ 3 00 $r^{2-2}$
$r^{\left(2g_{1}+2g_{2}\right)+2}$ 2 inconsistent \-- \-- 1
211200122100211200122100211200122100 $r^{2-0}$
$r^{\left(2g_{1}+2g_{2}\right)+0}$ 0 2121210012121200 $r^{2-0}$
$r^{\left(2g_{1}+2g_{2}\right)+0}$

Table 1: All limit roots on a nodal curve example taken from [50]. The
geometric multiplicity is given by
$\mu_{\Delta^{w}}=r^{b_{1}(\Gamma_{C})-b_{1}(\Gamma_{C}\setminus
E(\Delta^{w}))}$. The number of nodes left from the blow-up is denoted by $N$.
The number of limit roots is denoted by
$N_{P}=r^{2g_{\nu}+b_{1}(\Gamma_{C}\setminus E(\Delta^{w}))}$.

###  4.2 Errata, corrections and immediate extensions of earlier work

By taking the geometric multiplicity into account, we can refine the
statistics derived in the earlier publication [51]. There, lower bounds
$\check{N}^{(3)}_{P}$ on the number of limit root bundles with $h^{0}=3$ were
derived. By taking the geometric multiplicity into account, we can refine
these results. Our findings are listed in Table 2. Note that we have grouped
the spaces $\Delta^{\circ}_{128}$, $\Delta^{\circ}_{130}$,
$\Delta^{\circ}_{136}$, $\Delta^{\circ}_{236}$ together. This is because the
dual graphs are identical for these spaces. We point out a typo in [51] for
the result of $\Delta^{\circ}_{387}$. The correct result is highlighted in
green color in Table 2. There is also an incorrect result for
$\Delta^{\circ}_{856}$ in this earlier publication. The dual graph of
$\Delta^{\circ}_{856}$ is identical to that of $\Delta^{\circ}_{882}$, so the
lower bounds $\check{N}^{(3)}_{P}$ must agree for both spaces. A detailed
check shows that the results for $\Delta^{\circ}_{856}$ coincide with the
numbers listed for $\Delta^{\circ}_{882}$ in [51].

This brief study shows that roughly 50% of all limit root bundles are obtained
from a full blowup and have exactly three global sections. The largest
percentage is found for $\Delta_{88}$ (61.1%). In [51], the geometric
multiplicity was not taken into account. These earlier counts falsely led us
to the conclusion that for the polytope $\Delta_{8}$, the largest fraction of
full blowup roots with three global sections were found. However, as listed
above, the actual percentage is only 57.3% which is strictly smaller than the
result for $\Delta_{88}$.

Before we continue, let us point out that the observations from this initial
study are subject to the following subtleties. First, there are more limit
root bundles, resulting from partial blowups. Second, for physics
applications, we are eventually interested in physical roots. We will discuss
the first point momentarily and turn to the second in Section 5.

|  | $\check{N}^{(3)}_{P}$ | $\mu$ | $N_{\text{total}}$ |  $\check{N}^{(3)}_{P}\cdot\mu/N_{\text{total}}$ [%]
---|---|---|---|---|---
$\overline{K}_{B_{3}}^{3}=6$ | $\Delta^{\circ}_{8}$ | $142560$ | $12^{3}$ | $12^{8}$ | $57.3$
$\Delta^{\circ}_{4}$ | $11110$ | $12^{4}$ | $12^{8}$ | $53.6$
$\Delta^{\circ}_{134}$ | $10100$ | $12^{4}$ | $12^{8}$ | $48.7$
$\Delta^{\circ}_{128}$, $\Delta^{\circ}_{130}$, $\Delta^{\circ}_{136}$, $\Delta^{\circ}_{236}$ | $8910$ | $12^{4}$ | $12^{8}$ | $42.0$
$\overline{K}_{B_{3}}^{3}=10$ | $\Delta^{\circ}_{88}$ | $781.680.888$ | $20^{5}$ | $20^{12}$ | $61.1$
$\Delta^{\circ}_{110}$ | $738.662.983$ | $20^{5}$ | $20^{12}$ | $57.8$
$\Delta^{\circ}_{272}$, $\Delta^{\circ}_{274}$ | $736.011.640$ | $20^{5}$ | $20^{12}$ | $57.5$
$\Delta^{\circ}_{387}$ | $733.798.300$ | $20^{5}$ | $20^{12}$ | $57.3$
$\Delta^{\circ}_{798}$, $\Delta^{\circ}_{808}$, $\Delta^{\circ}_{810}$, $\Delta^{\circ}_{812}$ | $690.950.608$ | $20^{5}$ | $20^{12}$ | $54.0$
$\Delta^{\circ}_{254}$ | $35.004.914$ | $20^{6}$ | $20^{12}$ | $54.7$
$\Delta^{\circ}_{302}$ | $34.908.682$ | $20^{6}$ | $20^{12}$ | $54.7$
$\Delta^{\circ}_{52}$ | $34.980.351$ | $20^{6}$ | $20^{12}$ | $54.7$
$\Delta^{\circ}_{786}$ | $32.860.461$ | $20^{6}$ | $20^{12}$ | $51.3$
$\Delta^{\circ}_{762}$ | $32.858.151$ | $20^{6}$ | $20^{12}$ | $51.3$
$\Delta^{\circ}_{417}$ | $32.857.596$ | $20^{6}$ | $20^{12}$ | $51.3$
$\Delta^{\circ}_{838}$ | $32.845.047$ | $20^{6}$ | $20^{12}$ | $51.3$
$\Delta^{\circ}_{782}$ | $32.844.379$ | $20^{6}$ | $20^{12}$ | $51.3$
$\Delta^{\circ}_{377}$, $\Delta^{\circ}_{499}$, $\Delta^{\circ}_{503}$ | $30.846.440$ | $20^{6}$ | $20^{12}$ | $48.2$
$\Delta^{\circ}_{1348}$ | $30.845.702$ | $20^{6}$ | $20^{12}$ | $48.2$
$\Delta^{\circ}_{882}$, $\Delta^{\circ}_{856}$ | $30.840.098$ | $20^{6}$ | $20^{12}$ | $48.2$
$\Delta^{\circ}_{1340}$ | $28.954.543$ | $20^{6}$ | $20^{12}$ | $45.2$
$\Delta^{\circ}_{1879}$ | $28.950.852$ | $20^{6}$ | $20^{12}$ | $45.2$
$\Delta^{\circ}_{1384}$ | $27.178.020$ | $20^{6}$ | $20^{12}$ | $42.5$
Table 2: The results of [51] listed an incorrect result for
$\Delta^{\circ}_{856}$ and displayed a typo for $\Delta^{\circ}_{387}$. We fix
both and refine the earlier findings by use of the geometric multiplicity.

###  4.3 Statistics of partial blowup limit roots in QSM geometries

Let us employ these techniques to the nodal curves relevant for the QSMs. To
this end we employ the algorithms available in gap-4-package _QSMExplorer_.
The latter is part of the ToricVarieties$\\_$project [36]. Before we list the
results, let us discuss the limitations:

  1. 1.

Not all partial blowups come from tree-like subgraphs. Whenever this is the
case, the technology of Section 3 does not apply. Indeed, such configurations
matter for the QSM geometries but their number is typically small compared to
the tree-like cases. This includes circuits which admit a jump provided that
the nodes are moved in special alignment. We will analyze curves of this type
in more detail in Section 5. At this point, we cannot treat such curves
systematically, but rather rely on a case-by-case study. Therefore, our
algorithm merely provides a lower bound in such instances.

  2. 2.

If the nodal quark-doublet curve contains at least one elliptic curve, then we
face the following limitations for partial blowup root bundles. First, if the
elliptic curve contains one node that is not blown-up, then the technology of
Section 3 does not apply. Otherwise, if there is a $d=0$ line bundle on an
elliptic curve and we are looking for $r$-th roots, then we know that
$r^{2}-1$ of the $r^{2}$ roots are guaranteed to have no global sections. The
number of global sections on the remaining root depends on whether the
original bundle was trivial or not. Distinguishing this is highly non-trivial
and definitely goes beyond the topological data considered in our computer
scan. Thus, in those cases we provide a lower bound.

If we face one of these two cases, then we can only provide a lower bound to
the number of global sections. In particular, in non-generic setups, the
actual number of global sections could be strictly larger than this lower
bound. Therefore, we will list the roots for which we determine the exact
number of global sections and the roots for which we can currently only
provide a lower bound separately. For $\Delta_{4}^{\circ}$ with $\mu=12^{4}$,
our counter finds the following:

(81)

Note that we omit trivial results. Also, the percentages are rounded to one
decimal place. Hence, the number 0.0 indicates a non-zero result smaller than
$0.045$. We observe that $12^{4}\cdot\left(20537+199\right)=12^{8}$. Hence,
our algorithm indeed enumerated all limit root bundles, in particular partial
blowup limit roots. Therefore, it could tell that at least 99% of all limit
roots have exactly three sections on
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ irrespective of the complex
structure! The remaining percentage of limit roots begs to be investigated in
more detail. We will return to this task momentarily in Section 5. Counts
similar to Equation 81 are listed in Appendix B. For an overview, we will list
the percentages rather than the absolute numbers. For the families of QSM
spaces with $\overline{K}_{B_{3}}^{3}=6$, we summarize our findings in Table
3. Note that $E$ denotes the number of edges and that the nodal quark-doublet
curve in $\Delta_{8}^{\circ}$ contains an elliptic curve. This is the reason
why our ignorance is much larger for this curve. The results in Table 3 can be
computed in a few minutes on a personal computer. However, for the spaces with
$\overline{K}_{B_{3}}^{3}=10$, the computational challenge increases a lot:

  1. 1.

