# Ultrasound Image Enhancement using CycleGAN and Perceptual Loss

Shreeram Athreya  Department of Electrical and Computer Engineering, UCLA
Ashwath Radhachandran  Department of Bioengineering, UCLA  Vedrana Ivezić
Medical Informatics, UCLA  Vivek Sant  Department of Surgery, UT Southwestern
Medical Center  Corey W. Arnold  Department of Electrical and Computer
Engineering, UCLA  Department of Bioengineering, UCLA  Medical Informatics,
UCLA  Department of Radiological Sciences, UCLA  Department of Pathology, UCLA
Email<EMAIL_ADDRESS>William Speier  Medical Informatics, UCLA  Department
of Radiological Sciences, UCLA

###### Abstract

Purpose: The objective of this work is to introduce an advanced framework
designed to enhance ultrasound images, especially those captured by portable
hand-held devices, which often produce lower quality images due to hardware
constraints. Additionally, this framework is uniquely capable of effectively
handling non-registered input ultrasound image pairs, addressing a common
challenge in medical imaging.

Materials and Methods: In this retrospective study, we utilized an enhanced
generative adversarial network (CycleGAN) model for ultrasound image
enhancement across five organ systems. Perceptual loss, derived from deep
features of pretrained neural networks, is applied to ensure the human-
perceptual quality of the enhanced images. These images are compared with
paired images acquired from high resolution devices to demonstrate the model’s
ability to generate realistic high-quality images across organ systems.

Results: Preliminary validation of the framework reveals promising performance
metrics. The model generates images that result in a Structural Similarity
Index (SSI) score of 0.722, Locally Normalized Cross-Correlation (LNCC) score
of 0.902 and 28.802 for the Peak Signal-to-Noise Ratio (PSNR) metric.

Conclusion: This work presents a significant advancement in medical imaging
through the development of a CycleGAN model enhanced with Perceptual Loss
(PL), effectively bridging the quality gap between ultrasound images from
varied devices. By training on paired images, the model not only improves
image quality but also ensures the preservation of vital anatomic structural
content. This approach may improve equity in access to healthcare by enhancing
portable device capabilities, although further validation and optimizations
are necessary for broader clinical application.

##  1 Introduction

Ultrasound imaging holds a pivotal role in the field of medical diagnostics,
offering non-invasive, highly accurate point of care assessments that have
been increasingly adopted by healthcare professionals. Historically, the
technology has been limited to large, expensive devices typically found in
specialized medical settings. However, there has been a transformative shift
towards the development and adoption of compact, hand-held ultrasound devices.
These smaller devices promise to democratize access to medical imaging by
making it more affordable and widely available. Yet, the miniaturization and
cost-effectiveness often come at the expense of image quality, a trade-off
primarily attributable to hardware constraints [1].

Machine learning algorithms have been explored to enhance low-quality images
without hardware improvements. For instance, generative adversarial networks
(GANs) have been used to create high-quality reconstructions of ultrasound
images and videos, offering a cost-efficient avenue for the enhancement of
portable ultrasound devices [2, 3, 4]. The CycleGAN framework has become
increasingly popular for image-to-image translations, especially those not
requiring paired data [5]. The technology has been applied across a spectrum
of tasks including, style transfer [6] and object transfiguration [5, 7, 8].
In medical imaging, CycleGANs have been employed in tasks such as pixel-wise
translation in echocardiography [9]. CycleGANs have also been applied in
cross-modality medical image translation such as converting CT to MRI [10].
The architecture has even found utility in histopathology to standardize
microscopy staining for more accurate diagnoses [11].

We hypothesize that the integration of computational algorithms, particularly
CycleGAN, can mitigate the disparities in images acquired from different
medical imaging devices. Traditional training approaches for these models
artificially introduce corruptions into medical images to create pixel-wise
pairs [12, 13, 14]. However, these methods typically fail to encapsulate the
different characteristics of images acquired using different devices.
Acquiring paired images using different devices leads to technical issues as
images are captured at different time-instances with varying orientations,
leading to structural changes that cannot be completely resolved using image
registration.

To overcome these challenges, our approach leverages perceptual loss (PL),
which can eliminate the need for registration and more accurately relate
images from disparate domains. Traditional loss functions used in CycleGAN can
result in hallucinated features in the enhanced images [15]. By incorporating
PL, more interpretable images are generated that are more robust to
registration artifacts [16]. This method can enhance the reliability and
consistency of images from handheld ultrasound devices to bridge the gap with
expensive high-end systems for greater equity in access to healthcare.

##  2 Materials and Methods

###  2.1 Model overview

Our framework for generating high-quality images is a modification of the
CycleGAN architecture, designed to map between two distinct imaging domains.
In ultrasound image enhancement, these domains correspond to low-quality
(Domain L) and high-quality (Domain H) images. The model employs two
generators, $G_{L}$ and $G_{H}$ , and two discriminators, $D_{L}$ and $D_{H}$
(Fig. 1). The $G_{L}$ generator transforms an image from Domain H to align
with Domain L, whereas $G_{H}$ performs the inverse, which is crucial for
enhancing low-quality images to high-quality images. The discriminators aim to
distinguish real images in their respective domains from those transformed by
the generators. A unique feature of this approach is the cycle consistency
loss that ensures that an image translated to the other domain and then
reverted, closely resembles the original. After training, the $G_{H}$
generator is utilized to enhance images, maintaining essential structural
attributes while improving clarity and resolution.

###  2.2 Model description

GANs have seen transformative advancements, with CycleGAN representing a
significant milestone in facilitating unsupervised image-to-image
translations. The GAN architecture comprises two primary modules: the
Generator and the Discriminator.

The Generator is segmented into encoding, residual block, and decoding phases.
The encoder starts by applying $64$ channels of 2D convolution utilizing a
large kernel size to capture broader contextual details. Down-sampling layers
in the encoder reduce spatial dimensions while increasing depth, thereby
emphasizing hierarchical feature extraction. The skip connections in the
sequence of 15 residual blocks preserve information to address the vanishing
gradient problem. Each residual block contains two convolutional sub-layers
with a kernel size of 3 and padding of 1. The first layer includes Instance
Normalization and a $ReLU$ activation function, while the second omits the
activation function, thereby enabling a linear combination with the block’s
input. The Decoder performs spatial restoration and channel reduction using
nearest-neighbor interpolation for upsampling and skip connections to preserve
detailed features. We use the $tanh$ activation function to scale the output
values.

The Discriminator distinguishes between real and generated images. We use
spectral normalization to ensure stability during training. Its architecture
begins with a convolutional layer that compresses spatial information and
expands depth followed by a $LeakyReLU$ activation layer. Subsequent layers
maintain the use of spectral normalization to ensure $1-Lipschitz$ continuity.
This constraint on the spectral norm of each layer’s weights helps to balance
the generator and discriminator during training. The Discriminator concludes
by reducing spatial dimensions to a $30\times 30$ grid with a depth of $1$,
providing the final classification output.

Figure 1: The CycleGAN architecture. (a) An overview of the CycleGAN model
training and loss computation framework. (b) The Generator and (c) The
Discriminator model architectures. We note: the generators $G_{L}$ and $G_{H}$
share the same architecture. Similarly, the discriminators $D_{L}$ and $D_{H}$
share the same model architecture. The figure legend lists the different
layers in the model.

###  2.3 Perceptual loss

Conventional methodologies like MSE and SSI rely on pixelwise alignment which
makes them unsuitable for non-registered image pairs acquired using different
devices. The Learned Perceptual Image Patch Similarity (LPIPS) metric
addresses these constraints by evaluating the perceptual similarity between
images [17]. LPIPS leverages deep features extracted from pre-trained
convolutional networks, such as the VGG network [18]. The LPIPS metric
comparing images $X$ and $Y$ is given by:

$\Phi(X,Y)=\sum_{i}w_{i}\cdot\|F_{i}(X)-F_{i}(Y)\|_{2}$ (1)

where $F_{i}$ and $w_{i}$ denotes the feature maps and optimized weights from
the $i^{th}$ layer of the pretrained network. Deep feature maps are
systematically extracted from every layer within the network, ensuring a
comprehensive reflection of the multi-scale characteristics of human
perceptual judgment. These features are then unified through linear
combination, optimizing the weights to align with perceptual judgments
assessed by human evaluators. The LPIPS metric consistently outranks
traditional metrics, showcasing superior performance across an array of
perceptual judgment tasks [16]. This loss is calculated between real images
$L$ and $H$, and those generated through the CycleGAN framework’s generators
as:

$L_{per}(G_{H},G_{L})=\Phi(H,H^{\prime})+\Phi(L,L^{\prime})$ (2)

where $H^{\prime}$ represents $G_{H}(L)$ and $L^{\prime}$ represents
$G_{L}(H)$.

###  2.4 Loss function

#### Generator loss:

The generator’s loss function is a linear combination of several distinct loss
terms, each playing a pivotal role in optimizing image translation between the
two domains. First, adversarial loss $L_{adv}(D_{H},D_{L})$ induces the
discriminators to perceive generated images as genuine, whether they are
translated from low to high quality or vice versa.

$L_{adv}(D_{H},D_{L})=MSE(\mathbf{1},D_{H}(\overline{H}))+MSE(\mathbf{1},D_{L}(\overline{L}))$
(3)

Where $\overline{H}=G_{H}(G_{L}(H))$ and $\overline{L}=G_{L}(G_{H}(L))$.
Specifically, mean squared error ($MSE$) calculates the discrepancy between
the discriminator’s predictions and an array of ones. These terms push the
generator to produce images that can convince the discriminator they belong to
the high-quality domain. The cycle loss $L_{cycle}(G_{H},G_{L})$ prevents the
loss of critical image features by ensuring that an image translated to the
other domain and back yields the original image.

$L_{cycle}(G_{H},G_{L})=\|H-\overline{H}\|_{1}+\|L-\overline{L}\|_{1}$ (4)

The identity loss $L_{id}(G_{H},G_{L})$ ensures that an image from the source
domain fed into the generator remains unchanged.

$L_{id}(G_{H},G_{L})=\|H-G_{H}(H)\|_{1}+\|L-G_{L}(L)\|_{1}$ (5)

Finally, the aggregate generator loss, $L(G_{H},G_{L},D_{H},D_{L})$, is
computed by combining all individual loss terms weighed by their respective
lambda constants:

$L(G_{H},G_{L},D_{H},D_{L})=\lambda_{adv}L_{adv}(D_{H},D_{L})+\lambda_{cycle}L_{cycle}(G_{H},G_{L})+\lambda_{id}L_{id}(G_{H},G_{L})+\lambda_{per}L_{per}(G_{H},G_{L})$
(6)

By employing this multifaceted loss function, the model ensures that the
generators achieve high-quality image translations while preserving the
intrinsic characteristics of the source domain.

#### Discriminator loss:

The discriminator loss function is designed to evaluate the authenticity of
images, incorporating the principle of label smoothing to further enhance the
model’s generalizability. The discriminator is tasked with distinguishing
between real images from the dataset and fake images generated by the
corresponding generator.

$L(D_{H},D_{L})=\begin{cases}MSE(\mathbf{1},D_{H})+MSE(\mathbf{1},D_{L})&\text{if
}\text{mean}(D_{H})\text{ and }\text{mean}(D_{L})<0.9,\\\
MSE(\mathbf{0.9},D_{H})+MSE(\mathbf{0.9},D_{L})&\text{otherwise}.\end{cases}$
(7)

For each domain, the discriminator computes scores for both real and fake
images. Conventionally, discriminators are trained using hard labels, where
real images are labeled as ‘$1$’ and fake images as ‘$0$’. However, hard
labels can cause vulnerability to adversarial perturbations and lead to
overconfidence. Label smoothing addresses this issue by softening the labels;
if the mean scores of both real and fake images for the high-quality domain
are less than 0.9, a factor of 1.0 is used. Otherwise, a smoothing factor of
0.9 is applied, meaning the real images are given a target value slightly less
than 1, which prevents overconfidence and promotes model robustness. The total
discriminator loss, $L(D_{H},D_{L})$, is then computed by aggregating the
individual $MSE$ losses $L_{D_{H}}$ and $L_{D_{L}}$ for high and low-quality
domains respectively.

