# The role of the Heisenberg principle in Constrained Molecular Dynamics model

K. Wang<EMAIL_ADDRESS>Shanghai Institute of Applied Physics, Chinese
Academy of Sciences, Shanghai 201800, China  University of the Chinese Academy
of Sciences, Beijing 100080, China  School of Physical Science and Technology,
ShanghaiTech University, Shanghai 201203, China  A. Bonasera  Cyclotron
Institute, Texas A&M University, College Station, TX 77843, USA  Laboratori
Nazionali del Sud, INFN, via Santa Sofia, 62, 95123 Catania, Italy  H. Zheng
School of Physics and Information Technology, Shaanxi Normal University, Xi’an
710119, Shaanxi, China  G. Q. Zhang  Shanghai Institute of Applied Physics,
Chinese Academy of Sciences, Shanghai 201800, China  Y. G. Ma  Shanghai
Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800,
China  School of Physical Science and Technology, ShanghaiTech University,
Shanghai 201203, China  Institute of Modern Physics, Department of Nuclear
Science and Technology, Fudan University, Shanghai 200433, P.R. China  W. Q.
Shen  Shanghai Institute of Applied Physics, Chinese Academy of Sciences,
Shanghai 201800, China  School of Physical Science and Technology,
ShanghaiTech University, Shanghai 201203, China

###### Abstract

We implement the Heisenberg principle into the Constrained Molecular Dynamics
(CoMD) model with a similar approach to the Pauli principle using the one-body
occupation probability $\bar{f}_{i}$. Results of the modified and the original
model with comparisons to data are given. The binding energies and the radii
of light nuclei obtained with the modified model are more consistent to the
experimental data than the original one. The collision term and the density
distribution are tested through a comparison to p+12C elastic scattering data.
Some simulations for fragmentation and superheavy nuclei production are also
discussed.

###### pacs:

21.60.Ka, 24.10.Lx, 25.70.-z, 25.70.Jj

##  I Introduction

In recent decades, heavy-ion reactions have been an important approach to
study the properties of nuclear structure and reaction dynamics, such as the
properties of exotic nuclei, the nuclear equation of state, and superheavy
nuclei among other features. Besides experiments, theoretical models become
more and more useful in these researches. Among them, the molecular dynamics
models achieved great success since they dealt better with the $N$-body
problem than the one-body semiclassical transport models like the Vlasov-
Uehling-Uhlenbeck (VUU) bertsch1988  and the Boltzmann-Nordheim-Vlasov (BNV)
models bonasera1994 ; papa2001 . The Fermionic Molecular Dynamics model(FMD)
feldmeier1990 ; feldmeier1995  and the Antisymmetrized Molecular Dynamics
model(AMD) ono1992a ; ono1992b  have successfully preserved the Fermionic
nature of the many body system by expressing the wave function of the system
as a single Slater determinant of $N$ wave packets. As a result, the numerical
effort of these two models grows with $N^{3-4}$. It takes very long CPU time
for calculations of large mass systems ($\geq$80 nucleons) so that
approximations are needed feldmeier2000 . The Quantum Molecular Dynamics model
(QMD) aichelin1986 ; aichelin1991 , which needs only double-fold loops to
calculate two-body interactions, making its numerical effort grow with $N^{2}$
papa2001 , is much faster than FMD and AMD. It has made great achievements in
heavy-ion reactions from intermediate to high energies hartnack1998 ; ma2006 ;
kumar2010 ; gautam2012 ; feng2018 . However, the Fermionic feature of nucleons
is lacking in the QMD model, especially in the ground states or in low-energy
reaction phenomena papa2001 . Several modified quantum molecular dynamics
models have been proposed to solve this problem, like the extended Quantum
Molecular Dynamics (EQMD) model maruyama1996 , and the Constrained Molecular
Dynamics (CoMD) model papa2001 ; papa2005 .

The EQMD model tried to mimic the Fermionic features by introducing the Pauli
potential, which forbids nucleons with the same spin and isospin from coming
close to each other in phase space maruyama1996 . The model shows some good
features such as stable ground states and saturation properties of nuclear
matter, but it also has some problems, for instance a spurious repulsion in
the collisions wada1998 ; he2014 ; papa2001 .

By constraining the phase space to fulfill the Pauli principle at each time
step, the CoMD model has successfully introduced the effect of the Fermionic
nature to the nuclear many-body system. It is able to reproduce with the same
set of parameters not only the main characteristic of stable nuclei in a wide
mass region ($A$=30-208) but also the experimental fragment charge
distribution in many collisions such as 40Ca+40Ca, 197Au+197Au,
112,123Sn+58,64Ni systems at 35 MeV/nucleon, and 40Ca+48Ca system at 25
MeV/nucleon papa2001 ; papa2003 . However, the binding energies and
theoretical prediction of light nuclei, in particular the $Z$=2 yield, are not
consistent with experimental results papa2001 . In this paper, we try to
improve these features by introducing the Heisenberg principle as another
constraint into the CoMD model and study its effect on the finite nuclei.
Similar mechanism has already been successfully applied to study the
penetrability of the Coulomb barrier with molecular dynamics simulations
kimura2004 , and also the atomic structure and collision calculations
kimura2005 .

The paper is organized as follows. Section II gives a brief introduction of
the CoMD model as well as the formalism to implement the Heisenberg principle
into the original model. Section III presents the results and discussions of
the model. The binding energies of nuclei during initialization and time
evolution from the modified and the original models and the comparisons to
data are given. A brief summary is given in Sec. IV.

##  II Model and Formalism

The nucleon wave functions and the mean field used in this work are the same
as in the original CoMD model to maintain the same nuclear equation of state.
Details about the effective interaction were described in Ref. papa2001 .

In the original CoMD model, the constraints for the Pauli principle at each
time step are achieved via the following:

$\displaystyle{\bar{f}_{i}}$ $\displaystyle\leq$ $\displaystyle 1,$ (1)
$\displaystyle\bar{f}_{i}$ $\displaystyle\equiv$
$\displaystyle\sum_{j}\delta_{\tau_{i},\tau_{j}}\delta_{s_{i},s_{j}}\int_{h^{3}}f_{j}({\it
r,p})d^{3}rd^{3}p,$ (2)

in which $\tau_{i}$ represents the nucleon isospin and $s_{i}$ refers to the
spin projection quantum number papa2001 . At each time step and for each
particle $i$, an ensemble $K_{i}$ of nearest identical particles (including
the particle $i$) is determined within the distances
$\sqrt{\pi\hbar\sigma{{}_{r}}/2\sigma{{}_{p}}}$ and
$\sqrt{\pi\hbar\sigma{{}_{p}}/2\sigma{{}_{r}}}$ from particle $i$ in $r$ and
$p$ spaces, respectively papa2001 . Here $\sigma_{r}$ represents the width of
the Gaussian wave function in $r$ space and $\sigma_{p}$ is taken as the width
in $p$ space. Then the phase space occupation $\bar{f}_{i}$ is calculated for
the identical particles. If $\bar{f}_{i}$ is greater than 1, we change
randomly the momenta of the particles belonging to the ensemble $K_{i}$ to
make $\bar{f}_{i}$ less than 1. The mechanism we used is similar to a many-
body elastic scattering so that the total momentum and the total kinetic
energy of the newly generated sample are conserved. The new sample is accepted
only if the phase-space occupation $\bar{f}_{i}$ is smaller than the one
before papa2001 .

We adopt a similar approach to introduce the Heisenberg principle into the
modified model. After the check of the Pauli principle for identical particles
(same spin and isospin) as mentioned before, for each of those particles that
do not violate the Pauli principle, we search the ensemble $L_{i}$ of the
nearest non-identical particles (including the particle $i$) within the
distances $\sqrt{\pi\hbar\sigma{{}_{r}}/2\sigma{{}_{p}}}$ and
$\sqrt{\pi\hbar\sigma{{}_{p}}/2\sigma{{}_{r}}}$ from the particle $i$ in $r$
and $p$ spaces, respectively papa2001 . Then we calculate the average
$\bar{f}_{i}$ for the non-identical particles in the ensemble $L_{i}$. If the
average of $\bar{f}_{i}$ has a value greater than 25.0/93.0 (a value obtained
by calculating the $\bar{f}_{i}$ of two nucleons with their
$\Delta\it{r}\Delta\it{p}$ equals to $\hbar/2$), we change the momenta of the
particles belonging to the ensemble $L_{i}$ as we did for $K_{i}$ before.
Notably, those particles which have been scattered because of the Pauli
principle will not be included in the check of the Heisenberg principle. This
ensures that one particle will not be scattered twice.

Similar to the Pauli blocking, we introduced the ”Heisenberg blocking” in the
collision term in the modified model. For each $NN$ collision, we calculate
the occupation probability as we did before and check for both the Pauli
principle and the Heisenberg principle. If they are both satisfied, the
collision is accepted, rejected otherwise. In this way, the Heisenberg
principle is important for light nuclei ($Z\leq 4$) and the Pauli principle
becomes dominant for heavy nuclei.