The number of ways to conduct partial blowups grows as $2^{N}$ with $E$ the
number of edges. For $\overline{K}_{B_{3}}^{3}=6$ spaces, there are at most
edges 9 in the dual graph of $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$.
However, for spaces with $\overline{K}_{B_{3}}^{3}=10$, we can have up to 15
edges. Thus, the number of combinatorial possibilities increases from at most
512 for $\overline{K}_{B_{3}}^{3}=6$ to up to 32768 for
$\overline{K}_{B_{3}}^{3}=10$.

  2. 2.

For each of these partial blowups, determining the number of limit root line
bundles takes longer since the graph is more complicated.

  3. 3.

We repeat such a scan for each value of $h^{0}$ that we want to find the limit
root bundles for. For the spaces with $\overline{K}_{B_{3}}^{3}=6$, we find
all roots after running our algorithm for $h^{0}=3$. However, for
$\overline{K}_{B_{3}}^{3}=10$, we find limit root bundles with
$h^{0}\in\\{3,4,5,6\\}$, so that we have to repeat this step up to four times.

In consequence, even on the super-computer plesken.mathematik.uni-siegen.de,
the computations for the most complicated setups take almost four days.
Eventually, this leads to the results summarized in Table 4.

As we can see from Table 3 and Table 4, the biggest limitation for our
algorithm is the handling of elliptic curve components. Indeed, whenever at
least one $g=1$ component is present, we can only guarantee that roughly at
least $75\%$ of the limit root bundles have $h^{0}=3$. However, whenever no
such elliptic curve is present, our ignorance drops significantly and our
algorithm finds that at least about $95\%$ have $h^{0}=3$. With an eye towards
the F-theory QSMs, it is interesting to wonder if we can lower this ignorance
even further. Our immediate focus turns to the polytopes $\Delta_{4}^{\circ}$,
$\Delta_{134}^{\circ}$, $\Delta_{128}^{\circ}$, $\Delta_{130}^{\circ}$,
$\Delta_{136}^{\circ}$ and $\Delta_{236}^{\circ}$ in which we find that at
least $99\%$ of all limit root line bundles have $h^{0}=3$. We will study
$\Delta_{4}$ momentarily in Section 5.

Another interesting aspect is to compare our findings to the classical Brill-
Noether theory in which one has a continuous space of line bundles and
estimates the dimension of the variety of all line bundles with a certain
$h^{0}$. This concept does not apply for root bundles since there are only
finitely many of them in the first place. To the best of our knowledge, the
above tables Table 3 and Table 4 (more details in Appendix B) are the first
counts/estimates for such bundles. In this sense, we propose to consider them
a first response to the question ‘‘what is Brill-Noether theory for limit
roots?’’. In particular notice from Table 4, that for the QSM setups with
$\overline{K}_{B_{3}}^{3}=10$ we find limit root bundles which always have
more than minimal number of global sections.

In the next section, we will study limit roots on circuit-like subgraphs.
Since these setups lead to the computation of line bundle cohomology on
circuit-like nodal curves, we cannot handle them systematically but rather
conduct a case-by-case study. By focusing on $\Delta_{4}^{\circ}$, we then
find limit roots for which the number of global sections depends on the
relative position of the nodes, i.e. depends on the complex structure of this
nodal curve. For this reason, we term such limit roots _jumping-circuits_. In
particular, this phenomenon seems to resemble the classical Brill-Noether jump
on a smooth, irreducible curve. In the latter case, the points on the curve
corresponding to the line bundle divisor in question align such that the Serre
dual bundle admits a global section.

Polytope | $E$ | $h^{0}=3$ | $h^{0}\geq 3$ | $h^{0}=4$ | $h^{0}\geq 4$
---|---|---|---|---|---
$\Delta_{8}^{\circ}$ | 6 | 76.4 | 23.6 |  |
$\Delta_{4}^{\circ}$ | 7 | 99.0 | 1.0 |  |
$\Delta_{134}^{\circ}$ | 8 | 99.8 | 0.2 |  |
$\Delta_{128}^{\circ}$, $\Delta_{130}^{\circ}$, $\Delta_{136}^{\circ}$, $\Delta_{236}^{\circ}$ | 9 | 99.9 | 0.1 |  |
Table 3: Statistics of limit roots on
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ in all QSM spaces with
$\overline{K}_{B_{3}}^{3}=6$.

Polytope $h^{0}=3$ $h^{0}\geq 3$ $h^{0}=4$ $h^{0}\geq 4$ $h^{0}=5$ $h^{0}\geq
5$ $h^{0}=6$ $h^{0}\geq 6$ $\Delta_{88}^{\circ}$ 74.9 22.1 2.5 0.5 0.0 0.0
$\Delta_{110}^{\circ}$ 82.4 14.1 3.1 0.4 0.0 $\Delta_{272}^{\circ}$,
$\Delta_{274}^{\circ}$ 78.1 18.0 3.4 0.5 0.0 0.0 $\Delta_{387}^{\circ}$ 73.8
21.9 3.5 0.7 0.0 0.0 $\Delta_{798}^{\circ}$, $\Delta_{808}^{\circ}$,
$\Delta_{810}^{\circ}$, $\Delta_{812}^{\circ}$ 77.0 17.9 4.4 0.7 0.0 0.0
$\Delta_{254}^{\circ}$ 95.9 0.5 3.5 0.0 0.0 0.0 $\Delta_{52}^{\circ}$ 95.3 0.7
3.9 0.0 0.0 0.0 $\Delta_{302}^{\circ}$ 95.9 0.5 3.5 0.0 0.0
$\Delta_{786}^{\circ}$ 94.8 0.3 4.8 0.0 0.0 0.0 $\Delta_{762}^{\circ}$ 94.8
0.3 4.9 0.0 0.0 0.0 $\Delta_{417}^{\circ}$ 94.8 0.3 4.8 0.0 0.0 0.0 0.0
$\Delta_{838}^{\circ}$ 94.7 0.3 5.0 0.0 0.0 0.0 $\Delta_{782}^{\circ}$ 94.6
0.3 5.0 0.0 0.0 0.0 $\Delta_{377}^{\circ}$, $\Delta_{499}^{\circ}$,
$\Delta_{503}^{\circ}$ 93.4 0.2 6.2 0.0 0.1 0.0 $\Delta_{1348}^{\circ}$ 93.7
0.0 6.2 0.0 0.1 0.0 $\Delta_{882}^{\circ}$, $\Delta_{856}^{\circ}$ 93.4 0.3
6.2 0.0 0.1 0.0 0.0 $\Delta_{1340}^{\circ}$ 92.3 0.0 7.6 0.0 0.1 0.0
$\Delta_{1879}^{\circ}$ 92.3 0.0 7.5 0.0 0.1 0.0 $\Delta_{1384}^{\circ}$ 90.9
0.0 8.9 0.0 0.2 0.0

Table 4: Statistics of limit roots on
$C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$ in selected QSM spaces with
$\overline{K}_{B_{3}}^{3}=10$.

##  5 $\mathcal{O}(10^{11})$ F-theory QSM without vector-like quark-doublets

In this section, we focus on the F-theory QSMs associated to the polytope
$\Delta_{4}^{\circ}$ in the Kreuzer-Skarke database [52]. Specifically, we
wish to identify the number of global sections for all limit roots for which
our algorithm could only provide a lower bound in Table 3. The upshot of this
analysis is that we can identify a condition on the complex structure moduli
of those QSM geometries, such that all limit roots on
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$ have exactly three global
sections. This then implies that all physical roots on the smooth, irreducible
curve $C_{(\mathbf{3},\mathbf{2})_{1/6}}$ have exactly three global sections.
In other words, we can then guarantee that there are no vector-like exotics
present. To this end, we now study all setups in Equation 81 for which only a
lower bound is listed.

###  5.1 Stationary circuits

We begin by looking at two non-resolved nodes. It is not too hard to modify
our algorithm to print out the 110 limit roots, for which it can only provided
the lower bound $h^{0}(C^{\bullet},P^{\bullet})\geq 3$. It turns out that all
of them are associated to the following setup:

$h^{0}=1$$C_{1}$$h^{0}=2$$C_{2}$$h^{0}=2$$C_{3}$ (82)

To find out if the lower bound is saturated, we recall that a line bundle on a
nodal curve whose irreducible components are all isomorphic to
$\mathbb{P}^{1}$ is uniquely specified by two pieces of information [65]:

  * •

The degree of the line bundle on each irreducible component.