###  2.5 Implementation details

All models were trained for 300 epochs with a batch size of 24 images. We used
the Adam optimizer for model optimization, with separate learning rates set
for the discriminators (LR = $3\times 10^{-3}$) and generators (LR = $3\times
10^{-4}$) to implement the two time-scale update rule [4]. A beta value of 0.9
for the first and second moments in each optimizer. A learning rate scheduler
reduced learning rates by half ($\gamma=0.5$) every 100 epochs, to allow
adaptability during training. Weights were assigned to each loss term:
$\lambda_{adv}=1$ for adversarial loss, $\lambda_{cycle}=12$ for cycle-
consistency loss, $\lambda_{id}=1$ for identity loss, and $\lambda_{per}=1$
for PL. The training set was split with 90% for model training and 10% for
validation. Gradient scaling was used to optimize the model’s precision and
speed.

Figure 2: High quality images generated in comparison to reference during
validation. First row: Reference low-quality images, Second row: Generated
high-quality images, Third row: Reference high-quality images. Each column
represents scans from (i) Thyroid, (ii) Carotid, (iii) Liver, (iv) Kidney, and
(v) Breast.

###  2.6 Evaluation

Synthetic high-quality images are evaluated using three metrics: Structural
Similarity Index (SSI), Locally Normalized Cross-Correlation (LNCC), and Peak
Signal-to-Noise Ratio (PSNR). SSI estimates the structural fidelity between
generated and actual high-quality images. LNCC measures the local similarity
in intensity patterns between them. PSNR captures the error signal strength
derived from the mean squared error (MSE) between $H^{\prime}$ and $H$.

Table 1: Dataset summary across different ultrasound devices and organs [19].

Organ Low end device High end device Patients Ultrasound Image Pairs Training
Testing Total Training Testing Total Thyroid mSonics MU1 Toshiba Aplio 500 33
14 47 363 154 517 Carotid SSUN Toshiba Aplio 500 54 23 77 375 160 535 Abdomen
SSUN General Electric LOGIQ E9 21 9 30 210 90 300 Breast mSonics MU1 Aixplorer
ultrasound system 23 9 32 102 46 148 (SuperSonic Imaging S.A.) Overall 131 55
186 1050 450 1500

##  3 Results

Our study employs a dataset consisting of 3,000 ultrasound images, including
1,500 pairs of low and high-quality images (Table 1). These images were
collected from 186 patients with suspected thyroid tumors, carotid plaque, or
breast cancer, along with healthy volunteers. During scans, volunteers were
instructed to hold their breath for approximately 10 seconds to minimize
deformation, and landmark points were noted for nonrigid registration to
ensure the creation of accurate data pairs. This well-curated dataset provides
a robust foundation for this study. This dataset was provided by the
organizers of the MICCAI 2023 USEnhance challenge [19].

The CycleGAN without PL [4] shows improvement over Pix2Pix [20] in each metric
(Table 3). Notably, performance improves when PL is introduced in the CycleGAN
models. When increasing the number of residual blocks to 15 the highest
performance is observed, with an LNCC of $0.902$, an SSI of $0.722$, and a
PSNR of $28.802$. These results underscore our model’s superior capability in
retaining both local and global image similarities, ensuring a high-fidelity
transformation from low to high-quality ultrasound images while preserving
essential diagnostic features (Fig. 2).

We can notice a stable trend in both the SSI and LNCC scores after 150 epochs
of training (Fig. 3). Although the PSNR scores show some fluctuations, they
generally exhibit an upward trend, suggesting the model’s continuous effort to
better understand the relationship between low-quality and high-quality
domains.

Figure 3: Validation set results during model training. (a) Variation of LNCC scores across training epochs, (b) Variation of SSI scores across training epochs, (c) Variation of PSNR scores across training epochs. Table 2: Test set results. Model configurations | LNCC | SSI | PSNR
---|---|---|---
Pix2Pix | 0.669 | 0.343 | 19.732
CycleGAN (No PL) | 0.850 | 0.570 | 23.123
CycleGAN ($\Phi(H,\overline{H})$) | 0.861 | 0.579 | 22.697
CycleGAN ($\Phi(H,H^{\prime})$) | 0.884 | 0.702 | 26.024
Ours | 0.902 | 0.723 | 28.802

##  4 Discussion

In this work, we used the CycleGAN framework and PL to bridge the gap between
ultrasound images captured by different devices of varying quality. Using PL
to compare paired images led to superior results. These findings suggest the
possibility of bridging the gap between low and high-quality ultrasound
images, a crucial advancement for portable, handheld devices that often
struggle with image quality due to hardware limitations.

Using paired images from different devices allows for a more realistic
representation of the image quality differences that would be seen in actual
clinical practice. Previous studies have treated the low and high quality
domains as independent [4], which can address general image quality issues but
does not allow for more direct comparison of image content. The use of PL in
this study allows the model to directly compare image from the different
domains, improving quality while assuring the preservation of the anatomic
content.

One limitation is that the reliance on PL increases computational complexity
leading to longer training times. Furthermore, the findings need to be
validated through extensive real-world applications and on diverse datasets to
ascertain the model’s robustness and applicability across different ultrasound
imaging scenarios. While this model may work well across organ systems and
diseases, future work could also explore whether certain domains would benefit
from a more targeted model.

This study introduced an advanced CycleGAN model for ultrasound image
enhancement across domains by using PL to train on paired images. These
findings demonstrate the feasibility of bridging the image quality gap across
devices, leading to improved healthcare equity.

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# The Enumerative Geometry of Hyperplane Arrangements

Thomas Paul, Will Traves and Max Wakefield Department of Mathematics, U.S.
Naval Academy<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study enumerative questions on the moduli space $\mathcal{M}(L)$ of
hyperplane arrangements with a given intersection lattice $L$. Mnëv’s
universality theorem suggests that these moduli spaces can be arbitrarily
complicated; indeed it is even difficult to compute the dimension
$D=\dim\mathcal{M}(L)$. Embedding $\mathcal{M}(L)$ in a product of projective
spaces, we study the degree $N=\deg\mathcal{M}(L)$, which can be interpreted
as the number of arrangements in $\mathcal{M}(L)$ that pass through $D$ points
in general position. For generic arrangements $N$ can be computed
combinatorially and this number also appears in the study of the Chow variety
of zero dimensional cycles. We compute $D$ and $N$ using Schubert calculus in
the case where $L$ is the intersection lattice of the arrangement obtained by
taking multiple cones over a generic arrangement. We also calculate the
characteristic numbers for families of generic arrangements in
$\mathbb{P}^{2}$ with 3 and 4 lines.

##  1 Introduction

Enumerative geometry has a monumental history and continues to be an
inspiration for many different fields of research (for example see Katz [13]
and Kontsevich and Manin [17]). Solutions to enumerative problems have given
deep insight into the geometric nature of various algebraic varieties and
important spaces.

Realization spaces of matroids (or moduli spaces of hyperplane arrangements)
give a beautiful connection between combinatorics and algebraic geometry. In
particular, Mnëv’s universality theorem presents matroids as a kind of
dictionary for quasi-projective varieties (see Mnëv [18] or Vakil [27]). These
realization spaces have been studied from many different viewpoints. Kapranov
[12] showed the Hilbert compactification of the moduli space of generic
arrangements could be viewed as a Chow quotient. Hacking, Keel, and Tevelev
[10] applied the relative minimal model program to the moduli space of
arrangements. Terao [25] studied the closure of this moduli space in a product
of projective spaces and an associated logarithmic Gauss-Manin connection.
Speyer [24] proved that the $K$-theory class (actually a push forward-
pullback) of the inclusion of the moduli space into the appropriate
Grassmannian is actually the 2-variable Tutte polynomial of the associated
matroid. Then Fink and Speyer [6] generalized this result to non-realizable
matroids. In this note we study some of the geometry of this moduli space and
find another use of the Tutte polynomial.

One of our motivating problems is Terao’s conjecture which concerns the subset
of the realization consisting of free arrangements (see Orlik and Terao [19]
for a general reference on hyperplane arrangements). Yuzvinsky [28] showed
that this subset was Zariski-open. It is not known if this subset is also
closed. Our focus here is on computing the dimension and essentially the
degree of this realization space by using both combinatorial and geometric
methods.

To each hyperplane arrangement $\mathcal{A}=\\{H_{1},\ldots,H_{k}\\}$ in
$\mathbb{P}^{n}$ we associate an intersection lattice
$\mathcal{L}(\mathcal{A})$, the poset whose elements are intersections
$H_{i_{1}}\cap\cdots\cap H_{i_{t}}$ ordered by reverse inclusion, $B\leq C\iff
C\subseteq B$. Two arrangements $\mathcal{A}$ and $\mathcal{A}^{\prime}$ are
combinatorially equivalent if their intersection lattices are isomorphic, that
is, if there is a bijection $\phi:\mathcal{A}\rightarrow\mathcal{A}^{\prime}$
that preserves the lattice order, $B\leq C\iff\phi(B)\leq\phi(C)$. Let
$\mathcal{M}_{\mathcal{A}}$ be the set of arrangements that are
combinatorially equivalent to the arrangement $\mathcal{A}$. Identifying each
arrangement $\\{H_{1},\ldots,H_{k}\\}$ with its orbit
$\\{(H_{\sigma(1)},\ldots,H_{\sigma(k)}):\;\sigma\in\mathcal{S}_{k}\\}$ under
the permutation group $\mathcal{S}_{k}$, we obtain an embedding
$\mathcal{M}_{\mathcal{A}}\hookrightarrow(\mathbb{P}^{n*})^{k}/\mathcal{S}_{k}$
and we give $\mathcal{M}_{\mathcal{A}}$ the induced topology coming from the
Zariski topology on the quotient space. It is clear that
$\mathcal{M}_{\mathcal{A}}$ depends only on the intersection lattice of
$\mathcal{A}$. If $D=\dim\mathcal{M}_{\mathcal{A}}$ is the dimension of
$\mathcal{M}_{\mathcal{A}}$, then let $N_{\mathcal{A}}$ be the number of
arrangements in $\mathcal{M}_{A}$ passing through $D$ points in general
position in $\mathbb{P}^{n}$.

######  Question 1.

Given $\mathcal{A}$ compute $D$ and $N_{\mathcal{A}}$. Ideally these answers
should be given in terms of the combinatorics of the lattice
$\mathcal{L}(\mathcal{A})$.

In the next section we show that when $\mathcal{A}$ is a generic arrangement
of $k$ hyperplanes in $\mathbb{P}^{n}$, then
$\dim\mathcal{M}_{\mathcal{A}}=kn$ and we compute $N_{\mathcal{A}}$. The
characteristic number $N_{\mathcal{A}}(p,\ell)$ measures the number of
arrangements combinatorially equivalent to $\mathcal{A}$ that pass through $p$
points and are tangent to $\ell$ lines in general position (with
$p+\ell=\dim\mathcal{M}_{\mathcal{A}}$). We use intersection theory on the
correspondence between hyperplane arrangements and their dual arrangements to
compute the characteristic numbers for generic arrangements of three or four
lines in $\mathbb{P}^{2}$ (a good reference for the intersection theory that
we use is Eisenbud and Harris [4] or Fulton [7]). This seems to be the first
computation of these characteristic numbers for line arrangements though
characteristic numbers were computed for smooth curves of degrees 3 and 4 by
Zeuthen [29] in the 19${}^{\text{th}}$ century. As reported in Kleiman [15],
the 19${}^{\text{th}}$ century methods lacked adequate foundations prompting
Hilbert to ask for a rigorous computation of these characteristic numbers.
Aluffi [1] and Kleiman and Speiser [16] verified Zeuthen’s degree 3
predictions using intersection theory. Vakil [26] used intersection theory on
the moduli space of stable maps to verify the degree 4 predictions. The
importance of the characteristic numbers is suggested by a theorem originally
due to Zeuthen [30] (also see Fulton [7, section 10.4]) that shows that the
characteristic numbers for a family of curves determine the number of such
curves tangent to smooth curves of arbitrary degrees. We close Section 2 by
interpreting this theorem for line arrangements. It would be interesting to
use intersection theory on the moduli space of stable maps to recover our
results.