We also modified the cooling procedure in the initialization step to prepare
the ground state of the nucleus in the modified model. We use the constraints
to solve the equations of motion of the nucleons with friction terms. The
nucleons are first distributed in spheres of radius 1.2${\times}A^{1/3}$ fm
and $P^{nm}_{F}$ (Fermi momentum for infinite nuclear matter) in coordinate
and momentum space, respectively. Then at each time step, we first calculate
the value of $\bar{f}_{i}$ to fulfill the Pauli principle. If $\bar{f}_{i}$ is
greater than 1, the momenta of the particles belonging to the ensemble $K_{i}$
are scaled by a factor of 1.005. If $\bar{f}_{i}$ is less than 1, we calculate
the value of $\bar{f}_{i}$ to fulfill the Heisenberg principle for the
particles belonging to ensemble $L_{i}$. If the average $\bar{f}_{i}$ for the
Heisenberg principle is greater than 25.0/93.0, we set the scale factor to
1.0005. If this $\bar{f}_{i}$ is also less than the threshold value, the scale
factor is set to 0.99. Roughly, by iterating this cooling procedure up to 1000
fm/c, we can get stationary values of the total binding energy and the radius
of the nucleus for which the ground state is wanted for the heavy-ion reaction
simulation or other goals. After this cooling procedure, the friction term is
switched off and we test the stability of the ground state by checking the
time evolution of its root mean square radius $R$. The prepared ground state
will be accepted only if $R$ is stable for at least 1000 fm/c.

##  III Results and Discussions

In order to examine the CoMD model with the Heisenberg principle (CoMDH), we
compare some simulated results from CoMDH to the ones with the same conditions
from the original model and also to the experimental data.

FIG. 1: (Color online) Time evolution of the binding energies during the
cooling step for the original model without the Heisenberg principle (a) and
the modified model with the Heisenberg principle (b).

In the original model, the binding energies of the light particles are not
satisfactory, similarly to other approaches where the constant distribution
width is added to the binding energy feldmeier1990 ; ono1992a ; aichelin1986 ;
aichelin1991 ; giuliani2014 . Figure 1 shows the comparisons of the binding
energies of the light nuclei calculated with the original and the modified
models. From the results of the original model, we see that the binding
energies of the light nuclei keep going down to a very low value and they are
over bound. This is due to the fact that these light particles, i.e.,
deuteron, tritium, helium-3, and alpha particles do not violate the Pauli
principle thus their gaussian wave packets can move on top of each other in
the cooling procedure. We stress that this feature is common to all molecular
dynamics models but other authors choose to include the gaussian finite width
as the kinetic energy part which does not affect the motion of nucleons at
all. Thus their ground states are solids feldmeier2000 ; papa2005 ;
giuliani2014 . When adding the Heisenberg principle, we see that the binding
energies of the light nuclei oscillate around some average values. These are
the real ground state of the model depending only on the interaction
parameters and especially the surface term for light nuclei. The binding
energies of these light nuclei are now closer to the experimental values even
though deuterium, tritium, and 3He are still over bound. Figure 2 shows the
binding energies of some selected nuclei (the mass is ranging from 2 to 238)
from CoMD and CoMDH calculations. The experimental data are retrieved from the
National Nuclear Data Center pritychenko2012 . The modified model seems to
work better throughout the whole nuclei chart than the original one. We stress
also that in order to fulfill the Heisenberg principle, we enforce its effect
in the collision term as well for non-identical particles.

FIG. 2: (Color online) Binding energies of nuclei with mass ranging from 2 to
238 in initializations (full symbols) and collisions (open symbols). The
(black) full squares represent the experimental data. The (red) solid circles
and the (blue) solid triangles represent the binding energies of nuclei during
the initialization calculated with the modified model including the Heisenberg
principle (CoMDH) and with the original model (CoMD), respectively. The (red)
open circles and the (blue) open triangles represent the binding energies of
the fragments obtained in the nucleus-nucleus collisions. The latter fragments
might be stable but not in their ground state because they would normally
deexcite say by $\gamma$ rays at very long times, a mechanism not included in
the model.

To test the ability of the modified model to work with exotic nuclei, we
calculated the binding energies of the ground state of whole calcium isotopes
as shown in Fig. 3. Though the binding energies calculated by the modified
model are a little more bound, little differences (within the calculations
uncertainty) to the original approach are observed. Figure 4 shows the binding
energies of the nuclei with mass $A=40$. Both models reproduce the binding
energies in a similar fashion within the error bars, thus suggesting that with
larger nuclei the Heisenberg principle is less and less important than the
Pauli one as expected. Clearly the reproduction of the data is not perfect and
we did not try to adjust parameters since we believe that important effects,
such as (iso)spin-(iso)spin forces are missing and we plan to include those
effects in future works.

FIG. 3: (Color online) Binding energies of the calcium isotopes. FIG. 4:
(Color online) Binding energies of nuclei with mass $A=40$.

In order to test the reliability of the modified collision term coupled to the
mean field, we performed elastic scattering simulations for the p+12C system
at different energies. The collisions were performed in the range of impact
parameters from 0 to 6 fm. Since the proton is scattered from the target
without losing energy (we included a little energy loss below 4 MeV, the first
excited level of 12C), the reaction is rather fast and we followed its time
evolution up to 200 fm/c. In Fig. 5, the cross sections for the elastic
scattering data of 35 MeV pignanelli1986 , 51.93 MeV ohnuma1980 , 120 MeV
comfort1981 , and 250 MeV meyer1988  protons from 12C are compared with the
results of our model. As shown in the figure, the results of the modified
model are in good agreement with the experimental data except for the large
angle parts, mostly due to low statistics (1,000,000 events for each case).
Because of the large numerical effort, we performed calculations for the
modified model only. Inelastic scattering calculations required a much longer
CPU time and we decided to postpone them until (iso)spin-(iso)spin forces are
included in the model. Pion production is not included in the model, but its
effect might become important at the highest beam energy. However, since we
are looking at the elastic scattering data only, we feel our calculations are
reliable.

We also adopted another method usually used in the literature to test our
results preston1975 . First we calculated the charge density distribution in
the ground state of 12C displayed in Fig. 6. The Fourier transform of the
charge density times the Mott cross section preston1975  gives the elastic
cross sections reported in Fig. 5 as full lines (CoMDH) and dashed lines
(CoMD). The small difference in the densities from the two models does not
produce any appreciable difference in the elastic scattering. The agreement
with data is satisfactory and seems to agree better with increasing proton
beam energy, a similar feature noticed in the numerical simulations.

FIG. 5: (Color online) Comparison of calculated cross sections for the elastic
scattering of 35 MeV, 51.93 MeV, 120 MeV, and 250 MeV protons from 12C with
data pignanelli1986 ; ohnuma1980 ; comfort1981 ; meyer1988 ; kim2008 . The
(black) full squares represent the experimental data. The (red) solid circles
represent the calculated results with the CoMDH model. The (red) solid and the
(blue) dashed lines represent the cross sections calculated with the Fourier
transform of the charge density distribution (see the discussion in the
context) for the CoMDH and the CoMD models, respectively. FIG. 6: (Color
online) Comparison of calculated density distributions in the ground state of
12C.

Since we have tested the main ingredients of the model, we next turn to
fragmentation reactions. Figure 7 shows the charge distribution of 197Au+197Au
at 35 MeV/A with impact parameters ranging from 0 to 3.5 fm and t=1500 fm/c.
The modified model seems to work better than the original one, especially in
the light nuclei region, noticing that the original model gives too many
protons (not displayed in the figure - out of scale) papa2001 . At $Z=$20, the
data show a jump due to the different detectors used desequelles1998 . We
would get a better agreement to the data with the calculations if we slightly
shift the data yield up for $Z>$20 papa2001 . In fragmentation reactions,
fluctuations are important since they could signal critical phenomena such as
a liquid-gas phase transition. In order to test different physical
observables, we calculated the fragment ($Z\geq 4$) multiplicity
($N_{f4}$-Fig. 8a), the total charge bound in form of fragments ($Z_{b4}$-Fig.
8b), the fourth moment of the fragment charge distribution ($m_{4}$-Fig. 8c),
and the charge asymmetry distribution (a123-Fig. 8d) and compared to the
experimental data desequelles1998 . We took into account that the experimental
detection efficiency of the total charge is at least 70%, thus we modified the
total charge bound in form of fragments in our calculations by a factor of 85%
(the average of 70% and 100%). For all the charge correlations shown, the
modified model with the Heisenberg principle reproduced the data much better
than the original CoMD model. After adding the Heisenberg principle, the light
nuclei are less bound, thus the production of these particles is smaller while
that of the heavier fragments becomes larger. This will lead to an increase in
the total bounded charge and also in the fluctuation of the charge
distribution.

