# The Colorado Ultraviolet Transit Experiment ($CUTE$) Mission Overview

Kevin France Laboratory for Atmospheric and Space Physics, University of
Colorado Boulder, Boulder, CO 80303  Brian Fleming Laboratory for Atmospheric
and Space Physics, University of Colorado Boulder, Boulder, CO 80303  Arika
Egan Laboratory for Atmospheric and Space Physics, University of Colorado
Boulder, Boulder, CO 80303  Jean-Michel Desert Anton Pannekoek Institute for
Astronomy, University of Amsterdam, P.O. Box 94249, Noord Holland, NL-1090GE
Amsterdam, the Netherlands  Luca Fossati Space Research Institute, Austrian
Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria  Tommi T. Koskinen
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721–0092
Nicholas Nell Laboratory for Atmospheric and Space Physics, University of
Colorado Boulder, Boulder, CO 80303  Pascal Petit Institut de Recherche en
Astrophysique et Planétologie, Université de Toulouse, CNRS, CNES, 14 avenue
Edouard Belin, 31400 Toulouse, France  Aline A. Vidotto Leiden Observatory,
Leiden University, PO Box 9513, 2300 RA, Leiden, the Netherlands  Matthew
Beasley  Southwest Research Institute, Boulder, CO 80302  Nicholas DeCicco
Laboratory for Atmospheric and Space Physics, University of Colorado Boulder,
Boulder, CO 80303  Aickara Gopinathan Sreejith Laboratory for Atmospheric and
Space Physics, University of Colorado Boulder, Boulder, CO 80303  Space
Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042
Graz, Austria  Ambily Suresh  Laboratory for Atmospheric and Space Physics,
University of Colorado Boulder, Boulder, CO 80303  Jared Baumert  Laboratory
for Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO
80303  P. Wilson Cauley Laboratory for Atmospheric and Space Physics,
University of Colorado Boulder, Boulder, CO 80303  Carolina Villarreal
D’Angelo Instituto de Astronomía Teórica y Experimental, CONICET-UNC. Laprida
854, Còrdoba, Argentina  Keri Hoadley The University of Iowa, Dept. of Physics
& Astronomy, Van Allen Hall, Iowa City, IA 52242  Robert Kane  Laboratory for
Atmospheric and Space Physics, University of Colorado Boulder, Boulder, CO
80303  Blue Canyon Technologies 2550 Crescent Dr, Lafayette, CO 80026  Richard
Kohnert  Laboratory for Atmospheric and Space Physics, University of Colorado
Boulder, Boulder, CO 80303  Julian Lambert  Laboratory for Atmospheric and
Space Physics, University of Colorado Boulder, Boulder, CO 80303  Stefan
Ulrich  Laboratory for Atmospheric and Space Physics, University of Colorado
Boulder, Boulder, CO 80303 Kevin France<EMAIL_ADDRESS>
###### Abstract

Atmospheric escape is a fundamental process that affects the structure,
composition, and evolution of many planets. The signatures of escape are
detectable on close-in, gaseous exoplanets orbiting bright stars, owing to the
high levels of extreme-ultraviolet irradiation from their parent stars. The
Colorado Ultraviolet Transit Experiment ($CUTE$) is a CubeSat mission designed
to take advantage of the near-ultraviolet stellar brightness distribution to
conduct a survey of the extended atmospheres of nearby close-in planets. The
$CUTE$ payload is a magnifying NUV (2479 – 3306 Å) spectrograph fed by a
rectangular Cassegrain telescope (206mm $\times$ 84mm); the spectrogram is
recorded on a back-illuminated, UV-enhanced CCD. The science payload is
integrated into a 6U Blue Canyon Technology XB1 bus. $CUTE$ was launched into
a polar, low-Earth orbit on 27 September 2021 and has been conducting this
transit spectroscopy survey following an on-orbit commissioning period. This
paper presents the mission motivation, development path, and demonstrates the
potential for small satellites to conduct this type of science by presenting
initial on-orbit science observations. The primary science mission is being
conducted in 2022 – 2023, with a publicly available data archive coming on
line in 2023.