In Section 3 we consider cones over generic arrangements of hyperplanes.

######  Definition 2.

A $d$-coned generic arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$ is an
arrangement of $k>n$ hyperplanes obtained from a generic hyperplane
arrangement $\mathcal{B}$ in a linear subspace $H\cong\mathbb{P}^{n-(d+1)}$ by
taking the cone with a $d$-dimensional linear space (equivalently, with $d+1$
general points). That is, there exists a linear space $\Omega$ of dimension
$d$, disjoint from $H$ so that each hyperplane in $\mathcal{A}$ is the linear
span of both a hyperplane in $\mathcal{B}$ and $\Omega$.

In Section 3 we answer the enumerative problems from Question 1 for $d$-coned
generic arrangements. Here the methods of Schubert Calculus come into play and
the Catalan numbers make a cameo appearance.

Our approach to computing $N_{A}$ is to first compute the number of labeled
arrangements with intersection lattice isomorphic to
$\mathcal{L}(\mathcal{A})$ that pass through $D$ points in general position in
$\mathbb{P}^{n}$. Dividing by the number of ways to label the hyperplanes in
$\mathcal{A}$ gives $N_{A}$. This allows us to work in a product of polynomial
rings rather than its quotient. A recent paper by Fehér, Némethi, and Rimányi
[5] also studies enumerative problems involving hyperplane arrangements; they
embrace quotient varieties and work with equivariant cohomology.

Though we were not able to answer Question 1 in terms of the combinatorics of
$\mathcal{L}(\mathcal{A})$, we remain optimistic about this possibility for
special families of arrangements despite several warning signs that the
question may be very difficult in general. Mnëv’s Universality Theorem says
that each variety appears as the closure of $\mathcal{M}_{\mathcal{A}}$ for
some hyperplane arrangement $\mathcal{A}$, so the geometry of
$\mathcal{M}_{\mathcal{A}}$ can be arbitrarily complicated. Another warning
sign appears if we take a naive approach to the dimension problem in
$\mathbb{P}^{2}$. For each line arrangement $\mathcal{A}$ with $\ell$ labeled
lines $L_{1},\ldots,L_{\ell}$ and $k$ labeled points $p_{1},\ldots,p_{k}$ of
intersection among these lines, form the parameter space

$P_{\mathcal{A}}=\\{(H_{1},\ldots,H_{\ell},P_{1},\ldots,P_{k})\in(\mathbb{P}^{2*})^{\ell}\times(\mathbb{P}^{2})^{k}\;:\;P_{i}\in
H_{j}\text{ if $p_{i}\in L_{j}$ }\\}.$

Note that $\dim P_{\mathcal{A}}=\dim\mathcal{M}_{\mathcal{A}}$. If the
conditions $P_{i}\in H_{j}$ are algebraically independent, then this dimension
is $2(k+\ell)-\sum_{i=1}^{k}|\\{L_{j}:p_{i}\in L_{j}\\}|$ (sometimes this is
called the virtual dimension of $\mathcal{M}_{\mathcal{A}}$). Such a formula
holds for generic line arrangements and pencils as well as for many other
arrangements; however, it fails for the Pappus arrangement pictured in Figure
1, since one of the conditions $P_{i}\in H_{j}$ is implied by the others.
Indeed, the dimension of $\mathcal{M}_{\mathcal{A}}$ depends on the syzygies
among these incidence conditions and so any potential algorithms need to be
sensitive enough to recognize such syzygies from the combinatorial information
in $\mathcal{L}(A)$, a task that appears to be quite difficult. This example
(more precisely, its projective dual) was studied in detail by Fehér, Némethi,
and Rimányi [5] and by Ren, Richter-Gebert and Sturmfels [21].

Figure 1: The Pappus arrangement consists of nine lines. The 27 point-line
incidence conditions are dependent: the incidence correspondence has
codimension 26.

######  Remark 3.

We use the notation
$\\{(a)^{a^{\prime}},(b)^{b^{\prime}},\ldots,(k)^{k^{\prime}}\\}$ (sometimes
omitting the braces) to refer to the multiset consisting of $a^{\prime}$
copies of $a$, $b^{\prime}$ copies of $b$, et cetera. Similarly, we use
$\binom{d}{(a)^{a^{\prime}},(b)^{b^{\prime}},\ldots,(k)^{k^{\prime}}}$ to
refer to the multinomial that counts the ways of choosing $a^{\prime}$ groups
of $a$ objects, $b^{\prime}$ groups of $b$ objects, $\ldots$, and $k^{\prime}$
groups of $k$ objects from $d$ labeled objects. That is,
$\binom{d}{(a)^{a^{\prime}},(b)^{b^{\prime}},\ldots,(k)^{k^{\prime}}}=d!/(a!)^{a^{\prime}}\cdots(k!)^{k^{\prime}}$.

##  2 Generic Arrangements

An arrangement of $k$ hyperplanes in $\mathbb{P}^{n}$ is said to be generic if
$k>n$ and no point is in the intersection of more than $n$ of the hyperplanes.
Carlini discovered the following fact while studying the Chow variety of zero
dimensional degree-$k$ cycles in $\mathbb{P}^{n}$ [3, Proposition 3.4].

######  Theorem 4.

When $\mathcal{A}$ is a generic arrangement of $k$ hyperplanes in
$\mathbb{P}^{n}$ then the dimension $D$ of $\mathcal{M}_{\mathcal{A}}$ is $kn$
and the number of arrangements with lattice type isomorphic to
$\mathcal{L}(A)$ that pass through $D$ points in general position in
$\mathbb{P}^{n}$ is

$N_{\mathcal{A}}=\frac{1}{k!}\binom{kn}{n}\binom{(k-1)n}{n}\cdots\binom{n}{n}=\frac{(kn)!}{k!(n!)^{k}}.$

###### Proof.

The moduli space $\mathcal{M}_{\mathcal{A}}$ is a $k!$ to 1 cover of the
complement of a closed set in $(\mathbb{P}^{n*})^{k}$ that parameterizes the
sets of $k$ hyperplanes that contain a set of $n$ linearly dependent
hyperplanes. So
$D=\dim\mathcal{M}_{\mathcal{A}}=\dim(\mathbb{P}^{n*})^{k}=kn$. We count the
ordered (or labeled) generic arrangements passing through $kn$ points in
general position. Since the points are in general position, no more than $n$
can lie on any one hyperplane. So by the pigeon-hole principle, each
hyperplane contains precisely $n$ of the points and each hyperplane is
completely determined by these $n$ points. There are clearly
$\prod_{i=0}^{k-1}\binom{kn-in}{n}$ ways to distribute the points among the
labeled hyperplanes, but each hyperplane arrangement can be labeled in $k!$
ways so dividing the product of binomial coefficients by $k!$ gives
$N_{\mathcal{A}}$. ∎

######  Remark 5.

We remark that when $n=2$ the formula for $N_{\mathcal{A}}$ reduces to
$(2k-1)!!$. Aside from the nice simplicity of this result, this formulation
allows us to interpret the result in terms of the multivariate Tutte
polynomial of the lattice $\mathcal{L}(\mathcal{A})$. The multivariate Tutte
polynomial of $\mathcal{L}(\mathcal{A})$ (see Ardila [2] or Sokal [23]) is
defined as

$Z_{\mathcal{L}(\mathcal{A})}(q,x_{1},\ldots,x_{k})=\sum_{B\in\mathcal{L}(\mathcal{A})}(q^{-rk(B)})\prod_{H_{j}\in
B}x_{j},$

where if $B\subset\mathcal{A}$ then $rk(B)=\text{codim}\cap_{H\in B}H.$ When
$\mathcal{A}$ is a generic arrangement with $k$ hyperplanes,

$Z_{\mathcal{L}(\mathcal{A})}(1,x_{1},\ldots,x_{k})=\sum_{B\subset\mathcal{A}}\prod_{H_{j}\in
B}x_{j}=(1+x_{1})(1+x_{2})\cdots(1+x_{k}).$

In particular,

$Z_{\mathcal{L}(\mathcal{A})}(1,0,2,\ldots,2k-2)=(2k-1)!!=N_{\mathcal{A}}.$

######  Remark 6.

The intersection ring (or Chow ring) $A(X)$ of a variety $X$ can be defined as
the ring of equivalence classes of algebraic subvarieties of $X$ modulo
rational equivalence, graded by codimension [7]. Here $[Y_{1}\cup
Y_{2}]=[Y_{1}]+[Y_{2}]$. Also, $[Y_{1}\cap Y_{2}]=[Y_{1}]\cdot[Y_{2}]$ if
$Y_{1}$ and $Y_{2}$ intersect transversely. The intersection rings that we
consider can also be interpreted in terms of cohomology; for more on this
point of view, see Katz [13].

When $X\cong\mathbb{P}^{n_{1}}\times\cdots\times\mathbb{P}^{n_{k}}$ is a
product of projective spaces then

$A(X)\cong\frac{\mathbb{Z}[x_{1},\ldots,x_{k}]}{(x_{1}^{n_{1}+1},\ldots,x_{k}^{n_{k}+1})},$

where the $x_{i}$ are the pullbacks of the classes of hyperplanes on each
factor to the product.

Next we prove a Lemma that will be used in many of the following arguments.

######  Lemma 7.

Fix an arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$ with
$D=\dim\mathcal{M}_{\mathcal{A}}$ and $|\mathcal{A}|=k$. The conditions that
the arrangement contain $D$ specified points in general position are
transverse and the class corresponding to the intersection of these conditions
is

$\left(\sum_{i=1}^{k}x_{i}\right)^{D}.$

###### Proof.

The class of the condition that a point lies on the $i$th hyperplane is just
$x_{i}$. Then the class of the condition that a point lies on one of the $k$
hyperplanes is $\sum_{i=1}^{k}x_{i}$. Now, the action of $\mathrm{PGL}(n)^{k}$
is transitive on $\mathcal{M}_{\mathcal{A}}$. Since the $D$ point conditions
are assumed to be generic we can use this $\mathrm{PGL}(n)^{k}$ action to move
one point condition to another. Hence Kleiman’s Transversality Theorem [14]
says that these conditions are transverse and the corresponding Chow class is
the product of the classes. ∎

Note that Theorem 4 could also be easily proved by Lemma 7. Since we are not
putting any conditions on the generic lines we have nothing but point
conditions. Then the degree of the Chow class given in Lemma 4 counts the
number of ordered generic hyperplane arrangements passing through $D$ points
in general position. Dividing by $k!$ gives the number of such unordered
generic arrangements.

Continuing to focus on the case where $n=2$, we will compute the
characteristic numbers of generic line arrangements with $k=3$ or $k=4$ lines.
We define the characteristic number $N_{\mathcal{A}}(p,\ell)$ to be the number
of arrangements with the same lattice type as $\mathcal{A}$ that pass through
$p$ points and are tangent to $\ell$ lines in general position (with
$p+\ell=\dim\mathcal{M}_{\mathcal{A}}$). When $\mathcal{A}$ is a generic
arrangement of $k$ lines, we simplify the notation to $N_{k}(p,\ell)$. To get
these numbers we compute the degree of a related variety $M$ which is a
delicate intersection calculation. We do this in two different ways. First for
the case $k=3$ we preform an explicit calculation using the method of
undetermined coefficients. Then for the case $k=4$ we argue that the
corresponding intersection class is a product using transversality
considerations.