FIG. 7: (Color online) Comparison between the experimental isotope
distribution and the calculated results of the CoMDH and the CoMD models for
197Au+197Au at 35 MeV/A with impact parameters ranging from 0 to 3.5 fm and
t=1500 fm/c desequelles1998 ; papa2001 . FIG. 8: (Color online) Comparison of
the observable distributions for the experimental data (black full squares),
the CoMDH (red solid circles) and the CoMD (blue solid triangles) calculations
for the 197Au+197Au at 35 MeV/A with impact parameters ranging from 0 to 3.5
fm and t=1500 fm/c desequelles1998 ; papa2001 . (a) Fragment (Z $\geq$ 4)
multiplicity ($N_{f4}$), (b) Total charge bound in form of fragments
($Z_{b4}$), (c) Fourth moment $m_{4}$ of the fragment charge distribution with
the heaviest fragment excluded, (d) Charge asymmetry distribution. All the
vertical axes represent the mean multiplicity per event of the considered
observable desequelles1998 . FIG. 9: (Color online) Isotope distribution of
238U+238U at 15 MeV/A with impact parameter of 0 fm and t=4000 fm/c using the
CoMDH model. FIG. 10: (Color online) Time evolution of Livermorium-291 using
the CoMDH model. The red spheres represent protons, while the blue ones
represent neutrons. Only the largest clusters are shown.

We performed calculations at lower beam energies to test the probability of
superheavy formation in the collision of very heavy nuclei maruyama2002 ;
wuenschel2018 . Figure 9 displays the charge distribution of the 238U+238U at
15 MeV/A with zero impact parameter and t=4000 fm/c. The binding energies of
some isotopes coming out from the reaction region are shown in Fig. 2. We see
that all the binding energies of the nuclei from the collisions are smaller
than those in the initialization step, since they are still excited after the
4000 fm/c evolution (in the model they cannot emit say gamma rays to decay to
their ground states). The binding energies of deuterons and alpha particles of
the modified model are smaller than those of the original one, while for heavy
nuclei, they are more or less the same, consistent with the initialization
step. In the superheavy region, a peak around Z=116 was found. We selected one
nucleus with Z=116 and A=291, and let it evolve with time. One out of 500,000
events survived for a total time of another 6000 fm/c with only several
neutrons emitted as shown in Fig. 10, then finally it underwent fission and
broke into 136La+127Xe and one alpha particle with 15.78 MeV kinetic energy
(close to the value predicted by the nuclear density functional theory, which
is around 14 MeV staszczak2013 ; wuenschel2018 ), and several protons and
neutrons. This might be a different path to study super-heavy production
maruyama2002  and we will discuss more in detail in future works since a large
calculation effort is required.

##  IV Summary

In this paper, we have succeeded in modifying the CoMD model by introducing
the Heisenberg principle. The binding energies of the light particles are more
comparable to the experimental data in both initializations and collisions.
The modified model works well with exotic nuclei. We have also performed some
superheavy nuclear reactions with the new model. Though we got some events,
more efforts, both numerically and theoretically, need to be done and we
decided to postpone it until further refinements in the model are included
such as the (iso)spin-(iso)spin forces.

###### Acknowledgements.

This work is partially supported by the National Natural Science Foundation of
China under Contracts Nos. 11421505, 11305239, 11205230 and 11520101004, the
Major State Basic Research Development Program in China under Contracts Nos.
2014CB845401 and 2013CB834405, the CAS Project Grant No. QYZDJSSW-SLH002, and
also the Chinese Government Scholarship. This material is based upon work
supported by the Department of Energy, National Nuclear Security
Administration, under Award Number DE-NA0003841, the Center for Excellence in
Nuclear Training And University-based Research (CENTAUR).

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# Recent Developments in Brill-Noether Theory

David Jensen  David Jensen: Department of Mathematics, University of Kentucky
733 Patterson Office Tower, Lexington, KY 40506–0027, USA<EMAIL_ADDRESS>and  Sam Payne  Sam Payne: Department of Mathematics, University of Texas at
Austin
2515 Speedway, RLM 8.100, Austin, TX 78712, USA<EMAIL_ADDRESS>
(Date: October 18, 2021)

###### Abstract.

We briefly survey recent results related to linear series on curves that are
general in various moduli spaces, highlighting the interplay between algebraic
geometry on a general curve and the combinatorics of its degenerations.
Breakthroughs include the proof of the Maximal Rank Theorem, which determines
the Hilbert function of the general linear series of given degree and rank on
the general curve in $\mathcal{M}_{g}$, and complete analogs of the standard
Brill-Noether theorems for curves that are general in Hurwitz spaces. Other
advances include partial results in a similar direction for linear series in
the Prym locus of a general unramified double cover of a general $k$-gonal
curve and instances of the Strong Maximal Rank Conjecture.

##  1\. Linear Series on a Curve General in $\mathcal{M}_{g}$

In the early days of algebraic geometry, an algebraic curve was understood as
coming equipped with an embedding in projective space. In the late 19th and
early 20th century, however, mathematicians began to consider abstract curves
and Riemann surfaces, without reference to an ambient space. Since that time,
a central problem in algebraic geometry has been the following:

######  Problem 1.1.

Given a curve $C$, describe all maps from $C$ to projective space.

When approaching this problem, it is natural to fix discrete invariants, such
as the dimension of the projective space $\mathbb{P}^{r}$ and the degree $d$
of the maps. Passing to the language of line bundles and Picard varieties, we
are then interested in the geometry of the _Brill-Noether variety_

$W^{r}_{d}(C)\colon=\Big{\\{}L\in\operatorname{Pic}^{d}(C)\mbox{ }|\mbox{
}h^{0}(C,L)\geq r+1\Big{\\}}.$

A series of results in the 1980s describe these varieties when $C$ is general
in the moduli space $\mathcal{M}_{g}$. Throughout this survey, we assume that
the ground field is algebraically closed field of characteristic zero. For
extensions to positive characteristic, see Remark 1.3.

######  Theorem 1.2.

Let $C$ be a curve that is general in $\mathcal{M}_{g}$. Let
$\rho(g,r,d):=g-(r+1)(g-d+r)$.

  1. (1)

The variety $W^{r}_{d}(C)$ is of pure dimension $\min\\{\rho(g,r,d),g\\}$ when
this is non-negative, and empty otherwise [GH80];

  2. (2)

The locally closed variety $W^{r}_{d}(C)\smallsetminus W^{r+1}_{d}(C)$ is
smooth [Gie82];

  3. (3)

If $\rho(g,r,d)\geq 1$ then $W^{r}_{d}(C)$ is irreducible [FL81, Gie82];

  4. (4)

If $\rho(g,r,d)=0$ then
$|W^{r}_{d}(C)|=g!\cdot\prod_{j=0}^{r}\frac{j!}{(g-d+r+j)!}$ [Kem71, KL72,
GH80];

  5. (5)

If $\rho(g,r,d)=0$ then the universal $W^{r}_{d}$ over the locus of curves in
$\mathcal{M}_{g}$ with finitely many linear series of rank $r$ and degree $d$
is irreducible [EH87a];

This series of results about linear series on the generic curve in
$\mathcal{M}_{g}$, and more recent generalizations and extensions to general
curves in other moduli spaces, are broadly known as _Brill-Noether Theory_ ,
in honor of the celebrated work [BN74]. We will highlight the interplay
between algebraic geometry and the combinatorics of degenerations in this
subject. Note that the generic curve itself is a single object with no
combinatorial structure, and it is possible to prove much of Theorem 1.2
without degenerations, e.g., using vector bundle methods [Laz86]. Such an
approach is used in [ABFS16] to construct smooth Brill-Noether-Petri-general
curves of every genus defined over $\mathbb{Q}$.

In this brief survey, we emphasize the role of degenerations and their
associated combinatorics in Brill-Noether theory. The numerical invariants
appearing in Theorem 1.2 have rich combinatorial significance and intricate
combinatorial structures come to the surface in various proofs by degeneration
methods. For instance, when $\rho(g,r,d)=0$, the number of points in
$W^{r}_{d}(C)$ is equal to the number of standard tableaux on the rectangular
Young diagram $\lambda(g,r,d)$ with $r+1$ columns and $g-d+r$ rows. A
beautiful recent generalization says that the holomorphic Euler characteristic
of $W^{r}_{d}(C)$ is equal to the number of set-valued tableaux on
$\lambda(g,r,d)$ [ACT21, CP21].

In general, the codimension of $W^{r}_{d}(C)$ in $\mathrm{Pic}^{d}(C)$ is the
number of boxes in $\lambda(g,r,d)$. In some degeneration arguments, the limit
of $W^{r}_{d}(C)$ breaks into pieces that are naturally in bijection with the
standard fillings of $\lambda(g,r,d)$ with entries from $\\{1,\ldots,g\\}$. In
others, the limiting pieces are related to solutions of an intersection
problem on a Grassmannian, which is in turn related to the combinatorics of
Young tableaux via Schubert calculus. The degeneration methods that have
appeared in the Brill-Noether literature are themselves rich and varied. The
limits of $C$ that have been fruitfully considered include $g$-nodal [GH80,
Gie82] and $g$-cuspidal rational curves [EH83a], flag curves with $g$ elliptic
tails [EH83b, EH87a], chains of elliptic curves [Oss14], and tropical curves,
most often chains of $g$ loops [CDPR12, JP14].

Non-degenerative methods from algebraic geometry and the combinatorics of
various degenerations all continue to play key roles in current developments
in Brill-Noether theory. In some cases, similarities in the combinatorial
structures that appear through otherwise disjoint approaches are enough to
yield new insights and breakthroughs. We hope readers will be enticed to dive
more deeply into the details of this subject, to enjoy and appreciate the
interplay between algebraic geometry and combinatorics in the study of special
linear series on general curves.