  * •

Descent data which specifies the gluing conditions at the nodes. The latter
corresponds to a number $\lambda\in\mathbb{C}^{\ast}$ for each node. In
particular, the value of the global sections have to agree at each node up to
this factor $\lambda$.

So in an ideal world, we would specify the descent data of a line bundle
$\mathcal{L}^{\bullet}$ on a nodal curve $C^{\bullet}$, feed this information
into the root bundle description of [50] and would end up with the multidegree
_and_ descent data for each root. In this sense, the information in Equation
82 is incomplete, as it does not detail the descent data. Indeed, we will try
simpler. Instead of following the descent data through [50], we will now study
all different descent data that we can put on Equation 82. Equivalently, we
discuss all line bundles on this nodal curve with the indicated multidegree.
The rational behind this is that we can easily see that all of them admit
exactly two global sections on $C_{2}\cup C_{3}$, which is sufficient for our
study. The name _stationary circuit_ derives from this very observation that
_all_ of these line bundles have the same number of global sections.

To see that indeed all these line bundles have exactly two global sectiosn on
$C_{2}\cup C_{3}$, we pick homogeneous coordinates for $C_{2}$, $C_{3}$ and
parametrize the sections on the curves $C_{2}$ and $C_{3}$ by
$\left(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\right)\in\mathbb{C}^{4}$:

(86)

By use of a Möbius transformation, we can assume that the two nodes $C_{2}\cap
C_{3}$ are at the following positions:

(90)

We now enforce the gluing conditions. At $n_{1}$ we demand

$\displaystyle\left(\alpha_{1}a+\alpha_{2}b\right)([1:0])=\lambda_{1}\cdot\left(\alpha_{3}c+\alpha_{4}d\right)([1:0])\,,$
(91)

where $\lambda_{1}\in\mathbb{C}^{\ast}$. Similarly, we enforce at $n_{2}$ the
gluing by demanding

$\displaystyle\left(\alpha_{1}a+\alpha_{2}b\right)([0:1])=\lambda_{2}\cdot\left(\alpha_{3}c+\alpha_{4}d\right)([0:1])\,,$
(92)

where $\lambda_{2}\in\mathbb{C}^{\ast}$. It is not too hard to see that this
is equivalent to

$\displaystyle\alpha_{1}=\lambda_{1}\alpha_{3}\,,\qquad\alpha_{2}=\lambda_{2}\alpha_{4}\,.$
(93)

Hence, the global sections on $C_{2}\cup C_{3}$ are parametrized by
$(\alpha_{3},\alpha_{4})\in\mathbb{C}^{2}$ via

(97)

Hence, irrespective of $\lambda_{1},\lambda_{2}\in\mathbb{C}^{\ast}$, we find
exactly two global sections on $C_{2}\cup C_{3}$. Note that we could have
rescaled the sections on $C_{2}$ such that $\lambda_{2}\to 1$, so that only
$\lambda_{1}\in\mathbb{C}^{\ast}$ appeared in the analysis. We will make use
of this simplification in the next section.

In going back to the setup in Equation 82 we also take $C_{1}$ into account.
Thereby we find $h^{0}(C^{\bullet},P^{\bullet})=3$. Hence, the lower bound is
indeed saturated for this circuit. It is not too hard to repeat this analysis
for all circuits with up to five non-resolved nodes. In all of these cases, we
find that the lower bound is saturated. We list details in Appendix C and
suffice it to note that this improves Equation 81 as follows:

(108)

###  5.2 Jumping circuit

##### Jump from special node alignment

It remains to study the one circuit corresponding to seven remaining nodes:

$h^{0}=2$$C_{0}$$h^{0}=3$$C_{1}$$h^{0}=3$$C_{2}$$h^{0}=2$$C_{3}$ (109)

We parametrize the local sections by $\beta_{i}\in\mathbb{C}$ and pick
homogeneous coordinates of the $\mathbb{P}^{1}$s:

(115)

In the same spirit as in Section 5.1, it is in theory possible to follow the
descent data through [50]. Again, this is not necessary for this work and we
reserve this task for future investgation. For the time being, we thus again
study all descent data for Equation 109, respectively all line bundles on this
nodal curves with the displayed multidegree.

Recall that a Möbius transformation allows us to map any three distinct points
on a $\mathbb{P}^{1}$ to $0$, $1$ and $\infty$. Since there are four nodes on
$C_{1}$ and $C_{2}$ each, a Möbius transformation cannot fix the location of
that fourth node. This adds more parameters and allows to tune the location of
these nodes. This leads to jumping phenomenon -- there is descent data such
that the corresponding line bundle jumps from having exactly three global
sections to four global sections once the nodes are specially aligned. This
mirrors a classical Brill-Noether jump.

While we will not use this effect in the current work, it might be of use in
future F-theory MSSM constructions. Let us therefore briefly illustrate these
jumps. To this end, we use Möbius transformations and rescalings of the
sections on the irreducible components $C_{i}$ such that the locations of the
nodes and the descent data are as follows:

(124)

Recall that $\lambda_{i}\in\mathbb{C}^{\ast}$ are employed for the gluings of
the global sections at the nodes and that the position of the fourth node on
$C_{1}$ and $C_{2}$ is not uniquely specified by a Möbius transformation. The
positions of the latter are parametrized by
$a,b\in\mathbb{C}\setminus\\{0,1\\}$. It then follows that the null space of
the following matrix $A\in\mathbb{M}(7\times 10,\mathbb{C})$ describes the
global sections on this circuit:

$\displaystyle A=\left[\begin{array}[]{cccccccccc}\lambda_{1}&&-1\\\
1&1&&&&-1\\\ 0&1&&&&&&&-1\\\ &&1&1&1&-\lambda_{4}&-\lambda_{4}&-\lambda_{4}\\\
&&&&1&&&-\lambda_{5}\\\ &&1&a&a^{2}&&&&-1&-1\\\
&&&&&1&b&b^{2}&&-\lambda_{7}\end{array}\right]\in\mathbb{M}(7\times
10,\mathbb{C})\,.$ (132)

For generic values of
$\lambda_{1},\lambda_{4},\lambda_{5},\lambda_{7}\in\mathbb{C}^{\ast}$ and
$a,b\in\mathbb{C}\setminus\\{0,1\\}$ it is not too hard to see that
$N(A)\cong\mathbb{C}^{3}$. The key observation is that all $7\times 7$-minors
of this matrix vanish iff

$\displaystyle\lambda_{1}=\frac{1}{a-1}\,,\qquad\lambda_{4}=\frac{b}{a\left(b-1\right)}\,,\qquad\lambda_{5}=\frac{b}{a\left(a-1\right)}\,,\qquad\lambda_{7}=b-1\,.$
(133)

Hence, there is a unique line bundle with four global sections and multidegree
matching that of the canonical bundle on this curve. Therefore, Equation 133
list nothing other than the descent data of the canonical bundle on this
curve. To see the jumping phenomenon mentioend above, focus on a line bundles
with descent data

$\displaystyle\left(\lambda_{1},\lambda_{4},\lambda_{5},\lambda_{7}\right)=\left(\lambda_{1},\frac{\lambda_{1}\left(\lambda_{7}+1\right)}{\lambda_{7}\left(\lambda_{1}+1\right)},\frac{\lambda_{1}^{2}\cdot\left(\lambda_{7}+1\right)}{\lambda_{1}+1},\lambda_{7}\right)\,,$
(134)

where $\lambda_{1},\lambda_{7}\in\mathbb{C}\setminus\\{-1,0\\}$. For generic
$\lambda_{1},\lambda_{7}\in\mathbb{C}\setminus\\{-1,0\\}$, this line bundle
has exactly three global sections. However, once the nodes $n_{6}$, $n_{7}$
are positioned according to

$\displaystyle a=\frac{\lambda_{1}+1}{\lambda_{1}}\,,\qquad
b=\lambda_{7}+1\,,$ (135)

the number of global sections jumps to four and the line bundle in question
turns into the canonical bundle. Such a jump in the number of global sections
resembles a classical Brill-Noether jump. This is the reason why we propose to
call such a pair of a nodal curve and a multidegree a _jumping circuit_. While
we will not make use of such jumps in this article, it can be speculated that
they may be relevant to string model building. We hope to investigate such
applications, in particular to F-theory MSSMs, in future works.