######  Remark 8.

The only way that an arrangement of 3 lines
$\mathcal{A}=\\{\ell_{1},\ell_{2},\ell_{3}\\}$ can be tangent to a given line
$L$ is if the cubic defining the arrangement meets the line $L$ at a point of
multiplicity strictly greater than 1. If the arrangement is generic this means
that an intersection point $\ell_{i}\cap\ell_{j}$ of two of the 3 lines must
lie on $L$. This motivates the introduction of the points
$p_{ij}=\ell_{i}\cap\ell_{j}$ in the following lemma: a generic arrangement is
tangent to a line $L$ if and only if there exists a point $p_{ij}$ on $L$.

######  Lemma 9.

The set

$M=\left\\{(\ell_{1},\ell_{2},\ell_{3},p_{12},p_{13},p_{23})\in\left(\mathbb{P}^{2*}\right)^{3}\times\left(\mathbb{P}^{2}\right)^{3}\;:\;p_{ij}=\ell_{i}\cap\ell_{j}\text{
for $1\leq i<j\leq 3$}\right\\}$

is a quasi-projective variety of dimension 6 in
$(\mathbb{P}^{2*})^{3}\times(\mathbb{P}^{2})^{3}$. The Chow ring is

$A=A((\mathbb{P}^{2*})^{3}\times(\mathbb{P}^{2})^{3})\cong\mathbb{Z}[x_{1},x_{2},x_{3},y_{12},y_{13},y_{23}]/(x_{1}^{3},x_{2}^{3},x_{3}^{3},y_{12}^{3},y_{13}^{3},y_{23}^{3})$

where the $x_{i}$ are the classes of the lines and the $y_{ij}$ are the
classes of the points. The class of $M$ in $A$ is

$[M]=\prod_{1\leq i<j\leq 3}(x_{i}+y_{ij})(x_{j}+y_{ij}).$

###### Proof.

Since $M$ has codimension 6 the class $[M]$ is a degree 6 polynomial in the
Chow ring $A$. We write this class as the sum of all degree 6 monomials in
$A$:

$[M]=\sum\limits_{a_{1}+\cdots+a_{6}=6}c_{(a_{i})}x_{1}^{a_{1}}x_{2}^{a_{2}}x_{3}^{a_{3}}y_{12}^{a_{4}}y_{13}^{a_{5}}y_{23}^{a_{6}}.$

Now we determine all the coefficients. First we show which coefficients are
zero. To do this we first partition the set of degree-6 monomials that appear
in the expanded product into types. In the definitions that follow the
notation $p\in A_{i}/I$ denotes a degree-$i$ monomial in the variables
appearing in the quotient ring $A/I$.

The first type of monomials we define are

$T_{1}=\left\\{\\{x_{1}^{2}y_{12}^{2}p|p\in
A_{2}/(x_{1},y_{12})\\}\bigcup\\{x_{2}^{2}y_{12}^{2}p|p\in
A_{2}/(x_{2},y_{12})\\}\right.$ $\bigcup\\{x_{1}^{2}y_{13}^{2}p|p\in
A_{2}/(x_{1},y_{13})\\}\bigcup\\{x_{3}^{2}y_{13}^{2}p|p\in
A_{2}/(x_{3},y_{13})\\}$ $\left.\bigcup\\{x_{2}^{2}y_{23}^{2}p|p\in
A_{2}/(x_{2},y_{23})\\}\bigcup\\{x_{3}^{2}y_{23}^{2}p|p\in
A_{2}/(x_{3},y_{23})\\}\right\\}.$

The second type we define are

$T_{2}=\left\\{\\{x_{1}^{2}x_{2}^{2}y_{12}p|p\in
A_{1}/(x_{1},x_{2},y_{12})\\}\bigcup\\{x_{1}^{2}x_{3}^{2}y_{13}p|p\in
A_{1}/(x_{1},x_{3},y_{13})\\}\right.$
$\left.\bigcup\\{x_{2}^{2}x_{3}^{2}y_{23}p|p\in
A_{1}/(x_{2},x_{3},y_{23})\\}\right\\}.$

The third type we define are

$T_{3}=\left\\{\\{x_{1}y_{12}^{2}y_{13}^{2}p|p\in
A_{1}/(x_{1},y_{12},y_{13})\\}\bigcup\\{x_{2}y_{12}^{2}y_{23}^{2}p|p\in
A_{1}/(x_{2},y_{12},y_{23})\\}\right.$
$\left.\bigcup\\{x_{3}y_{23}^{2}y_{13}^{2}p|p\in
A_{1}/(x_{3},y_{23},y_{13})\\}\right\\}.$

The fourth type we define is

$T_{4}=\\{x_{1}^{2}y_{23}^{2}x_{2}y_{12},x_{1}^{2}y_{23}^{2}x_{3}y_{13},x_{2}^{2}y_{13}^{2}x_{1}y_{12},x_{2}^{2}y_{13}^{2}x_{3}y_{23},x_{3}^{2}y_{12}^{2}x_{1}y_{13},x_{3}^{2}y_{12}^{2}x_{2}y_{23}\\}.$

Note that $|T_{1}|=54$, $|T_{2}|=9$, $|T_{3}|=9$, and $|T_{4}|=6$.

For any monomial $m\in A_{6}$ we define the complement of $m$ by the unique
monomial $c(m)=m^{\prime}$ such that $mm^{\prime}\neq 0\in A_{12}$. Set
$Z_{1}=T_{1}\cup T_{2}\cup T_{3}\cup T_{4}$. Notice that $Z_{1}$ has the
property that $c(Z_{1})=Z_{1}$. Hence to show that the monomials in $Z_{1}$
have the coefficient of zero in $[M]$ we just check that multiplying $[M]$ by
each of the monomials in $Z_{1}$ gives zero. Also, notice that within each
type there is a union of smaller sets where each of those sets can be obtained
from one another by a permutation of the coordinates. Hence if we can show
that the coefficient is zero for one of the subsets then we have completed the
task for the entire type. In the following arguments we use duality on
$\mathbb{P}^{2}$ often.

First we argue $[M]m=0$ where $m\in T_{1}$. Let $m\in T_{1}$ be in the first
subset so $m=x_{1}^{2}y_{12}^{2}m^{\prime}$. Then in $A$ this monomial $m$
represents fixing a generic line for $\ell_{1}$ in the first factor of
$\mathbb{P}^{2*}$ and a generic point for $p_{12}$. Set
$\pi_{i}:\left(\mathbb{P}^{2*}\right)^{3}\times\left(\mathbb{P}^{2}\right)^{3}\to\mathbb{P}^{2*}$
to be the projection to the factor corresponding to the line $\ell_{i}$ and
$\pi_{ij}:\left(\mathbb{P}^{2*}\right)^{3}\times\left(\mathbb{P}^{2}\right)^{3}\to\mathbb{P}^{2}$
to be the projection to the factor corresponding to the point $p_{ij}$. With
this notion we can say $m=[\pi_{1}^{-1}(\text{generic
point})\cap\pi_{12}^{-1}(\text{generic point})\cap(m^{\prime}\text{
conditions})]$ where we will not need the conditions from $m^{\prime}$. Using
the Moving Lemma (see [4, Theorem 5.4])

$[M]m=[M\cap\pi_{1}^{-1}(\text{generic point})\cap\pi_{12}^{-1}(\text{generic
point})\cap(m^{\prime}\text{ conditions})].$

In $M$ the point $p_{12}$ must lie on $\ell_{1}$ and this cannot happen if
$\ell_{1}$ and $p_{12}$ are generic. Hence $[M]m=0.$

The other types have similar arguments. We include them for completeness, but
with briefer arguments. For type 2 we look again at the first subset and put
$m=x_{1}^{2}x_{2}^{2}y_{12}m^{\prime}$. Then $[M]m$ equals

$[M\cap\pi_{1}^{-1}(\text{generic point})\cap\pi_{2}^{-1}(\text{generic
point})\cap\pi_{12}^{-1}(\text{generic line})\cap(m^{\prime}\text{
conditions})].$

This means that we have fixed generic lines for lines 1 and 2. Hence the point
$p_{12}$ is fixed using $M$, but $p_{12}$ must also lie on a fixed line. We
cannot find such a configuration; hence $[M]m=0$.

For type 3 we again look at the first subset
$m=x_{1}y_{12}^{2}y_{13}^{2}m^{\prime}$. Then $[M]m$ equals

$[M\cap\pi_{1}^{-1}(\text{generic line})\cap\pi_{12}^{-1}(\text{generic
point})\cap\pi_{13}^{-1}(\text{generic point})\cap(m^{\prime}\text{
conditions})].$

Line $\ell_{1}$ must satisfy the condition coming from $x_{1}$, which can be
interpreted as saying that line $\ell_{1}$ lies in a fixed pencil of lines;
however, this is inconsistent with the conditions requiring points $p_{12}$
and $p_{13}$ to lie in fixed (and general) positions. Hence $[M]m=0$.

For type 4 we just look at the first monomial
$m=x_{1}^{2}y_{23}^{2}x_{2}y_{12}$. Then $[M]m$ equals

$[M\cap\pi_{1}^{-1}(\text{generic point})\cap\pi_{23}^{-1}(\text{generic
point})\cap\pi_{2}^{-1}(\text{generic line})\cap\pi_{12}^{-1}(\text{generic
line})].$

In this case $\ell_{1}$ must be a fixed generic line and $p_{23}$ must be a
fixed generic point. Since line $\ell_{2}$ must lie in a fixed pencil and
since $p_{23}\in\ell_{2}$, the line $\ell_{2}$ is fixed too. Then
$p_{12}=\ell_{1}\cap\ell_{2}$ is fixed and so it cannot satisfy the generic
linear condition in the class $[M]m$. Hence $[M]m=0$.

In this case we have fixed a generic line for line 1 and force $p_{23}$ to be
fixed point in general position. The condition corresponding to
$[\pi^{-1}_{2}(\text{generic line})]$ forces line 2 to lie in a fixed pencil.
Then line 2 is determined (since it passes through $p_{23}$) and so is
$p_{12}=\ell_{1}\cap\ell_{2}$. However, the condition corresponding to
$[\pi^{-1}_{12}(\text{generic line})]$ forces $p_{12}$ to lie on a fixed
general line (that is, not through $\ell_{1}\cap\ell_{2}$). Thus, we cannot
find such a configuration and so $[M]m=0$.

Now we define another set of monomials and prove that each of these monomials
have coefficient equal to 1 in $[M]$. Again we define this set as a union of a
few different types. Unfortunately, there are many more types in this set even
though there are less total monomials. We define the types in the following
table.