######  Remark 1.3.

Here, we mention extensions of the surveyed results to positive
characteristic. Theorems 1.2 and 4.3 hold in arbitrary characteristic. Theorem
3.1 holds in characteristic not 2. Theorem 2.7 pre-supposes irreducibility of
the Hurwitz space $\mathcal{H}_{k,g}$, which is known in characteristic
greater than $k$ [Ful69]. Nevertheless, the analog of parts (1)-(4) holds for
some component of $\mathcal{H}_{k,g}$ in arbitrary characteristic, as does the
analog of part (5) in any characteristic not dividing $k$. Theorem 3.3 pre-
supposes irreducibility of the moduli of chains of covers; we are unaware of
any extensions to positive characteristic. Nevertheless, the analog of Theorem
3.3 holds for some component of this moduli space in all characteristics not
equal to $2$, $3$, or $5$ and not dividing $k$.

Acknowledgments. We are grateful to Gavril Farkas, Eric Larson, Hannah Larson,
Yoav Len and Nathan Pflueger for helpful comments on an earlier draft of this
survey.

The first author was supported by NSF DMS–2054135. The second author was
supported by NSF DMS–2001502 and DMS–2053261.

##  2\. Hurwitz-Brill-Noether Theory

Theorem 1.2 answers many natural questions about the geometry of
$W^{r}_{d}(C)$ when $C$ is general in $\mathcal{M}_{g}$. But what about curves
that are not general? If a curve $C$ admits an atypical linear series, what
does that imply about the existence and behavior of other linear series on
$C$? One compelling special case is the following analog of Problem 1.1 for
curves with a special pencil.

######  Problem 2.1.

If $\pi\colon C\to\mathbb{P}^{1}$ is a general cover of genus $g$ and degree
$k$, describe $W^{r}_{d}(C)$.

Here, a general cover means that $C\to\mathbb{P}^{1}$ is general in an
appropriate Hurwitz space. At first glance, all of the beautiful structures
from classical Brill-Noether theory for curves general in $\mathcal{M}_{g}$
seem to break down. One can, without much difficulty, construct examples where
$W^{r}_{d}(C)$ is not equidimensional, and $W^{r}_{d}(C)\smallsetminus
W^{r+1}_{d}(C)$ is singular.

######  Example 2.2.

Let $\pi\colon C\to\mathbb{P}^{1}$ be a general cover of genus 6 and degree 3.
Then $W^{1}_{4}(C)$ is the disjoint union of a 1-dimensional component
consisting of line bundles of the form
$\pi^{*}\mathcal{O}(1)\otimes\mathcal{O}_{C}(p)$ for $p\in C$ and an isolated
point $K_{C}\otimes\pi^{*}\mathcal{O}(-2)$.

Nevertheless, special cases have been fully understood for many years,
including all cases where $C\to\mathbb{P}^{1}$ has degree $2$ or $3$ [Cli78,
Mar46]. In the 1990s, researchers considered higher degree covers, found upper
bounds for $\dim W^{r}_{d}(C)$, and proved the existence of components of
certain specified dimensions [Mar96, BK96, CM99]. More recently, Pflueger used
tropical methods on a chain of loops to produce an improved upper bound for
$\dim W^{r}_{d}(C)$, which he conjectured to be sharp [Pfl17]. Pflueger also
remarked that his results could be proved by similar combinatorial arguments
using degeneration to a chain of elliptic curves instead of tropical
degenerations, and both perspectives have been useful in subsequent work.

Pflueger’s work inspired a new wave of interest in this area. His conjecture
was proved by Ranganathan and the first author, using the algebraic geometry
of curves in toric varieties in combination with tropical geometry and log
deformation theory.

######  Theorem 2.3 ([JR21]).

Let $C\to\mathbb{P}^{1}$ be a general cover of genus $g$ and degree $k$. Then

$\dim
W^{r}_{d}(C)=\max_{\ell\in\\{0,\ldots,r^{\prime}\\}}\rho(g,r-\ell,d)-\ell k,$

where $r^{\prime}=\min\\{r,g-d+r-1\\}$.

Here, $W^{r}_{d}(C)$ may not have pure dimension, and $\dim W^{r}_{d}(C)$
means the maximum of the dimensions of its components. This analog of Theorem
1.2(1) has been followed by more comprehensive results, giving a description
of all components of $W^{r}_{d}(C)$, along with their dimensions and basic
properties. The starting point for this subsequent work is a stratification of
$W^{r}_{d}(C)$, using the following discrete invariants that generalize the
Maroni invariants of trigonal curves, introduced independently by H. Larson
[Lar21] and by Cook-Powell and the first author [CJ19].

We say that a line bundle has _splitting type_
$\bm{\mu}=(\mu_{1},\ldots,\mu_{k})$ if
$\pi_{*}L\cong\oplus_{i=1}^{k}\mathcal{O}(\mu_{i})$. The splitting type of a
line bundle $L$ is a more refined invariant than its rank and degree; it
encodes the rank of $L\otimes\pi^{*}\mathcal{O}(m)$ for all integers $m$.

######  Definition 2.4.

The _splitting type locus_ is

$W^{\bm{\mu}}(C,\pi)\colon=\Big{\\{}L\in\operatorname{Pic}(C)\mbox{ }|\mbox{
}\pi_{*}L\cong\bigoplus_{i=1}^{k}\mathcal{O}(\mu_{i})\Big{\\}}.$

######  Remark 2.5.

Note that $W^{\bm{\mu}}(C,\pi)\subset\operatorname{Pic}(C)$ is locally closed;
it is the analog of $W^{r}_{d}(C)\smallsetminus W^{r+1}_{d}(C)$ in the setup
for Theorem 1.2.

The analog of Theorem 1.2 for splitting type loci involves numerical
invariants that again have subtle and rich combinatorial significance, with
$k$-core partitions playing the role of arbitrary partitions in standard
Brill-Noether theory. While new in this context, $k$-core partitions have been
studied extensively in number theory, representation theory, and
combinatorics. We recall the definition and a few basic properties; see
[LLM+14] for a fuller treatment and further references.

We identify partitions with upper-left justified Young diagrams, so the
partition $4=3+1$ is depicted by the Young diagram in Figure 1.

&

Figure 1. The Young diagram corresponding to the parition $4=3+1$.

The Young lattice $\mathcal{Y}$ is the set of partitions, partially ordered by
inclusion of Young diagrams.

Each box in a partition determines a _hook_ , consisting of that box, together
with the boxes below it in the same column, and the boxes to the right of it
in the same row. The _hook length_ of a box is the number of boxes in the
associated hook.

######  Definition 2.6.

A _$k$ -core partition_ is a partition in which no box has hook length equal
to $k$.

For instance, the partition $4=3+1$ is $3$-core, but not $2$-core or $4$-core.

The $k$-core partitions $\mathcal{Y}_{k}\subset\mathcal{Y}$ form a sublattice
with respect to the induced partial order. For a $k$-core partition $\lambda$,
we write $r_{k}(\lambda)$ for the poset rank of
$[\hat{0},\lambda]\subset\mathcal{Y}_{k}$, i.e., the length of maximal chains
in the interval between the empty partition $\hat{0}$ and $\lambda$. Note that
$r_{k}(\lambda)$ is typically strictly smaller than the number of boxes in
$\lambda$, because deleting a corner box from a $k$-core partition typically
does not yield another $k$-core partition.

To each splitting type $\bm{\mu}=(\mu_{1},\ldots,\mu_{k})$, we associate a
$k$-core partition $\lambda(\bm{\mu})$ as follows:

$\lambda(\bm{\mu})\colon=\Big{\\{}(x,y)\in\mathbb{N}^{2}\mbox{ }|\mbox{
}\exists m\in\mathbb{Z}\text{ s.t.
}x\leq\sum_{i=1}^{k}\max\\{0,\mu_{i}+m+1\\},y\leq\sum_{i=1}^{k}\max\\{0,-\mu_{i}-m-1\\}\Big{\\}}.$

For example, Figure 2 depicts two 3-core partitions of the form
$\lambda(\bm{\mu})$.

&

&

$\lambda(\bm{\mu}_{0})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\lambda(\bm{\mu}_{1})$

Figure 2. The 3-core partitions associated to the splitting types
$\bm{\mu}_{0}=(-4,0,0)$ and $\bm{\mu}_{1}=(-3,-2,1)$.

We can now state the main results on splitting type loci.

######  Theorem 2.7.

Let $\pi\colon C\to\mathbb{P}^{1}$ be a general cover of genus $g$ and degree
$k$. Then

  1. (1)

The locally closed subvariety
$W^{\bm{\mu}}(C,\pi)\subset\operatorname{Pic}^{d}(C)$ has pure dimension
$g-r_{k}(\lambda(\bm{\mu}))$ when this is nonnegative, and is empty otherwise
[Lar21, CJ19];

  2. (2)

If $g-r_{k}(\lambda(\bm{\mu}))\geq 0$ then $W^{\bm{\mu}}(C,\pi)$ is smooth
[Lar21];

  3. (3)

If $g-r_{k}(\lambda(\bm{\mu}))\geq 1$ then $W^{\bm{\mu}}(C,\pi)$ is
irreducible [LLV20];

  4. (4)

If $g-r_{k}(\lambda(\bm{\mu}))=0$ then $|W^{\bm{\mu}}(C,\pi)|$ is the number
of maximal chains in the interval
$[\hat{0},\lambda(\bm{\mu})]\subset\mathcal{Y}_{k}$ [LLV20];

  5. (5)

If $g-r_{k}(\lambda(\bm{\mu}))=0$ then the universal $W^{\bm{\mu}}$ over the
locus of covers in the Hurwitz space with finitely many line bundles of
splitting type $\bm{\mu}$ is irreducible [LLV20].