##### Sections of the physically relevant root

Let us return to the F-theory QSMs. In this context, the parameters $a$ and
$b$ are not free, but dictated by the embedding of the nodal curve into the
QSM geometries $B_{3}(\Delta_{4}^{\circ})$. Hence, in order to analyze the
global number of sections, we wish to parametrize the location of the nodes
with the input parameters of the QSM. Recall that
$C_{(\mathbf{3},\mathbf{2})_{1/6}}=V(s_{3},s_{9})$ with $s_{3},s_{9}\in
H^{0}(B_{3},\overline{K}_{B_{3}})$. In the QSM geometries associated to
$\Delta_{4}^{\circ}$, we can write (cf. Appendix D)

$\displaystyle\begin{split}s_{9}&=\alpha_{1}\cdot
x_{0}^{6}x_{4}^{5}x_{5}^{4}x_{6}^{3}x_{7}^{2}x_{8}x_{9}^{4}x_{10}^{3}x_{11}^{2}x_{12}x_{14}^{2}x_{15}x_{17}^{4}x_{18}^{3}x_{19}^{2}x_{20}x_{22}^{2}x_{25}^{2}x_{26}\\\
&\qquad+\alpha_{2}\cdot
x_{0}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{3}x_{9}^{2}x_{10}^{2}x_{11}^{2}x_{12}^{2}x_{13}^{2}x_{14}x_{15}x_{16}x_{17}^{2}x_{18}^{2}x_{19}^{2}x_{20}^{2}x_{21}^{2}x_{22}x_{23}x_{25}x_{26}x_{27}\\\
&\qquad+\alpha_{3}\cdot
x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}x_{8}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}x_{15}x_{16}x_{17}x_{18}\\\
&\hskip 138.88916pt\times
x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}x_{26}x_{27}x_{28}\\\
&\qquad+\alpha_{4}\cdot
x_{3}^{6}x_{4}x_{5}^{2}x_{6}^{3}x_{7}^{4}x_{8}^{5}x_{10}x_{11}^{2}x_{12}^{3}x_{13}^{4}x_{15}x_{16}^{2}x_{18}x_{19}^{2}x_{20}^{3}x_{21}^{4}x_{23}^{2}x_{26}x_{27}^{2}\\\
&\qquad+\alpha_{5}\cdot
x_{2}^{3}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}^{2}x_{15}^{2}x_{16}^{2}x_{22}x_{23}x_{24}^{2}x_{28}\\\
&\qquad+\alpha_{6}\cdot
x_{1}^{3}x_{17}x_{18}x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}^{2}x_{26}^{2}x_{27}^{2}x_{28}^{2}\,.\end{split}$
(136)

We proceed under the assumption that the coefficients
$\alpha_{i}\in\mathbb{C}$ satisfy

$\displaystyle\alpha_{1},\alpha_{4},\alpha_{5}\neq
0\,,\qquad\alpha_{2}^{2}\neq 4\alpha_{1}\alpha_{4}\,.$ (137)

We have worked out the embedding of the nodal curve

$\displaystyle
C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}=V\left(\prod_{i=0}^{28}{x_{i}},s_{9}\right)=\bigcup_{i=0}^{28}{V\left(x_{i},s_{9}\right)}\
\,,$ (138)

including the location of the nodes of the circuit in Equation 109. The
details are listed in Appendix D. Suffice it to state that this leads to the
following parametrization of the nodes:

Node $C_{0}$ $C_{1}$ $C_{2}$ $C_{3}$ $n_{1}$
$[1:-\frac{\alpha_{4}}{\alpha_{5}}]$ $[0:1]$ $n_{2}$ $[1:0]$ $[0:1]$ $n_{3}$
$[0:1]$ $[0:1]$ $n_{4}$
$\left[1:\frac{-\alpha_{2}-\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{4}}}{2\alpha_{4}}\right]$
$\left[1:\frac{-\alpha_{2}-\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{4}}}{2\alpha_{4}}\right]$
$n_{5}$
$\left[1:\frac{\alpha_{2}-\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{4}}}{2\alpha_{4}}\right]$
$\left[1:\frac{\alpha_{2}-\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{4}}}{2\alpha_{4}}\right]$
$n_{6}$ $[1:0]$ $[1:-\frac{\alpha_{1}}{\alpha_{5}}]$ $n_{7}$ $[1:0]$ $[1:0]$
(147)

Note that these positions are rather different than those in Equation 124.
Hence, we look at a different matrix $A\in\mathbb{M}(7\times 10,\mathbb{C})$,
whose right null space is the vector space of global sections:

$\displaystyle
A=\left[\begin{array}[]{cccccccccc}\lambda_{1}&\lambda_{1}\cdot\left(-\frac{\alpha_{4}}{\alpha_{5}}\right)&0&0&-1&0&0&0&0&0\\\
1&0&0&0&0&0&0&-1&0&0\\\ 0&1&0&0&0&0&0&0&0&-1\\\
0&0&1&\frac{-\alpha_{2}-R}{2\alpha_{4}}&\left(\frac{\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&-\lambda_{4}&-\lambda_{4}\left(\frac{-\alpha_{2}-R}{2\alpha_{4}}\right)&-\lambda_{4}\left(\frac{\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&0&0\\\
0&0&1&\frac{-\alpha_{2}+R}{2\alpha_{4}}&\left(\frac{-\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&-\lambda_{5}&-\lambda_{5}\left(\frac{-\alpha_{2}+R}{2\alpha_{4}}\right)&-\lambda_{5}\left(\frac{-\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&0&0\\\
0&0&1&0&0&0&0&0&-1&\frac{\alpha_{1}}{\alpha_{5}}\\\
0&0&0&0&0&1&0&0&-\lambda_{7}&0\end{array}\right]$ (155)

In this matrix we use $R=\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{4}}$. For
generic parameters
$\lambda_{1},\lambda_{4},\lambda_{5},\lambda_{7}\in\mathbb{C}^{\ast}$, this
matrix has $N(A)\cong\mathbb{C}^{3}$. All $7\times 7$-minors vanish iff
$\lambda_{1}=\lambda_{4}=\lambda_{5}=\lambda_{7}=1$ and

$\displaystyle
N(A)=\mathrm{Span}_{\mathbb{C}}\left(\left[\begin{array}[]{c}0\\\ 0\\\ 0\\\
1\\\ 0\\\ 0\\\ 1\\\ 0\\\ 0\\\
0\end{array}\right]\,,\left[\begin{array}[]{c}1\\\ 0\\\ 0\\\ 0\\\ 1\\\ 0\\\
0\\\ 1\\\ 0\\\ 0\end{array}\right]\,,\left[\begin{array}[]{c}0\\\ 0\\\ 1\\\
0\\\ 0\\\ 1\\\ 0\\\ 0\\\ 1\\\
0\end{array}\right]\,,\left[\begin{array}[]{c}0\\\ 1\\\
-\frac{\alpha_{1}}{\alpha_{5}}\\\ -\frac{\alpha_{2}}{\alpha_{5}}\\\
-\frac{\alpha_{4}}{\alpha_{5}}\\\ 0\\\ 0\\\ 0\\\ 0\\\
1\end{array}\right]\right)\,.$ (196)

Hence, $\lambda_{1}=\lambda_{4}=\lambda_{5}=\lambda_{7}=1$ is nothing but the
descent data of the canonical bundle. Note that this parametrization differs
from Equation 133 since we are now employing the node positions in Equation
147. As a consistency check, recall that we are looking at the nodal curve

$\displaystyle C^{\bullet}=V\left(\prod_{i=1}^{28}{x_{i}},s_{9}\right)\,.$
(197)

This curve, and also the line bundle in question, remain completely unchanged
as we replace $s_{9}$ by $\mu\cdot s_{9}$, where $\mu\in\mathbb{C}^{\ast}$ is
arbitrary but fixed. Therefore, the null space computed above must be
invariant upon rescaling the coefficients of $s_{9}$ via
$\alpha_{i}\to\mu\cdot\alpha_{i}$. Indeed, this is the case for the null space
in Equation 196.

We must now wonder which descent data encodes the physically relevant root. To
this end, recall that in following [49], we are solving the root bundle
constraint $P_{\bullet}^{12}=K_{C^{\bullet}}^{12}$. We found in Section 5.1
that all solutions but the jumping circuit have exactly three global sections.
Apparently, $P_{\bullet}=K_{C^{\bullet}}$ is a solution and by theory it must
always hold $h^{0}(C^{\bullet},K_{C^{\bullet}})=4$. Hence, this solution was
not found before and must correspond to the jumping circuit. Therefore, the
physically relevant root bundle on this jumping circuit is nothing but the
canonical bundle.

We recall that the root bundle conditions are necessary conditions, in that it
is not yet clear which of these solutions are obtained top-down from an
F-theory gauge potential. Our analysis shows that more than $99.995\%$ of the
solutions to these necessary conditions have exactly three global sections. If
realized top-down from an F-theory gauge potential, this means that this
solution has no vector-like exotics in the representation
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$.