Type | Monomials
---|---
$S_{1}$ | $\\{x_{1}^{2}x_{2}^{2}x_{3}^{2}\\}$
$S_{2}$ | $\\{x_{1}^{2}x_{2}^{2}x_{3}y_{13},x_{1}^{2}x_{2}^{2}x_{3}y_{23},x_{1}^{2}x_{3}^{2}x_{2}y_{23},x_{1}^{2}x_{3}^{2}x_{2}y_{12},x_{2}^{2}x_{3}^{2}x_{1}y_{13},x_{2}^{2}x_{3}^{2}x_{1}y_{12}\\}$
$S_{3}$ | $\\{x_{1}^{2}x_{2}^{2}y_{13}y_{23},x_{1}^{2}x_{3}^{2}y_{12}y_{23},x^{2}_{2}x_{3}^{2}y_{12}y_{13}\\}$
$S_{4}$ | $\\{x_{1}^{2}x_{2}x_{3}y_{23}^{2},x_{2}^{2}x_{1}x_{3}y_{13}^{2},x_{3}^{2}x_{2}x_{1}y_{12}^{2}\\}$
$S_{5}$ | $\\{x_{1}^{2}x_{2}x_{3}y_{12}y_{13},x_{2}^{2}x_{1}x_{3}y_{12}y_{23},x_{3}^{2}x_{1}x_{2}y_{13}y_{23}\\}$
$S_{6}$ | $\\{x_{1}^{2}x_{2}x_{3}y_{23}y_{12},x_{1}^{2}x_{2}x_{3}y_{23}y_{13},x_{2}^{2}x_{1}x_{3}y_{13}y_{12},x_{2}^{2}x_{1}x_{3}y_{13}y_{23},$
| $x_{3}^{2}x_{2}x_{1}y_{12}y_{13},x_{3}^{2}x_{2}x_{1}y_{12}y_{23}\\}$
$S_{7}$ | $\\{x_{1}^{2}x_{2}y_{13}y_{23}^{2},x_{1}^{2}x_{3}y_{12}y_{23}^{2},x_{2}^{2}x_{1}y_{23}y_{13}^{2},x_{2}^{2}x_{1}y_{23}y_{13}^{2},x_{3}^{2}x_{1}y_{23}y_{12}^{2},x_{3}^{2}x_{2}y_{13}y_{12}^{2}\\}$
$S_{8}$ | $\\{x_{1}^{2}x_{2}y_{12}y_{13}y_{23},x_{1}^{2}x_{3}y_{12}y_{13}y_{23},x_{2}^{2}x_{1}y_{12}y_{13}y_{23},x_{2}^{2}x_{3}y_{12}y_{13}y_{23},$
| $x_{3}^{2}x_{2}y_{12}y_{13}y_{23},x_{3}^{2}x_{1}y_{12}y_{13}y_{23}\\}$
$S_{9}$ | $\\{x_{1}^{2}y_{23}^{2}y_{12}y_{13},x_{2}^{2}y_{13}^{2}y_{23}y_{12},x_{3}^{2}y_{12}^{2}y_{23}y_{13}\\}$
$S_{10}$ | $\\{x_{1}x_{2}x_{3}y_{12}^{2}y_{13},x_{1}x_{2}x_{3}y_{12}^{2}y_{23},x_{1}x_{2}x_{3}y_{13}^{2}y_{12},x_{1}x_{2}x_{3}y_{13}^{2}y_{23},$
| $x_{1}x_{2}x_{3}y_{23}^{2}y_{12},x_{1}x_{2}x_{3}y_{23}^{2}y_{13}\\}$
$S_{11}$ | $\\{x_{1}x_{2}y_{12}^{2}y_{13}y_{23},x_{1}x_{3}y_{12}y_{13}^{2}y_{23},x_{3}x_{2}y_{12}y_{13}y_{23}^{2}\\}$
$S_{12}$ | $\\{x_{1}x_{2}y_{12}y_{13}^{2}y_{23},x_{1}x_{2}y_{12}y_{13}y_{23}^{2},x_{1}x_{3}y_{12}^{2}y_{13}y_{23},x_{1}x_{3}y_{12}y_{13}y_{23}^{2},$
| $x_{3}x_{2}y_{12}y_{13}^{2}y_{23},x_{3}x_{2}y_{12}^{2}y_{13}y_{23}\\}$
$S_{13}$ | $\\{x_{1}x_{2}y_{13}^{2}y_{23}^{2},x_{1}x_{3}y_{12}^{2}y_{23}^{2},x_{2}x_{3}y_{12}^{2}y_{13}^{2}\\}$
$S_{14}$ | $\\{x_{1}y_{12}^{2}y_{13}y_{23}^{2},x_{1}y_{12}y_{13}^{2}y_{23}^{2},x_{2}y_{12}^{2}y_{13}^{2}y_{23},x_{2}y_{12}y_{13}^{2}y_{23}^{2},$
| $x_{3}y_{12}^{2}y_{13}y_{23}^{2},x_{3}y_{12}^{2}y_{13}^{2}y_{23}\\}$
$S_{15}$ | $\\{y_{12}^{2}y_{13}^{2}y_{23}^{2}\\}$

Now put $Z_{2}=S_{1}\cup\cdots\cup S_{15}$. In the next table we compute the
intersection of $M\cap m$ where $m\in Z_{2}$. We briefly describe, for each
type, how this intersection results in a unique arrangement of three lines
with the marked points $p_{ij}$. Again we use the argument that within each
$S_{i}$ we only need to check one monomial since all the others within that
class can be obtained from permuting coordinates.

Type | Monomial | Intersect this class with $M$ gives a unique point
---|---|---
$S_{1}$ | $x_{1}^{2}x_{2}^{2}x_{3}^{2}$ | Fixing three lines fixes the three intersection points.
$S_{2}$ | $x_{1}^{2}x_{2}^{2}x_{3}y_{13}$ | Fixing lines 1 and 2. Then fixing a linear condition
|  | point 13 gives a unique point and then putting
|  | a linear condition on line 3 fixes it.
$S_{3}$ | $x_{1}^{2}x_{2}^{2}y_{13}y_{23}$ | Fixing lines 1 and 2. Then giving linear conditions on points
|  | 13 and 23 fixes those points which fixes line 3.
$S_{4}$ | $x_{1}^{2}x_{2}x_{3}y_{23}^{2}$ | Fix line 1 and point 23. Then putting linear conditions on lines
|  | 2 and 3 fixes them.
$S_{5}$ | $x_{1}^{2}x_{2}x_{3}y_{12}y_{13}$ | Fix line 1. Then Putting a linear condition on points
|  | 12 and 13 fixes them on line 1.
|  | Then putting linear conditions on lines 2 and 3 fixes them.
$S_{6}$ | $x_{1}^{2}x_{2}x_{3}y_{12}y_{23}$ | Fixing line 1 and giving a linear condition on point 12
|  | fixes point 12. Together with putting a linear condition on
|  | line 2 fixes it. Then putting a linear condition on point 23
|  | fixes it and line 3.
$S_{7}$ | $x_{1}^{2}x_{2}y_{13}y_{23}^{2}$ | Fixing line 1, point 23, and putting a linear condition on
|  | line 2 fixes it. Then putting a linear condition on point 13
|  | fixes it on line one and hence fixes line 3.
$S_{8}$ | $x_{1}^{2}x_{2}y_{12}y_{13}y_{23}$ | Fix line 1 and putting liner conditions on points
|  | 12 and 13 fixes these points. Then putting a linear
|  | condition on line 2 fixes it. And then finally fixing a linear
|  | condition on point 23 fixes it and line 3.
$S_{9}$ | $x_{1}y_{23}^{2}y_{12}y_{13}$ | Fix line 1 and point 23. Putting linear conditions
|  | on points 12 and 13 fixes them on line 1. Then this
|  | fixes lines 2 and 3.
$S_{10}$ | $x_{1}x_{2}x_{3}y_{12}^{2}y_{13}$ | Fixing point 12 with putting linear conditions
|  | on lines 1 and 2 fixes them. Then putting a linear condition on
|  | point 13 fixes it on line 1. Then putting a linear condition
|  | on line 3 fixes it.
$S_{11}$ | $x_{1}x_{2}y_{12}^{2}y_{13}y_{23}$ | Fixing point 12 with putting linear conditions
|  | on lines 1 and 2 fixes them. Then putting linear conditions on
|  | points 13 and 23 fixes them and hence line 3.
$S_{12}$ | $x_{1}x_{2}y_{12}y_{13}^{2}y_{23}$ | Fixing point 13 and putting a linear condition on
|  | line 1 fixes it. Then putting a linear condition on point 12 fixes
|  | it on line 1. Then the linear condition on line 2 fixes it.
|  | Then the linear condition on point 23 fixes it and line 3.
$S_{13}$ | $x_{1}x_{2}y_{13}^{2}y_{23}^{2}$ | Fixing points 13 and 23 fixes line 3. Then putting linear
|  | conditions on lines 1 and 2 fixes them
$S_{14}$ | $x_{1}y_{12}^{2}y_{13}y_{23}^{2}$ | Fixing points 12 and 23 fixes line 2. Then giving a linear
|  | condition on line 1 fixes it. Then putting a linear condition
|  | on point 13 fixes it on line 1 which also fixes line 3.
$S_{15}$ | $y_{12}^{2}y_{13}^{2}y_{23}^{2}$ | The points 12, 13, and 23 are fixed. This fixes lines 1,2, and 3.

Again since the complement $c(Z_{2})=Z_{2}$ we have shown that the
coefficients of all these monomials in $[M]$ are equal to 1. There is just one
more monomial to consider. Let
$Z_{3}=\\{x_{1}x_{2}x_{3}y_{12}y_{13}y_{23}\\}$. This set also has the
property $c(Z_{3})=Z_{3}$. Now we compute its coefficient. The class
$x_{1}x_{2}x_{3}y_{12}y_{13}y_{23}$ puts a linear condition on each line and
point. For the lines we can think of this as fixing three points $q_{1}$,
$q_{2}$, and $q_{3}$ in general position and requiring that
$p_{i}\in\ell_{i}$, for $i\in\\{1,2,3\\}$. Similarly, to interpret the linear
conditions on the points $p_{ij}$ we fix three lines $L_{12}$, $L_{13}$, and
$L_{23}$ in general position, and require that $p_{ij}\in L_{ij}$.

Suppose now that we choose an arbitrary point $p_{12}$ on line $L_{12}$. Fix a
embedding $\phi:\mathbb{P}^{1}\to\mathbb{P}^{2}$ defined by
$\phi([a:b])=[f_{1}(a:b):f_{2}(a:b):f_{3}(a:b)]$ where
$\phi(\mathbb{P}^{1})=L_{12}$ and each $f_{i}$ is homogeneous of degree 1.
Then there exists $[a:b]\in\mathbb{P}^{1}$ such that $\phi([a:b])=p_{12}$. Now
since the points $q_{1}$ and $p_{12}$ are fixed we have that line 1,
$\ell_{1}$, is fixed. Line $\ell_{1}$ fixed means that we can get
$p_{13}=\ell_{1}\cap L_{13}$. Next we find $\ell_{3}$ is determined by the
points $p_{13}$ and $q_{3}$. With $\ell_{3}$ we can get $p_{23}=\ell_{3}\cap
L_{23}$. Now we make $\ell_{2}$ the line through the points $p_{23}$ and
$q_{2}$. Since we chose $p_{12}$ randomly at the beginning of this process we
have that $P=\ell_{2}\cap L_{12}$ may not be equal to $p_{12}$. Each step in
the above process to determine each line and point is a cross product
computation with a vector of constants (the line or point that is fixed before
the choice of $p_{12}$) with a vector of homogenous degree 1 in the functions
$f_{i}$. Hence there exist linear functions $g_{1}$ and $g_{2}$ such that
$P=\phi([g_{1}(a:b),g_{2}(a:b)])$. Now $P=p_{12}$ if and only if

$\det\left[\begin{array}[]{cc}g_{1}(a:b)&g_{2}(a:b)\\\
a&b\end{array}\right]=0.$

Since this a homogeneous quadratic polynomial of two variables there must be 2
solutions up to multiplicity. Hence the coefficient of the monomial
$x_{1}x_{2}x_{3}y_{12}y_{13}y_{23}$ is 2.

The total number of terms in $A_{6}$ 141 (this can be calculated by a standard
inclusion-exclusion argument). Since $|Z_{1}|=78$, $|Z_{2}|=62$, and
$|Z_{3}|=1$ we have computed the coefficients of all possible terms. Finally
with any computer software system one can expand the polynomial $\prod_{1\leq
i<j\leq 3}(x_{i}+y_{ij})(x_{j}+y_{ij})$ and see that the coefficients match
what we have just determined. This completes the proof. ∎

######  Remark 10.

A generic element of $M$ can be thought of as a generic arrangement of 3
lines. The complement inside $M$ of this generic locus consists of several 5
dimensional subvarieties. These consist of arrangements where two lines are
the same and arrangements consisting of a triple line, together with their
marked points.

Now we determine the characteristic numbers for 3 lines.

######  Theorem 11.

The characteristic numbers $N_{3}(p,6-p)$ for a generic arrangement of 3 lines
are given in the following table.