Theorem 2.7 is precisely analogous to Theorem 1.2, with the lattice of
$k$-core partitions $\mathcal{Y}_{k}$ playing the role of the full lattice of
partitions $\mathcal{Y}$. Indeed, Theorem 1.2 says that if $C$ is general in
$\mathcal{M}_{g}$, then the codimension of $W^{r}_{d}(C)$ in
$\operatorname{Pic}^{d}(C)$ is the rank of the interval
$[\hat{0},\lambda(g,r,d)]\subset\mathcal{Y}$ and if $\dim W^{r}_{d}(C)=0$ then
$|W^{r}_{d}(C)|$ is equal to the number of maximal chains in this interval.
When $k$ is sufficiently large, Theorem 2.7 specializes to Theorem 1.2.

######  Example 2.8.

Returning to Example 2.2, let $\pi\colon C\to\mathbb{P}^{1}$ be a general
cover of genus 6 and degree 3. If $L\in W^{1}_{4}(C)$ has splitting type
$\bm{\mu}=(\mu_{1},\mu_{2},\mu_{3})$, then

$\sum_{i=1}^{3}\max\\{0,\mu_{i}+1\\}=h^{0}(\pi_{*}L)=h^{0}(L)=2\mbox{ and }$
$\sum_{i=1}^{3}\mu_{i}=\deg(\pi_{*}L)=\chi(L)-3=-4.$

The two splitting types that satisfy both these conditions and the condition
that $6-r_{3}(\lambda(\bm{\mu}))\geq 0$ are $\bm{\mu}_{0}=(-4,0,0)$ and
$\bm{\mu}_{1}=(-3,-2,1)$. Each interval
$[\hat{0},\lambda(\bm{\mu}_{i})]\subset\mathcal{Y}_{3}$ has a unique maximal
chain, as shown in Figure 3. The lengths of these chains show that
$r_{3}(\lambda(\bm{\mu}_{0}))=6$ and $r_{3}(\lambda(\bm{\mu}_{1}))=5$. By
Theorem 2.7, this implies that $\dim W^{\bm{\mu}_{i}}(C,\pi)=i$ and moreover,
since the maximal chain in
$[\hat{0},\lambda(\bm{\mu}_{0})]\subset\mathcal{Y}_{3}$ is unique,
$W^{\bm{\mu}_{0}}(C,\pi)$ is a single point. A general trigonal curve is not
hyperelliptic, so $W^{2}_{4}(C)$ is empty and thus, as discussed in Example
2.2, $W^{1}_{4}(C)$ has two irreducible components, one of dimension one and
the other an isolated point of dimension 0.

1 & 2 3 4 5 6
3 4 5 6
5 6
1 & 3 5
2 4
3 5
4
5

Figure 3. Maximal chains of 3-core partitions in the intervals
$[\hat{0},\lambda(\bm{\mu}_{0})]$ and $[\hat{0},\lambda(\bm{\mu}_{1})]$.

These results are proven by degeneration to a chain of elliptic curves in
[Lar21, LLV20]. To ensure that this degenerate curve admits a map of degree
$k$ to a rational curve, one specifies that the difference between the two
attaching points on each component has torsion order $k$ in the group law on
the elliptic curve. Building on Pflueger’s earlier work, one can then classify
the limit linear series on this degenerate curve that are limits of line
bundles of a given splitting type. The tableaux that arise from this
construction are precisely those that come from maximal chains in the lattice
of $k$-cores. The same combinatorial structures arise when approaching these
problems via tropical degenerations, as in [CJ19, CJ20].

######  Remark 2.9.

Another interesting result in late 20th century Brill-Noether Theory says
that, if $\rho(g,r,d)<0$ then every component of the locus in
$\mathcal{M}_{g}$ of curves $C$ satisfying $W^{r}_{d}(C)\neq\emptyset$ has
codimension at most $-\rho(g,r,d)$ [Ste98]. To the best of our understanding,
the analogs of this statement in Hurwitz-Brill-Noether Theory, and in the
Prym-Brill-Noether theory discussed below, are open problems.

##  3\. Prym-Brill-Noether Theory

Let $C$ be a smooth curve of genus $g$, and let $\pi\colon\widetilde{C}\to C$
be an unramified double cover. In [Wel85], Welters defines the _Prym-Brill-
Noether locus_ to be

$V^{r}(C,\pi)\colon=\Big{\\{}L\in\operatorname{Pic}^{2g-2}(C)\mbox{ }|\mbox{
}\textrm{Nm}_{\pi}(L)=\omega_{C},h^{0}(C,L)\geq r+1,\text{ and
}h^{0}(C,L)\equiv r+1\pmod{2}\Big{\\}}.$

The following is an analog of parts (1)-(4) of Theorem 1.2 in this setting.

######  Theorem 3.1.

Let $\pi\colon\widetilde{C}\to C$ be a general unramified double cover, where
the base has genus $g$.

  1. (1)

The variety $V^{r}(C,\pi)$ is of pure dimension $g-1-{{r+1}\choose{2}}$ when
this is non-negative, and empty otherwise [Wel85, Ber87];

  2. (2)

The singular locus of $V^{r}(C,\pi)$ is $V^{r+2}(C,\pi)$ [Wel85] ;

  3. (3)

If $g-1-{{r+1}\choose{2}}\geq 1$ then $V^{r}(C,\pi)$ is irreducible [Deb00] ;

  4. (4)

If $g-1-{{r+1}\choose{2}}=0$, then
$|V^{r}(C,\pi)|=2^{{r}\choose{2}}\cdot(g-1)!\cdot\prod_{i=1}^{r}\frac{(i-1)!}{(2i-1)!}$
[DCP95].

It is possible to prove (3) and (4) using vector bundle techniques rather than
degenerations [DCP95]. As in Theorem 1.2, the numerical invariants that appear
have rich combinatorial significance, and intricate structures involving
partitions and tableaux emerge in proofs via degeneration.

######  Remark 3.2.

The Prym-Brill-Noether analog of Theorem 1.2 (5) is an open problem.

Let $\Delta_{r}=r+(r-1)+\cdots+1$ denote the isosceles triangular partition
with side length $r$.

&

Figure 4. The partition $\Delta_{3}$.

This partition plays a role in Prym-Brill-Noether theory analogous to that of
the rectangular partition $\lambda(g,r,d)$ in the Brill-Noether theory of a
general curve in $\mathcal{M}_{g}$. Indeed, Theorem 3.1 says that, if
$\pi\colon\widetilde{C}\to C$ is a general unramified double cover, then the
codimension of $V^{r}(C,\pi)$ in the Prym variety is the rank of the interval
$[\hat{0},\Delta_{r}]\subset\mathcal{Y}$, and if $\dim V^{r}(C,\pi)=0$, then
$|V^{r}(C,\pi)|$ is equal to the number of maximal chains in this interval.

Given Theorem 3.1 for unramified double covers of a general curve, it is
natural to ask next about the Prym-Brill-Noether theory of double covers of a
curve $C$ that is not necessarily general in $\mathcal{M}_{g}$. As in Section
2, a compelling special case is when $C$ is general in the $k$-gonal locus,
i.e., the irreducible locus of curves in $\mathcal{M}_{g}$ on which the
minimal degree of a pencil is $k$, for $k\geq 3$. When $k\geq 3$, there is an
irreducible moduli space of chains of covers $\widetilde{C}\to
C\to\mathbb{P}^{1}$, where the first map is an unramified double cover and the
second has degree $k$ [BF86], so the generic such cover is well-defined. In
this setting, recent work provides new bounds on the dimension of
$V^{r}(C,\pi)$, obtained for even $k$ in [LU20] and for all $k$ in [CLRW20].

######  Theorem 3.3.

[LU20, CLRW20] Let $\pi\colon\widetilde{C}\to C$ be an unramified double cover
of a general $k$-gonal curve of genus $g$, with $k\geq 3$. Let
$\ell=\left\lceil\frac{k}{2}\right\rceil$, and let

$n(r,k)\colon=\left\\{\begin{array}[]{ll}{{\ell+1}\choose{2}}+\ell(r-\ell)&\textrm{if
$\ell\leq r-1$}\\\ {{r+1}\choose{2}}&\textrm{if $\ell\geq
r$.}\end{array}\right.$

Then

$\dim V^{r}(C,\pi)\leq g-1-n(r,k).$

This upper bound is not sharp in general. For example, consider the case where
$g=6$ and $k=4$. A general curve of genus 6 is tetragonal, so by Theorem 3.1
we see that $V^{3}(C,\pi)$ is empty. Since $\ell=2=r-1$, however, Theorem 3.3
yields only

$\dim V^{3}(C,\pi)\leq 6-1-{{3}\choose{2}}-2=0.$

However, the bound is conjectured to be sharp when $g$ is sufficiently large.