###  5.3 Towards absence of vector-like exotics

Recall that in every QSM geometry, there are three curves with identical
topology:

$\displaystyle
C_{(\mathbf{\overline{3}},\mathbf{2})_{1/6}}=V(s_{3},s_{9})\,,\qquad
C_{(\mathbf{\overline{3}},\mathbf{1})_{-2/3}}=V(s_{5},s_{9})\,,\qquad
C_{(\mathbf{1},\mathbf{1})_{1}}=V(s_{1},s_{5})\,.$ (198)

Our arguments apply to each of these curves. In focusing on the spaces in
$B_{3}(\Delta_{4}^{\circ})$, let us explicitly parametrize $s_{1}$ and
$s_{9}$:

$\displaystyle\begin{split}s_{1}&=\alpha_{1}\cdot
x_{0}^{6}x_{4}^{5}x_{5}^{4}x_{6}^{3}x_{7}^{2}x_{8}x_{9}^{4}x_{10}^{3}x_{11}^{2}x_{12}x_{14}^{2}x_{15}x_{17}^{4}x_{18}^{3}x_{19}^{2}x_{20}x_{22}^{2}x_{25}^{2}x_{26}\\\
&\qquad+\alpha_{2}\cdot
x_{0}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{3}x_{9}^{2}x_{10}^{2}x_{11}^{2}x_{12}^{2}x_{13}^{2}x_{14}x_{15}x_{16}x_{17}^{2}x_{18}^{2}x_{19}^{2}x_{20}^{2}x_{21}^{2}x_{22}x_{23}x_{25}x_{26}x_{27}\\\
&\qquad+\alpha_{3}\cdot
x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}x_{8}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}x_{15}x_{16}x_{17}x_{18}\\\
&\hskip 138.88916pt\times
x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}x_{26}x_{27}x_{28}\\\
&\qquad+\alpha_{4}\cdot
x_{3}^{6}x_{4}x_{5}^{2}x_{6}^{3}x_{7}^{4}x_{8}^{5}x_{10}x_{11}^{2}x_{12}^{3}x_{13}^{4}x_{15}x_{16}^{2}x_{18}x_{19}^{2}x_{20}^{3}x_{21}^{4}x_{23}^{2}x_{26}x_{27}^{2}\\\
&\qquad+\alpha_{5}\cdot
x_{2}^{3}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}^{2}x_{15}^{2}x_{16}^{2}x_{22}x_{23}x_{24}^{2}x_{28}\\\
&\qquad+\alpha_{6}\cdot
x_{1}^{3}x_{17}x_{18}x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}^{2}x_{26}^{2}x_{27}^{2}x_{28}^{2}\,,\\\
s_{9}&=\beta_{1}\cdot
x_{0}^{6}x_{4}^{5}x_{5}^{4}x_{6}^{3}x_{7}^{2}x_{8}x_{9}^{4}x_{10}^{3}x_{11}^{2}x_{12}x_{14}^{2}x_{15}x_{17}^{4}x_{18}^{3}x_{19}^{2}x_{20}x_{22}^{2}x_{25}^{2}x_{26}\\\
&\qquad+\beta_{2}\cdot
x_{0}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{3}x_{9}^{2}x_{10}^{2}x_{11}^{2}x_{12}^{2}x_{13}^{2}x_{14}x_{15}x_{16}x_{17}^{2}x_{18}^{2}x_{19}^{2}x_{20}^{2}x_{21}^{2}x_{22}x_{23}x_{25}x_{26}x_{27}\\\
&\qquad+\beta_{3}\cdot
x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}x_{8}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}x_{15}x_{16}x_{17}x_{18}\\\
&\hskip 138.88916pt\times
x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}x_{26}x_{27}x_{28}\\\
&\qquad+\beta_{4}\cdot
x_{3}^{6}x_{4}x_{5}^{2}x_{6}^{3}x_{7}^{4}x_{8}^{5}x_{10}x_{11}^{2}x_{12}^{3}x_{13}^{4}x_{15}x_{16}^{2}x_{18}x_{19}^{2}x_{20}^{3}x_{21}^{4}x_{23}^{2}x_{26}x_{27}^{2}\\\
&\qquad+\beta_{5}\cdot
x_{2}^{3}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}^{2}x_{15}^{2}x_{16}^{2}x_{22}x_{23}x_{24}^{2}x_{28}\\\
&\qquad+\beta_{6}\cdot
x_{1}^{3}x_{17}x_{18}x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}^{2}x_{26}^{2}x_{27}^{2}x_{28}^{2}\,.\end{split}$
(199)

Now, we demand that

$\displaystyle\alpha_{1},\alpha_{4},\alpha_{5}\neq
0\,,\qquad\alpha_{2}^{2}\neq
4\alpha_{1}\alpha_{4}\,,\qquad\beta_{1},\beta_{4},\beta_{5}\neq
0\,,\qquad\beta_{2}^{2}\neq 4\beta_{1}\beta_{4}\,.$ (200)

Then, by the above arguments, we are guaranteed that for each of the
$\mathcal{O}(10^{11})$ toric spaces in $B_{3}(\Delta_{4}^{\circ})$, more than
$99.995\%$ of the limit root bundle on the nodal curves
$C^{\bullet}_{(\mathbf{\overline{3}},\mathbf{2})_{1/6}}$,
$C^{\bullet}_{(\mathbf{\overline{3}},\mathbf{1})_{-2/3}}$ and
$C^{\bullet}_{(\mathbf{1},\mathbf{1})_{1}}$ which satisfies the necessary
condition identified in [49] have exactly three global sections. Let us
emphasize that these root bundle conditions are necessary but not sufficient
for a physical (i.e. induced from an F-theory gauge potential) root bundle.
Hence, the physical roots are special among the ones covered in this article.
Since we have found that more than $99.995\%$ of the limit roots in this
superset have exactly three sections, this indicates that the chances are good
that this carries over to the physical roots in the representations
$(\mathbf{3},\mathbf{2})_{1/6}$, $(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$
and $(\mathbf{1},\mathbf{1})_{1}$.

##  6 Conclusion and Outlook

Root bundles appear prominently in studies of vector-like spectra of 4d
F-theory compactifications. Of particular importance to phenomenology are the
Quadrillion F-theory Standard Models (F-theory QSMs) [32]. In this work, we
have analyzed a superset of all physical root bundles for the matter
representations $(\mathbf{3},\mathbf{2})_{1/6}$,
$(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$ and $(\mathbf{1},\mathbf{1})_{1}$
in $\mathcal{O}(10^{11})$ F-theory QSM geometries. We have found that more
than $99.995\%$ of the roots in this superset have exatly three global
sections.

We arrived at this finding from a systematic study of a large fraction of QSM
geometries. While our results are strongest for the family
$B_{3}(\Delta_{4}^{\circ})$ of toric 3-folds obtained from full, regular, star
triangulations of the 4-th polytope in the Kreuzer-Skarke list of
3-dimensional, reflexive poytopes [52], the results for this large fraction of
QSM geometries provide statistical evidence that the absence of the vector-
like exotics in the representations $(\mathbf{3},\mathbf{2})_{1/6}$,
$(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$ and $(\mathbf{1},\mathbf{1})_{1}$
is a very likely scenario within the QSMs.

In the QSM geometries, vector-like spectra are counted by the cohomologies of
line bundles which are necessarily roots of (twists of certain powers of) the
canonical bundle [49]. Such root bundles $P_{\mathbf{R}}$ on the matter curve
$C_{\mathbf{R}}$ are by no means unique. Even more, it is not clear exactly
which ones are induced top-down from F-theory gauge potentials in the Deligne
cohomolgy. First, on one matter curve, only a subset of all mathematically
allowed root bundles could be induced from all physically allowed F-theory
gauge potentials. Secondly, it is conceivable that fluxes which induce a
specific root bundle on matter curve $C_{1}$, only induce a few selected root
bundles on another matter curve $C_{2}$. The study of these questions is
involved and currently beyond our abilities. While we hope to return to these
questions in future work, in the current work we opted for a local and bottom-
up analysis instead. In this sense, we followed the philosophy of the earlier
works [49, 51]. The presented study is bottom-up in that we studied _all_
mathematically allowed root bundles. In particular, our study covers all root
bundles that could possibly be induced from the $G_{4}$-flux and all spin
bundles on the matter curve in question. Our study is local in that we focused
on one matter curve at a time. Correlations among the vector-like spectra of
different matter curves were not taken into account. Hence, we study a set of
root bundles on each matter curve, which is a (proper) superset to all
physically allowed root bundles. For one family of QSM geometries, we argue
that all those roots in this superset have exactly three sections.
Consequently, since the physical roots are a subset of the set of roots that
we study, the absence of vector-like exotics follows.

A crucial step towards this goal was triangulation independence. Specifically,
each family $B_{3}(\Delta^{\circ})$ of toric 3-fold QSM base spaces admits a
triangulation independent nodal limit [51]. This observation allows us to
probe the vector-like spectra on the entire family $B_{3}(\Delta^{\circ})$
from vector-like spectra on just 5 nodal curves. Another motivation for this
deformation is that on smooth, irreducible curves, it is hard to construct
root bundles, not to mention count their global sections. On nodal curves
however, the construction of all limit root bundles has been detailed in [50].
For the family $B_{3}(\Delta_{4}^{\circ})$ \-- associated to the 4-th polytope
in the Kreuzer-Skarke list of 3-dimensional polytopes -- which consists of
$\mathcal{O}(10^{11})$ different toric spaces, we showed that all limit root
bundles on the nodal curves $C_{(\mathbf{3},\mathbf{2})_{1/6}}^{\bullet}$,
$C_{(\mathbf{\overline{3}},\mathbf{1})_{-2/3}}^{\bullet}$ and
$C_{(\mathbf{1},\mathbf{1})_{1}}^{\bullet}$ have exactly three global
sections. By upper semi-continuity, this must hold true on the corresponding
smooth, irreducible matter curves in each space of
$B_{3}(\Delta_{4}^{\circ})$. This then implies the absence of vector-like
exotics.