$p$ | 0 | 1 | 2 | 3 | 4 | 5 | 6
---|---|---|---|---|---|---|---
$N_{3}(p,6-p)$ | 15 | 30 | 48 | 57 | 48 | 30 | 15

###### Proof.

If we intersect $M$ with 6 generic conditions then we will obtain a finite
number of points none of which will lie on the 5-dimensional non-generic
subvarieties. So, when we multiply the class $[M]$ with that from Lemma 7 we
will only be counting generic arrangements of 3 lines. From Lemma 9 we have

$[M]=[\cap_{1\leq i<j\leq
3}(p_{ij}\in\ell_{i})\cap(p_{ij}\in\ell_{j})]=\prod_{1\leq i<j\leq
3}(x_{i}+y_{ij})(x_{j}+y_{ij}).$

Then the condition that the arrangement $\mathcal{A}$ passes through a given
point $p\in\mathbb{P}^{2}$ has class $x_{1}+x_{2}+x_{3}$. Similarly the
condition that a given line $L\in\mathbb{P}^{2}$ contains one of the $p_{ij}$
has class $y_{12}+y_{13}+y_{23}$. Using the Moving Lemma, Lemma 7, and Remark
8 the degree of the class
$[M](x_{1}+x_{2}+x_{3})^{p}(y_{12}+y_{13}+y_{23})^{6-p}$ in
$A_{*}\left(\left(\mathbb{P}^{2*}\right)^{3}\times\left(\mathbb{P}^{2}\right)^{3}\right)$
counts the number of labeled 3-generic arrangements that pass through $p$
general points and are tangent to $6-p$ general lines in $\mathbb{P}^{2}$. To
remove the effect of the labeling, we divide these numbers by $3!$ to obtain
the characteristic numbers. ∎

######  Remark 12.

The symmetry of $N_{3}(p,6-p)$ is easily explained: the dual of a 3-generic
arrangement through $p$ points $P_{1},\ldots,P_{p}$ and tangent to $6-p$ lines
$L_{1},\ldots,L_{6-p}$ is a 3-generic arrangement through the $6-p$ dual
points $\hat{L}_{1},\ldots,\hat{L}_{6-p}$ and tangent to the $p$ dual lines
$\hat{P}_{1},\ldots,\hat{P}_{p}$. This symmetry does not occur in the 4 lines
case as one can see below.

Now we consider the 4 line case. We compute the class $M$ as before but we use
transversality arguments in place of the method of undetermined coefficients.

######  Lemma 13.

The set

$M=\left\\{(\ell_{1},\dots,\ell_{4},p_{12},\dots,p_{34})\in\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}\;:\;p_{ij}\in\ell_{i}\cap\ell_{j}\text{
for $1\leq i<j\leq 4$}\right\\}$

is a quasi-projective variety of codimension 12 in
$\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}$. The
Chow ring of the ambient space is

$A=A\left(\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}\right)\cong\mathbb{Z}[x_{1},\dots,x_{4},y_{12},\dots,y_{34}]/(x_{1}^{3},\dots,x_{4}^{3},y_{12}^{3},\dots,y_{34}^{3})$

where the $x_{i}$ correspond to the lines and the $y_{ij}$ correspond to the
intersection points $p_{ij}$. In this ring the class of $M$ is

$[M]=\left[\cap_{1\leq i<j\leq
4}\left((p_{ij}\in\ell_{i})\cap(p_{ij}\in\ell_{j})\right)\right]=\prod_{1\leq
i<j\leq 4}(x_{i}+y_{ij})(x_{j}+y_{ij}).$

###### Proof.

As in Lemma 9 set
$\pi_{i}:\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}\to\mathbb{P}^{2*}$
to be the projection to the factor corresponding to the line $\ell_{i}$ and
$\pi_{ij}:\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}\to\mathbb{P}^{2}$
to be the projection to the factor corresponding to the point $p_{ij}$. The
condition $p_{ij}\in\ell_{i}$ is a bilinear hypersurface in
$\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}$ whose
class is $x_{i}+y_{ij}$. The closure of the variety $M$ is the intersection of
these 12 hypersurfaces. We will show that this is a local complete
intersection.

Put $\pi=\pi_{1}\times\pi_{2}\times\pi_{3}\times\pi_{4}$. Examine the
restriction $\pi:M\to\left(\mathbb{P}^{2*}\right)^{4}$. Now stratify
$\left(\mathbb{P}^{2*}\right)^{4}$ into the following quasi-projective
subvarieties $G=$ “generic arrangements ”, $T=$ “arrangements where two lines
are equal and the other two are generic”, $D=$ “arrangements where there are
two generic double lines”, $H=$“arrangements where three lines are equal and
the last one is generic”, and $F=$ “arrangements where all four lines are
equal”. So, $\left(\mathbb{P}^{2*}\right)^{4}=G\cup D\cup T\cup H\cup F$ and
all the unions are disjoint. The dimension of $G$ is 8 and for any $x\in G$
the fiber $\pi^{-1}(x)$ is a unique point. Hence $\pi^{-1}(G)$ is
8-dimensional. The dimension of $T$ is 6 and for any $x\in T$ the fiber
$\pi^{-1}(x)$ is 1-dimensional. Hence $\pi^{-1}(T)$ is 7-dimensional. The
dimension of $D$ is 4 and for $x\in D$ the fiber $\pi^{-1}(x)$ is
2-dimensional. Hence $\pi^{-1}(D)$ is 6-dimensional. The dimension of $H$ is 4
and for $x\in H$ the fiber $\pi^{-1}(x)$ is 3-dimensional. Hence $\pi^{-1}(H)$
is 7-dimensional. The dimension of $F$ is 2 and for $x\in F$ the fiber
$\pi^{-1}(x)$ is 6-dimensional. Hence $\pi^{-1}(F)$ is 8-dimensional. Since
$M=\pi^{-1}(G)\cup\pi^{-1}(T)\cup\pi^{-1}(D)\cup\pi^{-1}(H)\cup\pi^{-1}(F)$ we
have shown that $M$ is 8-dimensional.

Hence $M$ is a local complete intersection in
$\left(\mathbb{P}^{2*}\right)^{4}\times\left(\mathbb{P}^{2}\right)^{6}$. Using
Theorem 5.10 in [4] the class of the intersection $[M]$ is the product of the
classes $x_{i}+y_{ij}$. ∎

######  Theorem 14.

The characteristic numbers $N_{4}(p,8-p)$ for a generic arrangement of 4 lines
are given in the following table.

$p$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
---|---|---|---|---|---|---|---|---|---
$N_{4}(p,8-p)$ | 16695 | 17955 | 13185 | 8190 | 4410 | 2070 | 855 | 315 | 105

###### Proof.

We use an argument similar to the proof of Theorem 11 but there is a subtlety.
The component $F$ from Lemma 13 consisting of quadruple lines has the same
dimension as the component $G$ of generic arrangements of 4 lines. It follows
that $[M]=a[F]+b[G]$ for some integers $a$ and $b$. Specialize the equations
that cut out $M$ to a fixed generic arrangement $\mathcal{A}$ of 4 lines to
see that there is just one element of $M$ lying over $\mathcal{A}$, with
multiplicity one; this shows that $a=1$. Similarly, if we specialize the
equations that cut out $M$ to $\mathcal{A}$, a fixed quadruple line with six
fixed marked points $p_{ij}$, we again see that the fiber of $M$ lying over
$\mathcal{A}$ is a reduced point in $\mathcal{M}$; so $b$ = 1 and
$[M]=[F]+[G]$. Let
$C=(x_{1}+x_{2}+x_{3}+x_{4})^{p}(y_{12}+y_{13}+y_{14}+y_{23}+y_{24}+y_{34})^{8-p}$.
The degree of $[M]C=[F]C+[G]C$ also counts some instances in which all the
$\ell_{i}$ are equal (coming from the term $[F]C$). For instance, in the case
where $p=0$, we seek elements in $M$ with some $p_{ij}$ on each of $8$ lines
$L_{1},\ldots,L_{8}$. If we set all the $\ell_{i}$ equal to the line joining
$L_{i}\cap L_{j}$ to $L_{s}\cap L_{t}$ then the common line $\ell_{i}$ meets
the 8 lines $L_{j}$ in just $\binom{4}{2}=6$ points. Labeling these points
with the six points $p_{ij}$ gives an element of $M$ that is included in our
degree count but that doesn’t correspond to a 4-generic arrangement; this is
illustrated in the left picture of Figure 2, where the thick line represents
the common line $\ell_{i}$. There are $\binom{8}{2,2,4}6!/2$ ways to produce
such examples. Removing these from the degree count, we obtain the degree of
$[G]C$ and dividing the result by 4! to account for the effect of labeling, we
obtain the value of $N_{4}(0,8)$ in the Table.

Figure 2: Quadruple lines counted by the degree computation for $p=0$ (left),
$p=1$ (middle), $p=2$ (right).

Similarly, quadruple lines are counted in the degree computations for $p=1$
and $p=2$, as illustrated in the middle and right pictures of Figure 2. This
requires an adjustment of $\binom{8}{2,6}6!$ in the case of $p=1$ and $6!$ in
the case of $p=2$, giving rise to the values for $N_{4}(p,8-p)$ in the
statement of the theorem. ∎

######  Remark 15.

It would be interesting to have an explanation for the unimodality of the
characteristic numbers $N_{k}(p,2k-p)$. In particular, do the characteristic
numbers $\\{N_{k}(p,2k-p)\\}$ for generic arrangements of $k$ lines always
form a unimodal sequence?

######  Remark 16.

Taking the dual of the $4$-generic arrangement yields a braid line arrangement
$\mathcal{A}_{3}$, an arrangement of $6$ lines meeting in 4 triple points and
3 double points, pictured in Figure 3. This is related to the braid
arrangement (the reflecting hyperplanes of the action of $\mathcal{S}_{4}$ on
the variables of $\mathbb{C}^{4}$), $\mathcal{A}_{3}=\\{x_{i}=x_{j}\;:\;1\leq
i<j\leq 4\\}$ in $\mathbb{C}^{4}$ which consists of 6 hyperplanes, each
containing the line $\\{x_{1}=x_{2}=x_{3}=x_{4}\\}\subset\mathbb{C}^{4}$.
Quotienting out the common line and projectivizing gives the braid arrangement
in $\mathbb{P}^{2}$.

Figure 3: The braid arrangement $\mathcal{A}_{3}$

The moduli space $\mathcal{M}_{\mathcal{A}_{3}}$ has the same dimension, 8, as
the moduli space of the dual of $\mathcal{A}_{3}$. The characteristic numbers
$N_{\mathcal{A}_{3}}(p,8-p)$ counting the number of arrangements
combinatorially equivalent to the braid arrangement that pass through $p$
points and are tangent to $8-p$ lines in general position are given by
$N_{4}(8-p,p)$. For example, there are 16695 braid arrangements through 8
points in general position in $\mathbb{P}^{2}$, a result initially reported in
Paul [20].

The characteristic numbers are important because they determine the answer to
all enumerative questions involving points and tangency to arbitrary smooth
curves (see [7, section 10.4] and the references therein for the history of
this result).

######  Theorem 17.

Let $\mathcal{L}_{\mathcal{A}}$ be the intersection lattice of a line
arrangement and let $t=\dim\mathcal{M}_{\mathcal{A}}$. The number of line
arrangements with intersection lattice isomorphic to
$\mathcal{L}_{\mathcal{A}}$ through $p$ points and tangent to $t-p$ smooth
curves of degrees $n_{1},\ldots,n_{t-p}$ and classes111The class of a curve is
the number of lines passing through a given general point and tangent to the
curve at a simple point. For example, the class of a smooth curve of degree
$d$ is $d(d-1)$. $m_{1},\ldots,m_{t-p}$ in general position is obtained from
the product

$\mu^{p}\prod_{i=1}^{t-p}(m_{i}\mu+n_{i}\nu)$

by expanding the polynomial and replacing the monomial $\mu^{k}\nu^{t-k}$ by
the associated characteristic number – the number of arrangements with lattice
type $\mathcal{L}_{\mathcal{A}}$ passing through $k$ general points and
tangent to $t-k$ lines.