##  1 INTRODUCTION

The history of observational astronomy has been marked by the push to ever
larger and more capable telescopes and instruments. The 2010s witnessed the
development of a new generation of large astronomical observatories. Both on
the ground and in space, facilities 2 – 3 times the primary mirror diameter of
the previous state-of-the-art were brought closer to fruition for
implementation in the 2020s, 2030s, and 2040s, including the James Webb Space
Telescope (Gardner et al., 2006; Rigby et al., 2022), thirty-meter class
ground-based telescopes (e.g., Simard et al. 2016), and advanced
ultraviolet/optical (UV/O) facilities such as the Large Ultraviolet/ Optical/
Infrared Surveyor (LUVOIR; LUVOIR Final Report 2019). The large mission
studies conducted ahead of the 2020 Decadal Survey on Astronomy and
Astrophysics drove the recommendation for NASA’s suite of Future Great
Observatories, a series of probe- and flagship-class missions offering many
order-of-magnitude gains in the scientific grasp across numerous areas of
astrophysics. In parallel with this large observatory development, numerous
small telescope arrays have come on-line or have been expanded, and NASA’s
science divisions made significant new investments in small satellites
covering a range of scientific topics.

Small telescopes at ground-based sites have excelled at detecting and
characterizing new objects in the time-variable sky, including supernovae
eruptions (Dong et al., 2016) and tidal disruption events (Hammerstein et al.,
2022) from the Zwicky Transient Facility (Masci et al., 2019) and All Sky
Automated Survey for SuperNovae (ASAS-SN) (Holoien et al., 2017). The impact
of small telescopes has also been powerful for the detection of extrasolar
planets, including many Jovian-sized planets from Wide-Angle Search for
Planets (WASP) (Pollacco et al., 2006) and the Kilodegree Extremely Little
Telescope (KELT) (Pepper et al., 2007), and some of the most promising rocky
planets for study with $JWST$ from the MEarth (Charbonneau et al., 2009),
Transiting Planets and Planetesimals Small Telescope (TRAPPIST) (Gillon et
al., 2011), and Search for habitable Planets EClipsing ULtra-cOOl Stars
(SPECULOOS) (Burdanov et al., 2018) facilities.

The recent decadal survey highlighted the power of small space-based
telescopes, astronomical CubeSats and smallsats, for “monitoring of sources
for weeks or months at time, and at wavelengths not accessible from the
ground”, complementing the Hubble Space Telescope’s surveys in areas of
transmission spectroscopy (Sing et al., 2019; Cubillos et al., 2020) and
exoplanet host star radiation fields (France et al., 2016b; Loyd et al., 2018;
Ramiaramanantsoa et al., 2021). NASA has embraced this opportunity with a
dedicated funding line for astrophysics CubeSats (full mission life cycle cost
$<$ $10M) and the Pioneers program (mission cost $10M – $20M). In this paper,
we present an overview of NASA’s first UV astronomy CubeSat and the first
grant-funded small satellite dedicated to the characterization of extrasolar
planetary atmospheres, the Colorado Ultraviolet Transit Experiment (CUTE).

$CUTE$ conducts transit spectroscopy of short-period, giant planets in the
near-UV (2479 – 3306 Å) bandpass to access strong atomic transitions tracing
atmospheric escape and the near-UV spectral slope of giant planet atmospheres
that provide constraints on their composition. This paper presents the science
background for and the technical implementation of the mission. The manuscript
is laid out as follows: the scientific motivation for $CUTE$ and its science
objectives are presented in Section 2. Because $CUTE$ is one of the first
astronomy missions to be developed in a CubeSat framework, we present a
description of the mission development and implementation path in Section 3.
Section 4 presents the instrument design and high-level performance
specifications (see also Fleming et al. 2018 for a description of $CUTE$’s
science payload). Section 5 describes $CUTE$’s mission operations and we
present early-release examples of the mission’s on-orbit science data in
Section 6. We conclude with a brief summary in Section 7. A detailed
description of $CUTE$’s science instrument and on-orbit performance is
presented in a companion paper by A. Egan. Mission operations and on-orbit
commissioning (Suresh et al. – in prep), $CUTE$’s on-orbit data pipeline
(Sreejith et al. – in press), and Early Release Science results (Egan et al. –
in prep; Sreejith et al. – in prep) will be described in forthcoming papers.

Figure 1: The $CUTE$ instrument development from concept (instrument
schematic, top) to telescope characterization ($CUTE$ flight telescope in the
test facilities at the University of Colorado, middle), to pre-delivery in-
band spectral resolution test data (bottom).