######  Conjecture 3.4.

[CLRW20] Let $\pi\colon\widetilde{C}\to C$ be an unramified double cover of a
general $k$-gonal curve, with $k\geq 3$. If $g\gg n(r,k)$, then

$\dim V^{r}(C,\pi)=g-1-n(r,k).$

From a combinatorial perspective, Theorem 3.3 arises from studying the
partition $\Delta_{r}$ in the lattice of $k$-core partitions
$\mathcal{Y}_{k}$. Note that every box in $\Delta_{r}$ has odd hook length, so
if $k$ is even, then $\Delta_{r}$ is a $k$-core partition. When $k$ is even,
$n(r,k)$ is the rank of $\Delta_{r}$ in $\mathcal{Y}_{k}$. For example, Figure
5 depicts a maximal chain in the interval
$[\hat{0},\Delta_{3}]\subset\mathcal{Y}_{4}$. This chain has length 5,
corresponding to the fact that $n(3,4)=r_{4}(\Delta_{3})=5$.

1 & 3 4
2 5
4

Figure 5. A maximal chain in $\mathcal{Y}_{4}$.

A similar result holds when $k$ is odd, but because $\Delta_{r}$ is not a
$k$-core in this case, it requires the language of displacement tableaux from
[Pfl17]. A tableau $t$ is said to be a _$k$ -uniform displacement tableau_ if,
whenever $t(x,y)=t(x^{\prime},y^{\prime})$, we have $x-y\equiv
x^{\prime}-y^{\prime}\pmod{k}$. The value $n(r,k)$ is equal to the minimum
number of symbols in a $k$-uniform displacement tableau on $\Delta_{r}$.

Theorem 3.3 is proven via tropical methods. In [LU20], Len and Ulirsch show
that the skeleton of a Prym variety is the Prym variety of the skeleton. They
then construct an explicit double cover
$\varphi\colon\widetilde{\Gamma}\to\Gamma$ of tropical curves, which they call
the _$k$ -gonal uniform folded chain of loops_, and study the tropical Prym-
Brill-Noether variety $V^{r}(\Gamma,\varphi)$. These ideas are developed
further in [CLRW20], which establishes not only the dimension bound above, but
also various topological properties.

######  Theorem 3.5.

[CLRW20] Let $\varphi\colon\widetilde{\Gamma}\to\Gamma$ be a $k$-gonal uniform
folded chain of loops. Then the tropical Prym-Brill-Noether variety
$V^{r}(\Gamma,\varphi)$ has pure dimension $g-1-n(r,k)$ and, when this
dimension is positive, it is connected in codimension one.

Tropicalizations of irreducible varieties are of pure dimension and connected
through codimension 1 [CP12], so Theorem 3.5 points suggestively toward
analogs of parts (1) and (3) of Theorem 1.2.

Many open questions remain about the Prym-Brill-Noether variety
$V^{r}(C,\pi)$, where $\pi\colon\widetilde{C}\to C$ is an unramified double
cover of a general $k$-gonal curve.

######  Problem 3.6.

Let $\pi\colon\widetilde{C}\to C$ be an unramified double cover of a general
$k$-gonal curve, with $k\geq 3$.

  1. (1)

What is $\dim V^{r}(C,\pi)$?

  2. (2)

Is the locally closed variety $V^{r}(C,\pi)\smallsetminus V^{r+2}(C,\pi)$
smooth?

  3. (3)

If $\dim V^{r}(C,\pi)\geq 1$, is $V^{r}(C,\pi)$ irreducible?

  4. (4)

When $\dim V^{r}(C,\pi)=0$, what is $|V^{r}(C,\pi)|$?

  5. (5)

When $\dim V^{r}(C,\pi)=0$, what is the action of monodromy?

One might reasonably approach such problems starting from [CLRW20] via
tropical lifting, as in [CJP15], or via log deformation theory, as in [JR21].
Alternatively, one could look for analogous statements about limit linear
series on a double cover of a $k$-gonal chain of elliptic curves by a folded
chain of elliptic curves, and identify which of these are limits of line
bundles in the Prym-Brill-Noether locus of a degenerating family of covers, as
was done in [LLV20] for the Hurwitz-Brill-Noether theory.

##  4\. Multiplication Maps and Maximal Rank Results

A _linear series_ of rank $r$ on a curve $C$ is a pair $(L,V)$, where $L$ is a
line bundle on $C$ and $V\subseteq H^{0}(C,L)$ is a vector subspace of
dimension $r+1$. By Theorem 1.2, when $\rho(g,r,d)\geq 0$, the universal space
$G^{r}_{d}$ of linear series of degree $d$ and rank $r$ has a unique
irreducible component that dominates $\mathcal{M}_{g}$. In other words, there
is a well-defined generic linear series on the generic curve of genus $g$.
Natural directions for further study therefore include the syzygies of this
generic linear series, and the geometry of the locus of linear series with
special syzygies.

The most basic syzygies to consider in this context are the kernels of the
multiplication maps from symmetric powers of a linear series to global
sections of tensor powers of the underlying line bundle. Recall that a map
between finite dimensional vector spaces has _maximal rank_ if it is either
injective or surjective. Harris popularized the _Maximal Rank Conjecture_ ,
i.e., the prediction that the multiplication maps for the general linear
series on the generic curve should have maximal rank [Har82]. This is now a
theorem of E. Larson.

######  Theorem 4.1 ([Lar17]).

Suppose $C$ is general and $V\subset H^{0}(C,L)$ is a general linear series of
given degree and rank. Then the multiplication maps

$\mu_{m}:\operatorname{Sym}^{m}V\rightarrow H^{0}(C,L^{\otimes m})$

have maximal rank for all $m$.

The proof of this theorem relies on an elaborate induction involving
interpolation for vector bundles [ALY19] and an analogous statement for
hyperplane sections of the general curve of genus $g$ and degree $d$ in
$\mathbb{P}^{r}$ [Lar20]. It also uses novel degeneration techniques that
surpass what was previously achieved using embedded methods à la Hirschowitz-
Horace [Hir85], in part by exploiting properties of the Kontsevich moduli
space of stable maps. The essential ingredients span more than a half-dozen
articles, some of which remain unpublished, as does the final paper completing
the proof. For a short and helpful overview of this work, see [Lar18].

When $\rho(g,r,d)=0$, Theorems 1.2 and 4.1 together imply that every linear
series of degree $d$ and rank $r$ on a general curve has multiplication maps
of maximal rank for all $m$. When $\rho(g,r,d)$ is positive, other natural
questions arise about the geometry of the locus in $G^{r}_{d}(C)$, for a
general curve $C$, where a given multiplication map fails to have maximal
rank. In this direction, Aprodu and Farkas proposed a _Strong Maximal Rank
Conjecture_ that predicts the codimension of the locus in $G^{r}_{d}(C)$ where
$\mu_{m}$ fails to have maximal rank, when $C$ is general, for suitable
parameters $g,r,d,m$.

######  Conjecture 4.2 ([AF11]).

Fix integers $g,r,d\geq 1$ such that $0\leq\rho(g,r,d)<r-2$, and fix $m\geq
2$. Let $C$ be a general curve in $\mathcal{M}_{g}$. Then the determinantal
variety

$\\{(L,V)\in G^{r}_{d}(C)\mid\mu_{m}\colon\operatorname{Sym}^{m}V\to
H^{0}(C,L^{\otimes m})\text{ fails to have maximal rank }\\}$

is of the expected dimension

$\rho(g,r,d)-1-\left|{r+m\choose m}-(2d+1-g)\right|,$

where, by convention, the locus is empty when the expected dimension is
negative.

One motivation for pursuing this conjecture is that the images of the loci of
pairs $(C,L)$ for which $\mu_{m}$ fails to have maximal rank are candidates
for interesting effective cycles in the coarse moduli space
$\overline{M}_{g}$. When these syzygytic loci are of the expected dimension,
their classes can sometimes be computed via intersection theory and
computations with test curves. For instance, the counterexamples to the Slope
Conjecture in [FP05, Far09] are instances of such syzygytic loci that happen
to be effective divisors outside the cone spanned by previously known
effective divisors.

The cases of the Strong Maximal Rank Conjecture where $\rho(g,r,d)=0$ follow
from Theorem 4.1. Only a small selection of other cases are known; these
include the non-special cases where $r=d-g$, [AF11, Proposition 5.7], and the
cases where $m=2$ and $d\leq g+1$ [TiB03]. Note that, for $m=2$, and for each
fixed $r$, there are only finitely many other cases to consider. The cases
where $m=2$ and $r=4$ are discussed in [AF11, Remark 5.6].

Farkas and the authors recently proved a few new cases of the Strong Maximal
Rank Conjecture for $m=2$, starting with $(g,r,d)=(22,6,25)$ and $(23,6,26)$.
In these cases, every linear series of the given parameters on a general curve
$C$ of genus $g$ is complete, so $W^{r}_{d}(C)=G^{r}_{d}(C)$, and the expected
dimension of the locus where the multiplication map fails to have maximal rank
is negative.