Apparently, this argument relies on the ability to compute the cohomologies of
_all_ limit root bundles. This is exactly the domain of Brill-Noether theory.
By following the original work in [50], we find that for some limit roots,
this is equivalent to line bundle cohomology on tree-like curves. The
remaining limit roots require an understanding of line bundle cohomology on
circuit-like curves. We explained this in Section 2. In our applications to
the F-theory QSMs, the majority of limit roots arise from tree-like curves
with rational irreducible components. Therefore, we presented a computer
algorithm that counts line bundle cohomology on such curves in Section 3. The
few circuit-like cases are handled by hand.

The reason why there are so few circuits seems related to the fact that we are
looking for $12$-th and even $20$-th roots of the canonical bundle on nodal
curves with relatively small Betti numbers, namely 4 and 6. In fact, we
observe that lower root indices shift the ratio towards having more circuits.
For instance, if instead we were looking at second roots, one should expect
equal contributions from tree-like and circuit-like graphs. It would even be
conceivable that the circuits would dominate. We reserve a systematic study of
line bundle cohomology on circuit-like curves for future work.

Our algorithm simplifies the pair $(C^{\bullet},L)$ of a tree-like, rational,
nodal curve and a line bundle $L$ on it by removing components of
$C^{\bullet}$ and adjusting $L$ such that the number of global sections
remains unchanged. This simplification eventually terminates and leaves us
with a final configuration for which the number of global sections can be read
off easily. If we only want to tell if the number of sections is larger than
that on a smooth $\mathbb{P}^{1}$, then this algorithm can be optimized
further. While this optimization is not directly relevant in order to
establish the absence of vector-like exotics in $\mathcal{O}(10^{11})$ QSM
geometries, we outlined these steps in Appendix A.

In addition to integrating this algorithm into the algorithm used for [51], we
have also taken a combinatorial factor into account. In [50], this factor was
introduced as the _geometric multiplicity_. In layman’s terms, this
multiplicity ensures that we count as many limit root bundles as root bundles
on a smooth curve. Alternatively, under a deformation of a smooth to a nodal
curve, distinct root bundles on the smooth curve coalesce as we approach the
nodal curve. The exact number of distinct roots which coalesce is the
geometric multiplicity. In Section 4, we have elaborated in more detail on
this factor before pointing out two errors in [51]. The new and refined
results were listed in Table 3 and Table 4. The relevant computer algorithms
are available in the gap-4-package _QSMExplorer_ as part of the
ToricVarieties$\\_$project [36]. For convenience of the reader, let us
reproduce Table 3 (cf. Appendix B for more details):

(206)

Hence, by focusing on limit roots on tree-like partial blowups of
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$, we have established for the QSM
spaces with $\overline{K}_{B_{3}}^{3}=6$ that are not obtained from FRSTs of
$\Delta_{8}^{\circ}$, that at least 99% of the limit roots have exactly three
global sections. This is a massive improvement compared to [51] in which we
merely reached about $60\%$. Even more, these setups beg for a further
investigation of the remaining 1% of the limit roots, which are associated to
circuit-like partial blowups.

This is exactly what we have done in Section 5 for the family
$B_{3}(\Delta_{4}^{\circ})$ of QSM spaces obtained from triangulations of the
polytope $\Delta_{4}^{\circ}$. It turned out that, up to symmetries, only a
few line bundles on circuit-like, nodal curves had to be investigated. We
found that for some of these circuits, the line bundle cohomology is
independent of the relative position of the nodes. We call such circuit-like
curves _stationary circuits_ (cf. Appendix C for details). These results
mirror our finding that for the tree-like blow-up limit roots, the
cohomologies are independent of the relative position of the nodes. However,
there are special circuits for which such a dependence does exist. In
particular, there exists one jumping circuit for the limit roots on
$C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$ in $B_{3}(\Delta_{4}^{\circ})$.
In this case, the number of global sections of line bundles can jump from 3 to
4 as we move the nodes in special alignment. We propose to call such curves
_jumping circuits_. It can be speculated that such jumps may be relevant to
string model building. We hope to investigate such applications, in particular
to F-theory MSSMs, in future works.

To study the jumping circuit for $B_{3}(\Delta_{4}^{\circ})$ in full detail,
we investigated its embedding into all spaces of $B_{3}(\Delta_{4}^{\circ})$
in Appendix D. This embedding depends on the complex structure moduli chosen
for $C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}$. For the spaces in
$B_{3}(\Delta_{4}^{\circ})$, it holds that

$\displaystyle\begin{split}C^{\bullet}_{(\mathbf{3},\mathbf{2})_{1/6}}&=V\left(\prod_{i=0}^{28}{x_{i}},s_{9}\right)\,,\\\
s_{9}&=\alpha_{1}\cdot
x_{0}^{6}x_{4}^{5}x_{5}^{4}x_{6}^{3}x_{7}^{2}x_{8}x_{9}^{4}x_{10}^{3}x_{11}^{2}x_{12}x_{14}^{2}x_{15}x_{17}^{4}x_{18}^{3}x_{19}^{2}x_{20}x_{22}^{2}x_{25}^{2}x_{26}\\\
&\quad+\alpha_{2}\cdot
x_{0}^{3}x_{3}^{3}x_{4}^{3}x_{5}^{3}x_{6}^{3}x_{7}^{3}x_{8}^{3}x_{9}^{2}x_{10}^{2}x_{11}^{2}x_{12}^{2}x_{13}^{2}x_{14}x_{15}x_{16}x_{17}^{2}x_{18}^{2}x_{19}^{2}\\\
&\hskip 104.16687pt\times
x_{20}^{2}x_{21}^{2}x_{22}x_{23}x_{25}x_{26}x_{27}\\\ &\quad+\alpha_{3}\cdot
x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}x_{7}x_{8}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}x_{15}x_{16}x_{17}x_{18}\\\
&\hskip 104.16687pt\times
x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}x_{26}x_{27}x_{28}\\\
&\quad+\alpha_{4}\cdot
x_{3}^{6}x_{4}x_{5}^{2}x_{6}^{3}x_{7}^{4}x_{8}^{5}x_{10}x_{11}^{2}x_{12}^{3}x_{13}^{4}x_{15}x_{16}^{2}x_{18}x_{19}^{2}x_{20}^{3}x_{21}^{4}x_{23}^{2}x_{26}x_{27}^{2}\\\
&\quad+\alpha_{5}\cdot
x_{2}^{3}x_{9}x_{10}x_{11}x_{12}x_{13}x_{14}^{2}x_{15}^{2}x_{16}^{2}x_{22}x_{23}x_{24}^{2}x_{28}\\\
&\quad+\alpha_{6}\cdot
x_{1}^{3}x_{17}x_{18}x_{19}x_{20}x_{21}x_{22}x_{23}x_{24}x_{25}^{2}x_{26}^{2}x_{27}^{2}x_{28}^{2}\,.\end{split}$
(207)

For simplicity, we assumed $\alpha_{1},\alpha_{4},\alpha_{5}\neq 0$ and
$\alpha_{2}^{2}\neq 4\alpha_{1}\alpha_{4}$. The embedding was then worked out
and we found that the global sections on this circuit are encoded by the right
null space of

$\displaystyle
A=\left[\begin{array}[]{cccccccccc}\lambda_{1}&-\lambda_{1}\cdot\frac{\alpha_{4}}{\alpha_{5}}&0&0&-1&0&0&0&0&0\\\
1&0&0&0&0&0&0&-1&0&0\\\ 0&1&0&0&0&0&0&0&0&-1\\\
0&0&1&\frac{-\alpha_{2}-R}{2\alpha_{4}}&\left(\frac{\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&-\lambda_{4}&-\lambda_{4}\left(\frac{-\alpha_{2}-R}{2\alpha_{4}}\right)&-\lambda_{4}\left(\frac{\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&0&0\\\
0&0&1&\frac{-\alpha_{2}+R}{2\alpha_{4}}&\left(\frac{-\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&-\lambda_{5}&-\lambda_{5}\left(\frac{-\alpha_{2}+R}{2\alpha_{4}}\right)&-\lambda_{5}\left(\frac{-\alpha_{2}+R}{2\alpha_{4}}\right)^{2}&0&0\\\
0&0&1&0&0&0&0&0&-1&\frac{\alpha_{1}}{\alpha_{5}}\\\
0&0&0&0&0&1&0&0&-\lambda_{7}&0\end{array}\right]$ (215)

In this matrix we use $R=\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{4}}$. An
explicit computation reveals that this right null space is generically of
dimension $3$. All $7\times 7$ minors of this matrix vanish iff
$\lambda_{1}=\lambda_{4}=\lambda_{5}=\lambda_{7}=1$ and the corresponding line
bundle then has exactly four sections. Since the multidegrees match those of
the canoical bundle on the jumping circuit in Equation 109, we conclude that
this very line bundle is nothing but the canonical bundle on this nodal curve.
Indeed, we are solving $P_{\bullet}^{12}=K_{C^{\bullet}}^{12}$. One solution
is apparently $P_{\bullet}=K_{C^{\bullet}}$ and by theory
$h^{0}(C_{\bullet},K_{C^{\bullet}})=4$. We argued that all other solutions
other than the jumping circuit have exactly three global sections. Hence, in
this jumping circuit we are interested in $K_{C^{\bullet}}$ which by theory
has to have four global sections. Therefore, we conclude that for the family
of toric 3-folds $B_{3}(\Delta_{4}^{\circ})$, more than 99.995% of the
solutions to the necessary root bundle condition -- originally derived in [49]
\-- have exactly three global sections. We recall that it is not clear at this
point which of these solutions are realized top-down from an F-theory gauge
potential. This will likely select a subset of the set of root bundles
analyzed in this work. The fact, that more than 99.995% of the roots in this
superset have exactly three global sections, indicates that absence of vector-
like exotics in the representation $(\mathbf{3},\mathbf{2})_{1/6}$,
$(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$ and $(\mathbf{1},\mathbf{1})_{1}$
is a very likely scenario in the $\mathcal{O}(10^{11})$ QSM geometries
$B_{3}(\Delta_{4}^{\circ})$.