##  2 CUTE SCIENCE OBJECTIVES

Planetary escape processes play a key role in determining the chemical and
physical state of planets both within and beyond our solar system. Atmospheric
escape is thought to create the radius gap observed in the distribution of
short-period exoplanets (Fulton & Petigura, 2018), likely driven by a
combination of photoevaporative (Owen & Wu, 2017) and core-powered (Ginzburg
et al., 2018) mass loss. Escape is also a fundamental process in the evolution
of terrestrial worlds. For a planet to be habitable, our current view is that
it must lose its primordial hydrogen atmosphere and acquire/generate (and
retain) a secondary atmosphere (Lammer et al., 2018). Atmospheric escape is
known to have shaped the early atmospheres of Venus, Earth, and Mars, which
subsequently followed different evolutionary paths. The rapid hydrodynamic
escape that is believed to have affected Venus, Earth and Mars in the past no
longer takes place on any planet in the solar system. Therefore, we turn to
short-period extrasolar planets as laboratories on which to study vigorous
atmospheric loss.

The first detection of exoplanet atmospheric escape was achieved by Vidal-
Madjar et al. (2003) who used HI Ly$\alpha$ transit observations in the far-
ultraviolet (FUV) to observe the extended atmosphere of the Hot Jupiter
HD209458b. This was followed by the detection of O I, C II, Si III and Mg I on
the same planet (Vidal-Madjar et al., 2004, 2013; Linsky et al., 2010). These
initial observations inspired several independent groups to develop 1D and 3D
models to study both the physical characteristics of the upper atmospheres of
close-in planets and the escaping gas and plasma surrounding them (e.g.,
Koskinen et al. 2007; Murray-Clay et al. 2009; Koskinen et al. 2013a, b;
Bourrier & Lecavelier des Etangs 2013; Bourrier et al. 2016; Villarreal
D’Angelo et al. 2018; Carolan et al. 2021).

The interpretation of FUV transit measurements has often been controversial
(see Fossati et al. 2015 for a discussion). Recently, several atmospheric
escape studies have shifted to the near-ultraviolet (NUV), where the stellar
flux is much higher than in the FUV and the light curves are measured against
a better-understood intensity distribution from the stellar photosphere
(e.g.,Haswell et al. 2012; Llama & Shkolnik 2015). The NUV includes the Fe II
complexes near 2400 and 2600 Å, the Mg II doublet at 2796/2803 Å, the Mg I
line at 2852 Å, some of which have been detected on the Hot Jupiters WASP-12b,
HD209458b, and WASP-121b (Fossati et al., 2010; Sing et al., 2019; Cubillos et
al., 2020). We note that the Fe II and Mg II resonance lines in the near-UV
trace the highly extended (and potentially escaping) exoplanet atmosphere,
whereas optical band metal line detections made with ground-based telescopes
trace the lower, bound atmospheric layers (Hoeijmakers et al., 2019;
Casasayas-Barris et al., 2019; Cauley et al., 2019; Turner et al., 2020;
Hoeijmakers et al., 2020; Casasayas-Barris et al., 2021; Deibert et al.,
2021). The NUV also contains a pseudo-continuum that can probe scattering by
high altitude clouds and gas phase silicon and magnesium (Lothringer et al.,
2022), as well as the $A$ – $X$ bands of OH (3100 Å). Furthermore, NUV
transmission spectra give the unique opportunity to constrain the composition
of the aerosols lying in the lower atmospheres (Cubillos et al., 2020).

Depending on the temperature profile in the atmosphere, species like Si, Mg,
and Fe are expected to condense to form clouds in the lower atmosphere,
however, the calculations indicate that strong mixing, either by turbulence or
global circulation, can inhibit cloud formation or allow for these species to
be present in the upper atmosphere where they can escape (Koskinen et al.,
2013b; Cubillos et al., 2020; Koskinen et al., 2022). The comparison of
continuum and atomic line absorption therefore acts as a diagnostic of cloud
formation, elemental abundances and mass loss on close-in exoplanets
(Lothringer et al., 2020; Cubillos et al., 2020). Model outputs can be used to
translate observed planetary transit light curves into global mass-loss rates:
the depth and shape of the light curves directly relate to the atmospheric
parameters.

Finally, UV transits with $HST$ have provided evidence for time-variability,
potentially arising from changing stellar high-energy input, orbital timescale
changes in the planet’s atmosphere, or variation in the star-planet magnetic
environment. Lecavelier des Etangs et al. (2012) observed time-variable
neutral hydrogen absorption in FUV transit observations of HD 189733b,
possible due to the influence of high-energy stellar flares. NUV transit
observations of the close-in giant planet WASP-12b by Fossati et al. (2010)
found that the transit light curve of WASP-12b presents both an early ingress
when compared to its optical transit and excess absorption during the transit
(see also Haswell et al. 2012; Nichols et al. 2015). Possible explanations
include atmospheric hydrodynamic mass-loss supporting a shock upstream of the
planet’s orbit or generating an accretion stream that produces an early
ingress (Lai et al., 2010; Bisikalo et al., 2013; Turner et al., 2016) and a
magnetically supported bow-shock 4 – 5 planetary radii upstream of the
planet’s orbital motion (analogous to the Earth-Sun system; Vidotto et al.
2010; Llama et al. 2011).