###### Proof.

The proof follows immediately from the argument due to Fulton and MacPherson
in [7, section 10.4] if we interpret each point condition as a tangency
condition to a curve of degree 0 and class 1. Alternatively, we can work with
the dimension $(t-p)$ family of arrangements in $\mathcal{M}_{\mathcal{A}}$
that pass through the specified $p$ points, apply Fulton and MacPherson’s
theorem and then re-interpret the results in terms of the polynomial displayed
above. ∎

######  Example 18.

To compute the number of generic arrangements of 4 lines through 3 points,
tangent to a given line and tangent to four smooth conics, we expand the
product $\mu^{3}(2\mu+2\nu)^{4}\nu^{1}$ and replace the monomials with the
appropriate quantities from Theorem 14. The answer is

$2^{4}(315)+2^{4}\binom{4}{1}(855)+2^{4}\binom{4}{2}(2070)+2^{4}\binom{4}{3}(4410)+2^{4}(8190)=671760.$

######  Remark 19.

The computation of $N_{k}(p,2k-p)$ becomes more intricate as $k$ increases.
For instance, using the method of proof from Theorem 14 to compute
$N_{5}(0,10)$, we see that every quintuple line determines a point in the
incidence correspondence $M$. This higher-dimensional component of the
solution space (isomorphic to $\mathbb{P}^{2}$) “counts” as a finite number of
points in the standard intersection theory computation of $N_{5}(0,10)$. The
appropriate count is called the excess intersection of this component and can
be computed using the Excess Intersection Formula [7, section 6.3]. We do not
pursue this approach further here.

##  3 Pencils and Cones Over Generic Arrangements

An arrangement of lines is said to be a pencil if all lines pass through a
common point. Such an arrangement is also a $0$-coned generic arrangement. We
start this section with a simple enumerative result about pencils.

######  Theorem 20.

If $\mathcal{A}$ is a pencil of $k$ lines in $\mathbb{P}^{2}$ then
$\dim\mathcal{M}_{\mathcal{A}}=k+2$ and the characteristic numbers are
$N_{\mathcal{A}}(k+2,0)=3\binom{k+2}{4}$,
$N_{\mathcal{A}}(k+1,1)=\binom{k+1}{2}$, $N_{\mathcal{A}}(k,2)=1$. All other
characteristic numbers are 0.

###### Proof.

By the Pigeonhole Principle, if a pencil $\mathcal{A}$ of $k$ lines passes
through more than $k+3$ points in general position then 3 of the lines must
each contain 2 points and all three lines must be coincident, but this means
that these 6 points must satisfy an algebraic relation and thus violates the
assumption that the points were in general position. As well, it is easy to
see that there are a finite number of pencils $\mathcal{A}$ of $k$ lines
passing through $k+2$ points in general position (two lines contain 2 points
each and intersect in the node of the pencil, the other $k-2$ lines each
contain one of the remaining points) so $\dim\mathcal{M}_{\mathcal{A}}=k+2$.
There are $N_{\mathcal{A}}(k+2,0)=\binom{k+2}{2,2,k-2}/2=3\binom{k+2}{4}$ ways
to create such an arrangement.

To compute $N_{\mathcal{A}}(k+1,1)$ note that the node of the pencil must lie
on the given line. The arrangement must have one line containing $2$ of the
$k+1$ points and this line intersects the given line at the node of the
pencil; the remaining $k-1$ lines in the pencil each contain 1 point and the
node. There are $N_{\mathcal{A}}(k+1,1)=\binom{k+1}{2}$ ways to choose the
first line in the pencil; the rest of the construction is determined. Finally,
to compute $N_{\mathcal{A}}(k,2)$ note that the node must lie at the
intersection of the two given lines and that each line in the pencil must pass
through the node and one of the $k$ given points. So $\mathcal{A}$ is uniquely
determined. ∎

We proceed to study $d$-coned generic arrangements with $d\geq 1$. Recall from
Definition 2 that an arrangement $\mathcal{A}$ of $k>n$ hyperplanes in
$\mathbb{P}^{n}$ is a $d$-coned generic arrangement if $\mathcal{A}$ can be
obtained from a generic hyperplane arrangement $\mathcal{B}$ in a linear
subspace $H\cong\mathbb{P}^{n-d-1}$ by taking the cone with a $d$-dimensional
linear space (equivalently, with $d+1$ general points). Given a $d$-coned
generic arrangement $\mathcal{A}$ in $\mathbb{P}^{n}$, the intersection with a
general linear space $H$ of dimension $n-d-1$ is a generic hyperplane
arrangement in $\mathbb{P}^{n-d-1}$.

######  Lemma 21.

Let $\mathcal{A}$ be a $d$-coned generic arrangement of $k$ hyperplanes in
$\mathbb{P}^{n}$. Then the dimension of the moduli space
$\mathcal{M}_{\mathcal{A}}$ is $D=(d+1)(n-d)+k(n-d-1)$.

###### Proof.

Note that an arrangement $\mathcal{B}\in\mathcal{M}_{\mathcal{A}}$ is
determined by $\Lambda$ in the Grassmannian $\mathbb{G}(d,n)$ of
$d$-dimensional linear spaces of $\mathbb{P}^{n}$, together with $k$ points in
$\mathbb{P}(\mathbb{C}^{n+1}/\Lambda)\cong\mathbb{P}^{n-d-1}$ (since $\Lambda$
determines a subspace of dimension $d+1$ in $\mathbb{C}^{n+1}$). So the
dimension of $\mathcal{M}_{\mathcal{A}}$ is $D=\text{dim
}\left[\mathbb{G}(d,n)\times(\mathbb{P}^{n-d-1})^{k}\right]=(d+1)(n-d)+k(n-d-1)$.
∎

As a warm-up, we count the number of $0$-coned generic arrangements through a
set of points.

######  Theorem 22.

Let $\mathcal{A}$ be a $0$-coned generic arrangement of $k\geq n$ hyperplanes
in $\mathbb{P}^{n}$. Then the number of arrangements combinatorially
equivalent to $\mathcal{A}$ that pass through $kn+n-k$ points in general
position is

$N_{\mathcal{A}}=\frac{1}{k!}\binom{k}{n}\binom{kn+n-k}{(n)^{n},(n-1)^{k-n}}=\frac{(kn+n-k)!}{(n!)^{n+1}((k-n)!)((n-1)!)^{k-n}}.$

###### Proof.

By Lemma 21, $\dim\mathcal{M}_{\mathcal{A}}=kn+n-k$. The only way to obtain an
arrangement combinatorially equivalent to $\mathcal{A}$ through the $kn+n-k$
points is to have $n$ hyperplanes pass through $n$ of the points and meet in a
new point $p$; then each of the remaining $k-n$ of the hyperplanes pass
through the point $p$ and $n-1$ of the remaining given points. Labeling the
set of hyperplanes with $n$ points as $H_{1},\ldots,H_{n}$ and the remaining
hyperplanes $H_{n+1},\ldots,H_{k}$, we have
$\binom{kn+n-k}{(n)^{n},(n-1)^{k-n}}$ labeled arrangements. Dividing by
$n!k-n!$ gives the number of unlabeled arrangements and rearranging terms
gives the desired formula. ∎

It is tempting to think that the number of $d$-coned generic arrangements of
$k$ hyperplanes in $\mathbb{P}^{n}$ that pass through $D=(d+1)(n-d)+k(n-d-1)$
points in general position is just

$N_{\mathcal{A}}=\frac{1}{k!}\binom{k}{n}\binom{D}{(n)^{n-d},(n-d)^{k-n+d}},$

but in fact this grossly undercounts the arrangements. In this sense, the
$0$-coned generic arrangement result in Theorem 22 is misleading.

Before stating our result on the enumeration of $d$-coned generic
arrangements, we quickly review the structure of the intersection ring
$A=A(\mathbb{G}(d,n))$ of the Grassmannian $\mathbb{G}(d,n)$ of $d$-planes in
$\mathbb{P}^{n}$. Given any complete flag of linear spaces $F:F_{0}\subsetneq
F_{1}\subsetneq\ldots\subsetneq F_{n}=\mathbb{P}^{n}$ with $\dim F_{j}=j$, we
have a cellular decomposition of $\mathbb{G}(d,n)$ into affine cells; for each
$(d+1)$-tuple $(\alpha_{0},\alpha_{1},\ldots,\alpha_{d})$ of non-increasing
non-negative integers $\alpha_{i}\leq n-d$ we have an affine cell
$X_{\alpha}(F)$ of codimension $|\alpha|=\sum\alpha_{i}$. To describe the
$d$-planes $\Lambda$ in $X_{\alpha}(F)$, note that for each $d$-plane
$\Lambda$ the sequence $\dim\Lambda\cap F_{j}$ is completely characterized by
the $d+1$ places where the dimension increases, the so-called rank jumps. Now
$\Lambda\in X_{\alpha}(F)$ precisely when the collection of rank jumps is
$\\{n-d+j-\alpha_{j}\;:\;0\leq j\leq d\\}$. For instance the largest open cell
$X_{(0)^{d+1}}$ consists of $d$-planes in general position with respect to the
flag $F$. See Hatcher [11] for more details on the cellular decomposition of
$\mathbb{G}(d,n)$. Define the intersection class $\sigma_{\alpha}\in A$ to be
the class of the closure of $X_{\alpha}(F)$. The set of $\sigma_{\alpha}$ form
an additive basis for the intersection ring $A(\mathbb{G}(d,n))$. The
multiplicative structure of the intersection ring is described by the Pieri
and Giambelli formulas (see e.g. Gatto [9, 8] for statements of these results
and an interesting description of the multiplicative structure of
$A(\mathbb{G}(d,n))$ in terms of Hasse-Schmidt derivations). The class
$\sigma_{(0)^{d+1}}$ is the multiplicative identity in $A(\mathbb{G}(d,n))$.
We denote the tuple consisting of $i$ 1’s followed by $d+1-i$ 0’s by $1^{i}$.
Similarly, the tuple consisting of $d+1$ copies of $n-d$ is denoted
$(n-d)^{d+1}$. The class $\sigma_{(n-d)^{d+1}}$ is the class of a point and
the class $\sigma_{1^{d+1}}$ is the class of the set of $d$-planes contained
in $F_{n-1}\cong\mathbb{P}^{n-1}$. Though we will not need the full power of
the Pieri and Giambelli formulas, we note that there is a duality between the
classes $\sigma_{\alpha}$ in the sense that

$\sigma_{\alpha}\cdot\sigma_{r((n-d)^{d+1}-\alpha)}=\sigma_{(n-d)^{d+1}},$

where $r$ is the operation that reverses a tuple:
$r(\beta_{0},\ldots,\beta_{d})=(\beta_{d},\ldots,\beta_{0})$. As well, if
$\sigma_{\alpha_{1}},\ldots,\sigma_{\alpha_{t}}$ are intersection classes with
$\alpha_{1}+\cdots+\alpha_{t}=\dim\mathbb{G}(d,n)=(n-d)(d+1)$ then their
product has a well-defined degree, denoted
$\int_{\mathbb{G}(d,n)}\sigma_{\alpha_{1}}\cdots\sigma_{\alpha_{t}}$, which
can be interpreted as the number of $d$-planes in
$X_{\alpha_{1}}\cap\cdots\cap X_{\alpha_{t}}$ if the classes $X_{\alpha_{i}}$
meet transversally (here the $X_{\alpha_{i}}$ can be interpreted as coming
from different flags, in general position with respect to each other).