It is, at least in principle, possible to extend this analysis to other QSM
spaces. Based on Table 3, one should focus on $B_{3}(\Delta_{134}^{\circ})$,
$B_{3}(\Delta_{128}^{\circ})$, $B_{3}(\Delta_{130}^{\circ})$,
$B_{3}(\Delta_{136}^{\circ})$, $B_{3}(\Delta_{236}^{\circ})$ next. These
geometries and our limit root counts are very similar to those of
$B_{3}(\Delta_{4}^{\circ})$. We should therefore expect similar findings,
possibly establishing an even larger class of F-theory Standard Models without
vector-like exotics. However, once we turn to spaces with
$\overline{K}_{B_{3}}^{3}=10$, then our argument breaks down. Namely, we can
read off from Table 4 that there are root bundles on the canonical nodal
matter curves that admit more than three global sections. To understand such
setups, one has to formulate conditions that guarantee that the number of
sections remains constant/drops down as one deforms back to a smooth curve.
Such conditions will also be vital to guarantee exactly one Higgs pair. It can
be speculated that the physics of Yukawa interactions, which informs us how,
when and which fields acquire masses upon deformations, will be a good guide
towards such conditions. A quest for F-theory MSSM must therefore address this
question eventually. We hope to investigate this fascinating and challenging
interplay in future works.

As far as the techniques in the current work are concerned, the
$\mathcal{O}(10^{11})$ QSM geometries $B_{3}(\Delta_{4}^{\circ})$ behaved
identically. To some extent, this must be seen as very fortunate
circumstances. However, many of the intersection numbers -- and this applies
more generally for all QSM geometries -- are triangulation-dependent. This
affects the cosmological aspects of these models, including inflation. Namely,
as already mentioned in [32], the classical Kähler potential depends on
exactly these triangulation-dependent intersection numbers.

While our analysis is streamlined towards the particle phenomenology,
specifically the vector-like spectra, of the F-theory QSMs, we were led to
establish -- what we believe to be the first -- computational approaches
towards Brill-Noether theory of limit root bundles. The classical Brill-
Noether theory concerns a classification of a continuous space of line bundles
by their global sections. However, there always exists a finite number
$N_{\text{total}}$ of root bundles. In this regard, their Brill-Noether theory
becomes a question of finding the partition
$N_{\text{total}}=N_{0}+N_{1}+\dots$ in which $N_{i}$ is the number of root
bundles with $i$ global sections. This is exactly how we propose to read our
central results in Table 3 and Table 4. Also, we propose to consider a
_jumping circuit_ as the analogue of a classical Brill-Noether jump.

The ignorances in Table 3 and Table 4 stem from two sources. First, we only
provide a lower bound if a node remains on an elliptic curve. For this, one
could use theta characteristics to identify a basis of the sections on the
elliptic curve in question and then work out the conditions imposed by the
gluings at the node. The second limitation arises due to the lack of a
systematic treatment of line bundle cohomology on circuit-like nodal curves
(with rational or even elliptic) components. In this work, we have treated the
relevant cases by hand and proposed to distinguish stationary and jumping
circuits. In particular, one could speculate that a circuit of rational curves
is jumping if there are at least four nodes on one curve component, owing to
the fact that the action of $SL(2,\mathbb{Z})$ can be employed to fix the
location of three points. Such a study is reserved for future works and should
help to further sharpen our understanding of Brill-Noether theory of limit
roots.

Our approach towards Brill-Noether theory of limit root bundles also opens an
avenue for a potential machine learning application. Indeed when focusing on
(powers of) the canonical bundle, a major part of the information is contained
in the dual graph of the nodal curve. For sufficiently simple graphs, our
algorithm can -- modulo the above-mentioned ignorances -- compute an
approximation of the Brill-Noether theory of the corresponding limit roots. It
would be fascinating to seek a pattern that relates those graphs and the
corresponding Brill-Noether approximations (such as Table 3 and Table 4). For
a mathematical perspective, this should ideally lead to insights that
establish a connection among dual graphs and counts of limit root bundles in a
provable manner. With a phenomenological application in mind, one could
instead hope to acquire a well-trained algorithm that can estimate the limit
root bundles on the Higgs curve, which is currently in most setups beyond our
computational abilities. Of course, such counts are then best repeated with
conditions that preserve (or explicitly enumerate the defect of) the number of
global sections upon deformations, as elaborated upon above.

Finally, it must be mentioned that we have avoided to answer a key question in
this program, namely which mathematical roots are induced from an F-theory
gauge field in the Deligne cohomology. We anticipate that the answer to that
question is involved and have therefore, decided to analyze the full set of
mathematically admissible root bundles instead. In this current work, we have
found that in the family $B_{3}(\Delta_{4}^{\circ})$ of $\mathcal{O}(10^{11})$
different QSM geometries, more than 99.995% of the root bundles in this
superset of mathematically admissible root bundles do not have vector-like
exotics. This is indicative that absence of vector-like exotics in the
representations $(\mathbf{3},\mathbf{2})_{1/6}$,
$(\mathbf{\overline{3}},\mathbf{1})_{-2/3}$ and $(\mathbf{1},\mathbf{1})_{1}$
is a very likely scenario. We reserve a detailed study of the top-down
conditions for future work.

##### Acknowledgement

We are grateful to Muyang Liu for past collaboration, insightful discussions
and ongoing collaboration on limit roots on the Higgs curve. The OSCAR
computer algebra system was used for key computations in this work, and we
thank Lars Kastner and Benjamin Lorenz for valuable discussions. The reliable
computations conducted by the supercomputer plesken.mathematik.uni-siegen.de
allowed us to count limit root bundles. This we truly appreciate. M.B. and
M.C. thank the _Ludwig-Maximilians-Universitaet Muenchen_ for hospitality
during early stages of this project. M.B., R.D. and M.O. are partially
supported by the NSF grant DMS 2001673 and by the Simons Foundation
Collaboration grant #390287 on ‘‘Homological Mirror Symmetry’’. M.O. is
grateful for the support by the Ph.D. Presidential Fellowship research fund.
The work of M.C. is supported by DOE Award DE-SC0013528Y. M.B. and M.C.
further acknowledge support by the Simons Foundation Collaboration grant
#724069 on ‘‘Special Holonomy in Geometry, Analysis and Physics’’. M.C. thanks
the Slovenian Research Agency No. P1-0306 and the Fay R. and Eugene L.
Langberg Chair for support.

##  Appendix A Speciality

Consider a rational, connected, nodal curve $C:=\bigcup_{i\in I}{C_{i}}$ and
$L\in\operatorname{Pic}(C)$.

######  Definition A.1.

We say that $(C,L)$ with $\deg(L)\geq-1$ is _special_ if the following
equivalent conditions hold.

  1. (i)

$h^{0}(C,L)$ jumps (down) under deformation of $C$,

  2. (ii)

$h^{1}(C,L)\neq 0$,

  3. (iii)

$h^{0}(C,L)\neq\chi(C,L)=\deg(L)+1$.

######  Definition A.2.

Let $d_{i}=\deg(L|_{C_{i}})$. Write $I=I_{+}\cup I_{-}$, where

$\displaystyle I_{-}:=\left\\{i\in I|d_{i}<0\right\\}\,,\qquad
I_{+}:=\left\\{i\in I|d_{i}\geq 0\right\\}\,.$ (216)

Let

$\displaystyle C_{+}:=\bigcup_{i\in I_{+}}{C_{i}}\,,\qquad
C_{-}:=\bigcup_{i\in I_{-}}{C_{i}}\,.$ (217)

Note that the curves $C_{+}$, $C_{-}$ could be disconnected. So, let $k_{+}$
and $k_{-}$ be the numbers of connected components of $C_{+}$ and $C_{-}$. Let
$e_{i}$ be the intersection number of $C_{i}$ with $C_{-}$ and let $L_{+}$ be
the line bundle on $C_{+}$ with $\deg(L_{+}|_{C_{i}})=d_{i}-e_{i}$.

######  Proposition A.3.

The pair $(C,L)$ with $\deg(L)\geq-1$ is non-special iff the following
conditions hold:

  1. (i)

$h^{0}(C_{+},L_{+})=\chi\left(C_{+},L_{+}\right)$,

  2. (ii)

$d_{i}\geq-1$ for all $i\in I$ and $C_{i}\cap C_{j}=\emptyset$ for all
distinct $i,j\in I_{-}$.