######  Theorem 4.3 ([FJP20]).

Let $g=22$ or $23$. For a general curve $C$ in $\mathcal{M}_{g}$, the
multiplication map

$\mu_{2}\colon\operatorname{Sym}^{2}H^{0}(C,L)\to H^{0}(C,L^{\otimes 2})$

is injective for _all_ line bundles $L\in W^{6}_{g+3}(C)$.

Once again, the syzygytic loci obtained as images of pairs $(C,L)$ where the
multiplication map fails to be injective turn out to be effective divisors of
smaller slope than any previously known effective divisors on
$\overline{M}_{22}$ and $\overline{M}_{23}$, respectively. Moreover, the
slopes are small enough to conclude that these moduli spaces are of general
type.

######  Theorem 4.4 ([FJP20]).

The moduli space of curves $\overline{M}_{g}$ is of general type for $g\geq
22$.

This improves the bound of $g\geq 24$ proved decades ago by Harris, Mumford,
and Eisenbud [HM82, Har84, EH87b]; they used Theorem 1.2 to construct the
requisite divisors of small slope in those cases. The case $g=22$ also
disproves the conjecture of Eisenbud and Harris that $\overline{M}_{g}$ should
be uniruled for $g<23$ [EH89]. The Kodaira dimensions and uniruledness of
$\overline{M}_{g}$ for $17\leq g\leq 21$ remain open problems.

Farkas and the authors also applied the techniques from [FJP20], with some
small but important improvements, to prove the Strong Maximal Rank Conjecture
for $m=2$ and $(g,r,d)=(13,5,16)$, and thereby construct divisors of small
slope on $\overline{M}_{13}$.

######  Theorem 4.5 ([FJP21]).

For a general curve $C$ in $\mathcal{M}_{13}$, the multiplication map

$\mu_{2}\colon\operatorname{Sym}^{2}H^{0}(C,L)\to H^{0}(C,L^{\otimes 2})$

is surjective for all line bundles $L\in W^{5}_{16}(C)$.

The closure of the locus of Brill-Noether general curves of genus 13 with at
least one $L\in W^{5}_{16}$ such that $\mu_{2}$ is not surjective supports an
effective divisor in $\overline{M}_{g}$. This is the first known effective
divisor on a moduli space $\overline{M}_{g}$ for $g\geq 10$ with slope less
than $6+10/g$; it had previously been believed that such divisors may not
exist [CFM13]. The class of this divisor is used to compute the class of yet
another divisor of even smaller slope, the _non-abelian Brill-Noether divisor_
, which is used to show that the Prym moduli space
$\overline{\mathcal{R}}_{13}$ is of general type; 13 is now the smallest $g$
for which $\overline{\mathcal{R}}_{g}$ is known to be of general type.

This survey is far from exhaustive, and highlights only a portion of the
recent results in this active area. It inevitably reflects the biases and
limitations of the authors. We hope the references provided will be a useful
starting point for those who wish to learn more.

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UDC 519.24

Equilibrium in Wright-Fisher models of population genetics D.Koroliouk†,
V.S.Koroliuk∗
†Institute of Telecommunications and Global Information Space

∗Institute of Mathematics

†∗Ukrainian Academy of Sciences, Kiev, Ukraine

Abstract. For multivariant Wright-Fisher models in population genetics we
introduce equilibrium states, expressed by fluctuations of probability
relations, in distinction of the traditionally used fluctuations, expressed by
the difference between the current value of a random process and its
equilibrium value.
Then the drift component of the gene frequencies dynamic process, primarily
espressed as a ratio of two quadratic forms, is transformed in a cubic
parabola with a certain normalization factor.

Keywords: Wright-Fisher model, population genetics, evolutionary process,
equilibrium state, fluctuations of probability relations.

##  1 Introduction

The population genetics models by Wright-Fisher are defined by regression
functions which are determined by a ratio of two quadratic forms [1, Ch.10].

However, equilibrium state, defined by equilibrium point of regression
function, requires additional analysis (see, for ex., [2, 3]).

At the same time, the equilibrium is easily determined for the incremental
regression function at each stage [4, 5, 6].

In the present work, the models of population genetics of genotypes
interaction are determined by difference evolution equations with regression
functions of increments for the frequency probabilities of genotypes.

In this case, the equilibrium state of the probabilities frequency is given by
the equilibrium of the regression function of increments, which is postulated
by the form of such a function.

##  2 Regression function of increments

The probabilities of genotype frequencies at each stage $k\geq 0$ are
determined by the evolutionary process $P(k)=(P_{m}(k),\ 0\leq m\leq M)$ with
$M+1$ ($M\geq 1$) finite number of the state set $E=\\{e_{0},e_{1},\dots
e_{M}\\}$.

The dynamics of the frequency probabilities at the next $k+1$-th stage ($k\geq
0$) is given by the regression function [1, Ch.10]

$\displaystyle P_{m}(k+1):=W_{m}(p)/W(p)\ ,\ \ 0\leq m\leq M\ ,\ \ k\geq 0,$
(1) $\displaystyle W_{m}(p):=p_{m}\sum_{n=0}^{M}W_{mn}p_{n}\ ,\ \ 0\leq m\leq
M,$ (2) $\displaystyle W(p):=\sum_{n=0}^{M}W_{m}(p).$ (3)

The probabilities of frequencies obey the usual restrictions $0\leq p_{m}\leq
1$, $\sum_{n=0}^{M}p_{n}=1$. The respective restrictions for the survival
parameters are $0\leq W_{mn}\leq 1$, $0\leq m,n\leq M$.

The increment of probability at each stage

$\Delta P_{m}(k+1):=P_{m}(k+1)-P_{m}(k)\ ,\ \ 0\leq m\leq M\ ,\ \ k\geq 0,$
(4)

is given by the incremental regression function

$\Delta P_{m}(k+1)=W_{0}^{(m)}(p)\ ,\ \ 0\leq m\leq M,$ (5)
$W_{0}^{(m)}(p)=V_{0}^{(m)}(p)/W(p)\ ,\ \ V_{0}^{(m)}(p):=W_{m}(p)-p_{m}W(p)\
,\ \ 0\leq m\leq M.$ (6)

Let us introduce new parameters of survival:

$V_{mn}:=1-W_{mn}\ ,\ \ 0\leq m,n\leq M.$ (7)

Then the numerator of incremental regression function (6) is transformed to
the form:

$V_{0}^{(m)}(p)=p_{m}\left[\sum_{n=0}^{M}p_{n}(V_{n},p)-(V_{m},p)\right]\ ,\ \
0\leq m\leq M,$ (8)

and the normalizing denominator (3) has the form:

$W(p)=1-\sum_{n=0}^{M}p_{n}(V_{n},p).$ (9)

where the scalar product

$(V_{m},p):=\sum_{n=0}^{M}V_{mn}p_{n}\ ,\ \ 0\leq m\leq M.$ (10)

Introduce the equilibriums of incremental regression functions (8) by the
relations:

$(V_{m},\rho)=\pi\ ,\ \ 0\leq m\leq M\ ,\ \ \pi:=\prod_{n=0}^{M}\rho_{n}.$
(11)

The normalized constant $\pi$ is also generated by equilibriums
$\rho=(\rho_{m}$, $0\leq m\leq M)$.

######  Lemma 1.

The equilibriums of incremental regression functions (6) - (10) are given by
the relation:

$\rho_{m}=\pi\overline{V}_{m}\ ,\ \ 0\leq m\leq M\ ,\ \
\pi:=\prod_{n=0}^{M}\rho_{n},$ (12)

where $\overline{V}_{m}:=\sum_{n=0}^{M}\overline{V}_{mn}$, $0\leq m\leq M$
with the summands which are the elements of inverse matrix
$\mathbb{V}^{-1}:=[\overline{V}_{mn}$, $0\leq m,n\leq M]$ with respect to the
directing parameters matrix $\mathbb{V}=[V_{mn}$; $0\leq m,n\leq M]$, under
the additional normalization condition
$\sum_{m=0}^{M}\overline{V}_{m}=\sum_{m,n=0}^{M}\overline{V}_{mn}=\pi^{-1}$.

###### Proof.

The relation (11) means that

$\mathbb{V}\rho=\pi\textbf{1}\ ,\ \ \pi\textbf{1}:=(\pi\ ,\ \ 0\leq n\leq M).$

Hence the vector of equilibriums has the following representation:

$\rho=\pi\mathbb{V}^{-1}\textbf{1}\ ,\ \ \rho_{m}=\overline{V}_{m}\pi\ ,\ \
0\leq m\leq M,$ (13)

that is, the assertion of Lemma (12). ∎

######  Corollary 1.

The equilibria (12) provide the equilibrium state of the probability frequency
(1):

$V_{0}^{(m)}(\rho)\equiv 0\ ,\ \ 0\leq m\leq M.$ (14)

######  Corollary 2.

The equilibria (12) generate a representation of the scalar products (10) by
fluctuations of the probability relations:

$(V_{m},p)=\pi p_{m}/\rho_{m}\ ,\ \ 0\leq m\leq M.$ (15)

The normalizing constant $\pi$ is defined in (11).