######  Theorem 23.

If $\mathcal{A}$ is a $d$-coned generic arrangement of $k$ hyperplanes in
$\mathbb{P}^{n}$, then the number of pencils combinatorially equivalent to
$\mathcal{A}$ that pass through $D=(d+1)(n-d)+k(n-d-1)$ points in general
position is given by

$N_{\mathcal{A}}=\frac{1}{k!}\sum_{\Gamma}\sigma^{s}\binom{k}{s_{0},s_{1},\ldots,s_{d+1}}\binom{D}{(n-(d+1))^{s_{0}},(n-d)^{s_{1}},\ldots,(n)^{s_{d+1}}},$

where

$\Gamma=\left\\{(s_{0},\ldots,s_{d+1})\in\mathbb{N}^{d+2}:\sum_{i=0}^{d+1}is_{i}=(d+1)(n-d)=\dim\mathbb{G}(d,n),\sum_{i=0}^{d+1}s_{i}=k\right\\}$

and

$\sigma^{s}=(\sigma_{0})^{s_{0}}(\sigma_{1})^{s_{1}}(\sigma_{(1)^{2}})^{s_{2}}\cdots(\sigma_{(1)^{d+1}})^{s_{d+1}},$

using the notation of Remark 3.

###### Proof.

We first note that the intersection ring
$A(\mathbb{G}(d,n)\times(\mathbb{P}^{n*})^{k})$ is just the tensor product of
$A(\mathbb{G}(d,n))$ with $A((\mathbb{P}^{n*})^{k})$. Now we determine the
class $[Z]\in A(\mathbb{G}(d,n)\times\mathbb{P}^{n*})$, where $Z$ is the
closed set determining an incidence correspondence,

$Z=\\{(\Lambda,H)\in\mathbb{G}(d,n)\times\mathbb{P}^{n*}\;:\;\Lambda\subset
H\\}.$

Note that $\mathbb{G}(d,n)\times\mathbb{P}^{n*}$ has dimension $(d+1)(n-d)+n$
and $Z$ has dimension $(d+1)(n-d)+(n-d-1)$ so $Z$ has codimension $d+1$. It
follows that

$[Z]=\sum_{\\{\alpha:\;|\alpha|\leq
d+1\\}}a_{\alpha}\sigma_{\alpha}h^{d+1-|\alpha|},$

where $\sigma_{\alpha}$ and $h$ denote the pullbacks of these classes to
$A(\mathbb{G}(d,n)\times\mathbb{P}^{n*})$ and the $a_{\alpha}$ are integers.

We multiply $[Z]$ by $\sigma_{r((n-d)^{d+1}-\alpha)}h^{n-d-1+|\alpha|}$ to
obtain $a_{\alpha}\sigma_{(n-d)^{d+1}}h^{n}$, representing $a_{\alpha}$
elements $(\Lambda,H)\in Z$ that also meet the flag $F$ in a manner prescribed
by the class $\sigma_{r((n-d)^{d+1}-\alpha)}h^{n-d-1+|\alpha|}$. That is,
$\Lambda$ has rank jumps at positions
$n-d-(n-d-\alpha_{d})=\alpha_{d},1+\alpha_{d-1},\ldots,d+\alpha_{0}$ (or
earlier) and we can assume that $H$ must pass through $n-d-1+|\alpha|$ points
in general position.

Since $|\alpha|\leq d+1$ and $\alpha\in\mathbb{N}^{d+1}$, we see that if any
$\alpha_{i}>1$ then there are at least $d-|\alpha|+2$ trailing zeros in
$\alpha$. It follows that the first set of rank jumps occur at positions
$0,1,2,\ldots,d-|\alpha|+1$. Now $H\supset\Lambda\supset F_{d-|\alpha|+1}$,
which forces $H$ to pass through $d-|\alpha|+2$ points in general position in
$F_{d-|\alpha|+1}$. Together with the previous $n-d-1+|\alpha|$ points in $H$,
this forces $H$ to pass through $n+1$ points in general position in
$\mathbb{P}^{n}$. Of course this is not possible, so $a_{\alpha}=0$ if any
entry of $\alpha$ is greater than $1$.

On the other hand, if $\alpha=(1)^{d+1-\ell}$ then we multiply $[Z]$ by
$\sigma_{(n-d)^{\ell}(n-d-1)^{d+1-\ell}}h^{n-\ell}$ to get
$a_{(1)^{d+1-\ell}}\sigma_{(n-d)^{d+1}}h^{n}$, representing
$a_{(1)^{d+1-\ell}}$ elements $(\Lambda,H)\in Z$ such that $\Lambda$ has rank
jumps at positions in $0,1,\ldots,\ell-1,\ell+1,\ldots,d+1$ (or earlier) and
$H$ contains $n-\ell$ points in general position. But then
$H\supset\Lambda\supset F_{\ell-1}$ and together with the conditions imposed
by the $n-\ell$ points, $H$ is uniquely determined. Then $\Lambda=H\cap
F_{d+1}$ is also uniquely determined so $a_{\alpha}=a_{(1)^{d+1-\ell}}=1$ (in
the case $\ell=d+1$, i.e. $\alpha=(0)^{d+1}$, $\Lambda=F_{d}$ and $H$ is
uniquely determined by $H\supset\Lambda=F_{d}$ and the point conditions). It
follows that the class $[Z]\in A(\mathbb{G}(d,n)\times\mathbb{P}^{n*})$ is

$[Z]=\sum_{\ell=0}^{d+1}\sigma_{(1)^{d+1-\ell}}h^{\ell}.$

Now we return to our main result and determine the number of hyperplane
arrangements consisting of $k$ hyperplanes in $\mathbb{P}^{n}$ that all
contain a $d$-dimensional linear space and pass through $D$ points in general
position. Let

$Z_{i}=\\{(\Lambda,H_{1},\ldots,H_{k})\in\mathbb{G}(d,n)\times(\mathbb{P}^{n*})^{k}\;:\;\Lambda\subset
H_{i}\\}$

and write $h_{i}$ and $\sigma_{\alpha}$ for the pull-backs of the obvious
classes to $\mathbb{G}(d,k)\times(\mathbb{P}^{n*})^{k}$. Note that
$[Z_{i}]=\sum_{\ell=0}^{d+1}\sigma_{(1)^{d+1-\ell}}h_{i}^{\ell}\in
A(\mathbb{G}(d,n)\times(\mathbb{P}^{n*})^{k})$ and $Z_{i}$ is a local complete
intersection.

Now we show that $\bigcap_{i=1}^{k}Z_{i}$ is a local complete intersection.
The dimension of the ambient space is $\dim\mathbb{G}(d,n)+nk$ and the
dimension of $\bigcap_{i=1}^{k}Z_{i}$ is $\dim\mathbb{G}(d,n)+(n-(d+1))k$,
showing that $\bigcap_{i=1}^{k}Z_{i}$ has codimension $k(d+1)$. Since each
$Z_{i}$ has codimension $d+1$ and is a local complete intersection, their
intersection is also a local complete intersection. Again using Theorem 5.10
of [4], we have that on the level of classes

$\left[\bigcap_{i=1}^{k}Z_{i}\right]=[Z_{1}][Z_{2}]\cdots[Z_{k}].$

Let $P_{1},\ldots,P_{D}$ be points in general position in $\mathbb{P}^{n}$ and
for each $P_{i}$ let

$Y_{i}=\\{(\Lambda,H_{1},\ldots,H_{k})\in\mathbb{G}(d,n)\times(\mathbb{P}^{n*})^{k}\;:\;P_{i}\in
H_{1}\cup\cdots\cup H_{k}\\}.$

Note that $[Y_{i}]=h_{1}+\cdots+h_{k}.$ Again from Lemma 7 and using Kleiman’s
Transversality Theorem (see Kleiman’s fundamental paper [14]) we have that at
the level of classes

$\left[\left(\cap_{i=1}^{k}Z_{i}\right)\cap\left(\cap_{j=1}^{D}Y_{j}\right)\right]=\left[\left(\cap_{i=1}^{k}Z_{i}\right)\right]\left[\left(\cap_{j=1}^{D}Y_{j}\right)\right]=[Z_{1}][Z_{2}]\cdots[Z_{k}][Y_{1}][Y_{2}]\cdots[Y_{D}].$

Now

$\begin{array}[]{lll}\left[\left(\cap_{i=1}^{k}Z_{i}\right)\cap\left(\cap_{j=1}^{D}Y_{j}\right)\right]&=&[Z_{1}]\cdots[Z_{k}][Y_{1}]\cdots[Y_{D}]\\\
&=&\left[\prod_{i=1}^{k}\left(\sum_{\ell=0}^{d+1}\sigma_{(1)^{d+1-\ell}}h_{i}^{\ell}\right)\right]\left(h_{1}+\cdots+h_{k}\right)^{D}\\\
&=&\sum_{\Gamma}\sigma^{s}\binom{k}{s_{0},s_{1},\ldots,s_{d+1}}\binom{D}{(n-(d+1))^{s_{0}},(n-d)^{s_{1}},\ldots,(n)^{s_{d+1}}},\end{array}$

where
$\Gamma=\\{(s_{0},\ldots,s_{d+1})\in\mathbb{N}^{d+2}:\sum_{i=0}^{d+1}is_{i}=(d+1)(n-d)=\dim\mathbb{G}(d,n),\sum_{i=0}^{d+1}s_{i}=k\\}$
and

$\sigma^{s}=(\sigma_{0})^{s_{0}}(\sigma_{1})^{s_{1}}(\sigma_{(1)^{2}})^{s_{2}}\cdots(\sigma_{(1)^{d+1}})^{s_{d+1}}.$

Since each unordered arrangement of hyperplanes gives $k!$ arrangements of
labeled hyperplanes, we divide by $k!$ to obtain the correct degree
$N_{\mathcal{A}}$. Since the points $P_{i}$ are in general position, Kleiman’s
transversality theorem assures us that
$\left(\cap_{i=1}^{k}Z_{i}\right)\cap\left(\cap_{j=1}^{D}Y_{j}\right)$ is a
finite collection of points, each appearing with multiplicity one. So we can
interpret the above computation as saying that there are $N_{\mathcal{A}}$
$d$-coned generic arrangements of $k$ hyperplanes in $\mathbb{P}^{n}$ that
pass through $D$ points in general position. ∎

It would be interesting to study the enumerative geometry of hyperplane
arrangements in a combinatorial equivalence class different from the generic
arrangements and $d$-coned generic arrangements. In general this seems to
require some subtle intersection theory computations. In particular, it would
be interesting to see if intersection theory computations on the moduli space
of stable maps can deal with the enumerative geometry of more general families
of line arrangements.

######  Remark 24.

The result in Theorem 23 takes a particularly nice shape when $d=1$. In this
case
$\sigma^{s}=(\sigma_{0})^{s_{0}}(\sigma_{1})^{s_{1}}(\sigma_{11})^{s_{2}}$
counts the number of lines that are in $s_{2}$ general hyperplanes in
$\mathbb{P}^{n}$ and meet $s_{1}$ general codimension-$2$ planes. Cutting down
by the hyperplanes, this is the number of lines in $\mathbb{P}^{n-s_{2}}$ that
meet $s_{1}=2(n-1)-2s_{2}$ codimension-$2$ planes. This was one of the
classical problems studied by Schubert [22], who showed that
$\sigma^{s}=C_{n-s_{2}}$, where $C_{N}=\frac{1}{N}\binom{2N-2}{N-1}$ defines
the sequence of Catalan numbers (with the index shifted to start at 1).

######  Example 25.

The number of $1$-coned generic arrangements of $9$ hyperplanes in
$\mathbb{P}^{5}$ that go through $35$ points is

$\begin{array}[]{lll}N_{\mathcal{A}}&=&\frac{1}{9!}\left[\binom{9}{5,0,4}\binom{35}{(3)^{5}(4)^{0}(5)^{4}}+\binom{9}{4,2,3}\binom{35}{(3)^{4}(4)^{2}(5)^{3}}+2\binom{9}{3,4,2}\binom{35}{(3)^{3}(4)^{4}(5)^{2}}+\right.\\\
&&\phantom{\frac{1}{9!}[[}\left.5\binom{9}{2,6,1}\binom{35}{(3)^{2}(4)^{6}(5)^{1}}+14\binom{9}{1,8,0}\binom{35}{(3)^{1}(4)^{8}(5)^{0}}\right]\\\
&=&148,467,792,706,702,950,173,442,750.\end{array}$

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