###### Proof.

Since $C$ is a rational nodal curve, we have the identity

$\displaystyle\sum_{i\in I_{+}}{e_{i}}=k_{+}+k_{-}-1\,.$ (218)

Since Euler characteristics are additive under disjoint union, applying
Riemann-Roch to components $C_{i}$ of $C_{+}$ where $g(C_{i})=0$ yields

$\displaystyle\begin{split}\chi(C_{+},L_{+})&=\deg(L_{+})+k_{+}=\sum_{i\in
I_{+}}{d_{i}}-\sum_{i\in I_{+}}{e_{i}}+k_{+}\,.\end{split}$ (219)

Substituting (219) into the above gives

$\displaystyle\begin{split}\chi(C_{+},L_{+})=\sum_{i\in
I_{+}}{d_{i}}-k_{-}+1=\mathrm{deg}(L)+1-\Bigl{(}k_{-}+\sum_{i\in
I_{-}}{d_{i}}\Bigr{)}\geq\chi(C,L)\,,\end{split}$ (220)

The last inequality follows because $d_{i}\leq-1$ for all $i\in I_{-}$ and so,

$k_{-}+\sum_{i\in I_{-}}{d_{i}}\leq k_{-}-|I_{-}|\leq 0\,.$ (221)

By 3.2, $h^{0}(C,L)=h^{0}(C_{+},L_{+})$. Therefore,

$\displaystyle
h^{0}(C,L)=h^{0}(C_{+},L_{+})\geq\chi(C_{+},L_{+})\geq\chi(C,L)\,.$ (222)

Condition (i) ensures that $h^{0}(C_{+},L_{+})=\chi(C_{+},L_{+})$. Also,
$\chi(C_{+},L_{+})=\chi(C,L)$ iff condition (ii) holds. Indeed, if two curves
in $C_{-}$ intersect, then $k_{-}$ is strictly less than $|I_{-}|$. Therefore,
the right inequality in Equation 221 is strict. On the other hand, if
$d_{i}<-1$ for some $i\in I_{-}$, then the left inequality in Equation 221 is
strict. Conversely, if condition (ii) holds, then Equation 221 is an equality
and $\chi(C_{+},L_{+})=\chi(C,L)$. ∎

######  Algorithm 2 ($\deg(L)\geq-1$).

Start with $(C,L)$, where $C$ is a rational, connected, nodal curve and
$L\in\operatorname{Pic}(C)$ with $\deg(L)\geq-1$

  1. 1.

Check condition (ii) of A.3, which involves the following.

    1. (i)

Are there any degrees smaller than $-1$? If yes, then $(C,L)$ is special.
Otherwise, proceed.

    2. (ii)

Is $I_{-}$ trivial? If yes, then $(C,L)$ is not special. Otherwise, proceed.

    3. (iii)

Do curves with negative degrees intersect? If yes, then $(C,L)$ is special.
Otherwise, proceed.

  2. 2.

Form $I^{(1)}=I^{(1)}_{+}\cup I^{(1)}_{-}$, where $d_{i}=\deg(L|_{C_{i}})$,

$\displaystyle I^{(1)}_{+}=\\{i\in I:d_{i}\geq 0\\}\,,\quad
I^{(1)}_{-}=\\{i\in I:d_{i}<0\\}\,.$ (223)

If $C_{i}$ is a curve component of $C$, then set

$\displaystyle C^{(1)}_{+}=\bigcup_{i\in I^{(1)}_{+}}C_{i}\,,\quad
C^{(1)}_{-}=\bigcup_{i\in I^{(1)}_{-}}C_{i}\,.$ (224)

Define a line bundle $L^{(1)}_{+}$ over $C^{(1)}_{+}$ such that
$\deg(L^{(1)}_{+}|_{C_{i}})=d_{i}-e_{i}$ where $e_{i}$ is the intersection
number between $C_{i}$ and $C^{(1)}_{-}$.

  3. 3.

Is $C^{(1)}_{+}\cap C^{(1)}_{-}=\emptyset$? If not, feed
$(C^{(1)}_{+},L^{(1)}_{+})$ back into step 2 and iterate. Otherwise, proceed.

  4. 4.

Stop iterating when $C^{(n)}_{+}\cap C^{(n)}_{-}=\emptyset$, which means that
$(C^{(n)}_{+},L^{(n)}_{+})$ is terminal. Since $C$ has finitely many
components, the algorithm will eventually terminate.

  5. 5.

Check condition (i) of A.3 by computing $\chi(C,L)=\deg(L)+1$ and

$\displaystyle h^{0}(C_{+}^{(n)},L_{+}^{(n)})=\sum_{i\in
I^{(n)}_{+}}d_{i}+k_{+}(C^{(n)}_{+}).$ (225)

If $h^{0}(C_{+}^{(n)},L_{+}^{(n)})\neq\chi(C,L)$, then $(C,L)$ is special.
Otherwise, $(C,L)$ is non-special.

This algorithm ultimately seeks to determine if $h^{0}(C,L)=\chi(C,L)$ by
simplifying the curve and checking the conditions of A.3. Whenever condition
(ii) is satisfied in step 1,
$\chi(C^{(i)}_{+},L^{(i)}_{+})=\chi(C^{(i+1)}_{+},L^{(i+1)}_{+})$ by A.3. In
particular, if the algorithm reaches a terminal curve and condition (ii) is
satisfied all throughout, then

$\displaystyle\chi(C,L)=\cdots=\chi(C_{+}^{(n)},L_{+}^{(n)}).$ (226)

Thanks to 3.2, we can use the terminal curve to compute
$h^{0}(C,L)=h^{0}(C_{+}^{(n)},L_{+}^{(n)})$ in step 5. We check if condition
(i) is satisfied by comparing $h^{0}(C_{+}^{(n)},L_{+}^{(n)})$ with
$\chi(C_{+}^{(n)},L_{+}^{(n)})$.

######  Example A.4.

Consider the following pair $(C,L)$ with $\deg(L)=0$.

$2$$1$$1$$-1$$-1$$-1$$-1$$0$$-1$$1$$-1$$0$ (227)

Since $I^{(2)}_{+}\cap I^{(2)}_{-}=\emptyset$, the algorithm terminates. Since
$h^{0}(C^{(2)}_{+},L^{(2)}_{+})=1=\chi(C,L)$, the curve is non-special.

######  Definition A.5.

The pair $(C,L)$ with $\deg(L)<-1$ is _special_ iff $h^{0}(C,L)>0$.

This is because $\chi(C,L)=\deg(L)+1<0$.

######  Algorithm 3 ($\deg(L)<-1$).

Start with $(C,L)$, where $C$ is a rational, connected, nodal curve and
$L\in\operatorname{Pic}(C)$ with $\deg(L)<-1$.

  1. 1.

Form $I^{(1)}=I^{(1)}_{+}\cup I^{(1)}_{-}$, where $d_{i}=\deg(L|_{C_{i}})$,

$\displaystyle I^{(1)}_{+}=\\{i\in I:d_{i}\geq 0\\}\,,\quad
I^{(1)}_{-}=\\{i\in I:d_{i}<0\\}\,.$ (228)

If $C_{i}$ is a curve component of $C$, then set

$\displaystyle C^{(1)}_{+}=\bigcup_{i\in I^{(1)}_{+}}C_{i}\,,\quad
C^{(1)}_{-}=\bigcup_{i\in I^{(1)}_{-}}C_{i}\,.$ (229)

Define a line bundle $L^{(1)}_{+}$ over $C^{(1)}_{+}$ such that
$\deg(L^{(1)}_{+}|_{C_{i}})=d_{i}-e_{i}$ where $e_{i}$ is the intersection
number between $C_{i}$ and $C^{(1)}_{-}$.

  2. 2.

After forming $(C^{(1)}_{+},L^{(1)}_{+})$, we perform the following checks:

    1. (i)

Is $I^{(1)}_{+}=\emptyset$? If yes, then $(C,L)$ is non-special. Otherwise,
proceed.

    2. (ii)

Is $I^{(1)}_{+}\cap I^{(1)}_{-}=\emptyset$? If not, then feed
$(C^{(1)}_{+},L^{(1)}_{+})$ into back into Step 2 and iterate. Otherwise
proceed.

  3. 3.

Stop iterating when $C^{(n)}_{+}\cap C^{(n)}_{-}=\emptyset$, which means that
$(C^{(n)}_{+},L^{(n)}_{+})$ is terminal. Since $C$ has finitely many
components, the algorithm will eventually terminate. Compute

$\displaystyle h^{0}(C^{(n)}_{+},I^{(n)}_{+})=\sum_{i\in
I_{+}}d_{i}+k_{+}(C^{(n)}_{+})\,.$ (230)

  4. 4.

If $h^{0}(C^{(n)}_{+},I^{(n)}_{+})>0$, then $(C,L)$ is special. If not, then
$(C,L)$ is non-special.

This algorithm simplifies $(C,L)$ and allows us to compute