First of all, note that relation (12) coincides with the definition of the
equilibrium (11), under additional assumption that the directing parameters
matrix $\mathbb{V}=V_{m}\delta_{mn}$; $0\leq m,n\leq M$, is diagonal. Hence we
have the following

######  Lemma 2.

There takes place the following relation:

$\pi\rho^{-1}=\mathbb{V}\textbf{1},$ (16)

which coincides with formula (15).

Now the incremental regression functions (6) - (9) with the relations (14)
generate the following

######  Proposition 1.

The incremental regression functions with Wright-Fisher normalization is given
by the relations:

$\displaystyle W_{0}^{(m)}(p)=V_{0}^{(m)}(p)/W(p),$ (17) $\displaystyle
V_{0}^{(m)}(p)=\pi\rho_{m}[\sum_{m=0}^{M}p_{n}^{2}/\rho_{n}-p_{m}/\rho_{m}],$
(18) $\displaystyle W(p)=1-\pi\sum_{m=0}^{M}p_{n}^{2}/\rho_{n}.$ (19)

It is obvious the balance condition:

$\sum_{m=0}^{M}V_{0}^{(m)}(p)=0,$ (20)

which in scalar form is the following:

$\sum_{m=0}^{M}p_{m}\sum_{n=0}^{M}p_{n}^{2}/\rho_{n}-\sum_{m=0}^{M}p_{m}^{2}/\rho_{m}\equiv
0.$ (21)

##  3 Equilibrium state

The presence of equilibrium state is provided by equilibrium point of the
incremental regression function:

$V_{0}^{(m)}(\rho)=0\ ,\ \ 0\leq m\leq M\ ,\ \ \rho=(\rho_{m}\ ,\ \ 0\leq
m\leq M).$ (22)

The normalizing Wright-Fisher factor has the form:

$W(\rho)=1-\pi\ ,\ \ \pi=\prod_{m=0}^{M}\rho_{m}.$ (23)

The equilibrium generated by the state $\rho=(\rho_{m}\ ,\ \ 0\leq m\leq M)$,
is interpreted by the convergence of evolutionary processes (1).

######  Theorem 1.

For any initial data: $0<P_{m}(0)<1$, $0\leq m\leq M$, evolutionary processes
$P_{m}(k)$, $0\leq m\leq M$, $k\geq 0$, which are determined by solutions of
difference evolutionary equation (5) with the incremental regression function
(16) - (18) converge, by $k\to\infty$, to equilibrium

$\lim_{k\to\infty}P_{m}(k)=\rho_{m}\ ,\ \ 0\leq m\leq M.$ (24)

###### Proof.

The property of the main components is used, which is specified by the sum

$\sum_{n=0}^{M}p_{n}^{2}/\rho_{n}=\sum_{n=0}^{M}p_{n}(p_{n}/\rho_{n}),$ (25)

This means averaging the fluctuations of the probability relations
$p_{n}/\rho_{n}$, $0\leq m\leq M$ on the distribution of frequencies at the
current stage. In this case, the fluctuations of the ratios are equal to one
for $p_{n}=\rho_{n}$, $0\leq m\leq M$, and at the same time, the main
component of the incremental regression function is also equal to one.
Consequently, the possible values of the frequency probabilities can be split
into three zones:

$(+)\quad p_{n}<\rho_{n}\quad;\qquad(-)\quad
p_{n}>\rho_{n}\quad;\qquad(0)\quad p_{n}=\rho_{n}\quad.$

The signs of the incremental regression functions (16) - (18) in such a zones
is the same:
in zone (+) the probabilities increase, in zone (-) they are decrease.

Therefore, there exists a limit (23) whose value is ensured by the necessary
condition for the existence of a limit:

$\lim_{k\to\infty}\Delta P_{m}(k+1)=0.$ (26)

∎

##  4 Binary evolutionary process

The binary EP $P_{\pm}(k)$, $k\geq 0$, are determined by the following
regression functions [1]:

$\displaystyle P_{\pm}(k+1)=W_{\pm}(p)/W(p)\ ,\ \ k\geq 0,$ (27)
$\displaystyle W_{\pm}(p)=P_{\pm}(W_{\pm}p_{\pm}+p_{\mp}),$ (28)
$\displaystyle
W(0)=W_{+}(p_{+})+W_{-}(p_{-})=W_{+}p_{+}^{2}+2p_{+}p_{-}+W_{-}p_{-}^{2}.$
(29)

The frequencies probabilities at $k$-th stage satisfy the usual conditions
$0\leq p_{\pm}\leq 1$, $p_{+}+p_{-}=1$. The survival parameters are also
limited by the relation $0<W_{\pm}<1$.
For probability increments

$\Delta P_{\pm}(k+1):=P_{\pm}(k+1)-P_{\pm}(k)\ ,\ \ k\geq 0,$ (30)

the corresponding regression functions of increments can be represented as
follows:

$W_{0}^{\pm}(p_{\pm})=W_{\pm}(p_{\pm})/W(p)-p_{\pm},$ (31)

or equivalently

$\displaystyle W_{0}^{\pm}(p_{\pm})=V_{0}^{\pm}(p_{\pm})/W(p),$ (32)
$\displaystyle V_{0}^{\pm}(p_{\pm})=W_{\pm}(p_{\pm})-p_{\pm}W(p).$ (33)

Now introduce direction parameters, based on the survival ones:
$V_{\pm}:=1-W_{\pm}$ The relative equilibriums will be
$\rho_{\pm}=V_{\pm}^{-1}$ with the normalization condition $V_{+}+V_{-}=1$.
Then the numerators (27) of the regression function has the following form:

$W_{\pm}(p_{\pm})=p_{\pm}(1-\pi p_{\mp}/\rho_{\pm})\ ,\ \
\pi:=\rho_{+}\rho_{-}.$ (34)

Therefore the numerator (32) transforms into the following:

$V_{0}^{\pm}(p_{\pm})=p_{\mp}W_{\pm}(p_{\pm})-p_{\pm}W_{\mp}(p_{\mp})=p_{+}p_{-}(\rho_{\pm}p_{\mp}-\rho_{\mp}p_{\pm}).$
(35)

The linear component has the following representation:

$\rho_{+}p_{-}-\rho_{-}p_{+}=-(p_{+}-\rho_{+})=p_{-}-\rho_{-}.$ (36)

So the regression functions of the increments of binary evolutionary processes
are represented by the probability of fluctuations:

$V_{0}^{\pm}(p_{\pm})=-p_{+}p_{-}(p_{\pm}-\rho_{\pm}).$ (37)

The Wright-Fisher normalizing factor has the following representation:

$W(p)=1-\pi[p_{+}^{2}/\rho_{+}+p_{-}^{2}/\rho_{-}]=1-[\rho_{-}p_{+}^{2}+\rho_{+}p_{-}^{2}].$
(38)

The balance condition is also evident:

$V_{0}^{+}(p_{+})+V_{0}^{-}(p_{-})\equiv 0.$ (39)

##  5 Conclusion

The considered in the present work evolution processes serve as predictable
components of stochastic models in population genetics and are represented by
conditional mathematical expectations:

$\begin{split}P_{m}(k+1):=E[S_{N}^{(m)}(k+1)\,|\,S_{N}(k)=P(k)]\ ,\ \ 0\leq
m\leq M\ ,\ \ k\geq 0.\end{split}$ (40)

The stochastic models in population genetics are determined by averaged sums

$S_{N}(k):=\frac{1}{N}\sum^{N}_{n=1}\delta_{n}(k)\ ,\ \ k\geq 0.$ (41)

of random sample variables $\delta_{n}(k)$, $1\leq n\leq N$, which take values
in a finite set with $M+1$ ($M\geq 1$) states
$E=\\{e_{0},e_{1},\dots,e_{M}\\}$ (see [7]). So the stochastic models (41) are
defined by the sum of two components:

$S_{N}(k+1)=V(S_{N}(k))+\Delta\mu_{N}(k+1)\ ,\ \ k\geq 0.$ (42)

The first, predictable component is generated by conditional mathematical
expectations:

$V_{m}(P_{m}(k))=P_{m}(k)+V_{0}^{(m)}(P_{m}(k))/W(P_{m}(k))\ ,\ \ 0\leq m\leq
M\ ,\ \ k\geq 0.$ (43)

The second component forms a martingale differences

$\Delta\mu_{N}(k+1)=S_{N}(k+1)-V(S_{N}(k))\ ,\ \ k\geq 0,$ (44)

characterized by the first moments:

$\begin{split}&E\Delta\mu_{N}^{(m)}(k+1)=0\ ,\ \ 0\leq m\leq M,\\\
&E[(\Delta\mu_{N}^{(m)}(k+1))^{2}\,|\,S_{N}(k)]=\sigma_{m}^{2}(S_{N}(k))\ ,\ \
0\leq m\leq M\ ,\ \ k\geq 0.\end{split}$ (45)

The conditional dispersion is determined by regression functions:

$\sigma_{m}^{2}(p)=V_{m}(p)[1-V_{m}(p)]\ ,\ \ 0\leq m\leq M.$ (46)

The asymptotical properties of stochastic models (42) - (45) by $N\to\infty$
as well as by $k\to\infty$, will be investigated in our next paper. The
algorithms of phase merging [8] and statistical estimation of drift parameter
[9], [10] can be directly applied.

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