$CUTE$’s primary science goal is to provide new constraints on the physics and
chemistry of hot, Jovian-size exoplanets. The $CUTE$ mission addresses this
goal with the following observing program:

  1. 1.

Measure NUV transmission spectra for a small survey of approximately 10 short
period planets

  2. 2.

Infer atmospheric escape rates and constrain the composition of the upper
atmospheres of hot giant planets

  3. 3.

Measure temporal variability in UV transit light curves by observing 6 – 10
transit observations per planet

  4. 4.

Measure out-of-transit baseline fluxes to better characterize the stellar
inputs to the planet’s atmosphere and to capture light curve asymmetries

$CUTE$’s instrument design and mission implementation was developed to enable
the four key goals of the observing program. The spectral coverage and
resolution of the $CUTE$ ($\Delta$ $v$ $<$ 300 km s-1) spectrograph provides
ample separation of the relevant atomic, molecular, and continuum bands in
this range (see, e.g., Figure 8 of Sing et al. 2019). $CUTE$’s mission design
complements the instrument to meet the science goals of the mission. (1) We
couple observations of the NUV continuum opacity, individual ionic tracers (Fe
II, Mg II) with atmospheric chemistry, and hydrodynamic escape models to
determine mass loss rates for $CUTE$’s targets. The sample size is driven by a
combination of mission lifetime and instrumental sensitivity considerations.
(2) $CUTE$ measures the amplitude and slope of the NUV transmission curve to
provide constraints on the chemistry and structure of the escaping atmosphere.
The instrumental effective area was specified to enable multiple, wavelength
resolved, NUV bands with sufficient photometric precision to distinguish the
NUV transit radius from the white-light radius of the planet on all 10 targets
and isolate transit spectra of the strongest absorption lines (e.g., Fe II and
Mg II) on the brightest targets (addressing goals 1 and 2). The target sample
was defined by estimating the detectability of excess NUV absorption; a
combination of stellar brightness (V-magnitude), spectral type (A- and F-type
stars have spectral energy distributions peaked in the NUV), planetary radius,
effective planetary surface temperature, and gravity (hotter, lower-mass
planets being more likely to exhibit extended atmospheres). (3) $CUTE$’s
point-stare-repeat concept of operations is designed to make numerous visits
to the same planet over the course of 4 to 8 weeks, building signal-to-noise
for fainter targets and enabling measurements of light curve variability for
brighter targets. (4) The same point-stare-repeat observing mode provides a
wide stellar baseline to measure changes in the Mg II activity and the
increased dispersion of the photospheric and chromospheric continuum flux that
indicate variability in the star’s escape-driving XUV output.

##  3 Mission Implementation Path

$CUTE$ is NASA’s first grant-funded UV/ Optical/ Infrared small satellite and
first dedicated exoplanet spectroscopy mission. Given the novelty of this
mission format for astrophysics science missions, we present a brief overview
of the process, schedule, and cost of the mission here. The initial motivation
for $CUTE$ was discussed at a Keck/KISS workshop on exoplanet magnetic fields
in August 2013, with the final science and measurement concept in place by the
summer of 2015 following numerous informal discussions at science conferences.
Fall 2015 was spent on science measurement definition and the development of
the $CUTE$ instrument design.

$CUTE$ was proposed as a four-year program through NASA’s ROSES2015 call
(submitted in March 2016), at an initial cost-to-launch of $3.3M, comparable
to an astrophysics sounding rocket proposal but considerably lower cost than a
stratospheric balloon program. $CUTE$ was proposed and selected prior to the
initiation of dedicated funding for astrophysics CubeSats, leading to a long
delay between proposal submission and the start of funding (approximately 16
months; there was no Phase A or Concept Study period for $CUTE$). Long lead-
time items such as the $CUTE$ spacecraft bus, the rectangular telescope,
holographically-ruled diffraction grating, and NUV-optimized CCD detectors
were ordered a few months after selection in fall 2017. The $CUTE$ spacecraft
(Blue Canyon Technology, BCT) costs increased relative to the quote provided
for the proposal. To accommodate the cost increases in a mission class without
reserves, several descopes were implemented, including scaling back the
spacecraft’s attitude control system to a single star-tracker and eliminating
engineering model radios. In 2018 and 2019, we developed the hardware test
facilities that complemented the University of Colorado’s existing UV vacuum
calibration facilities (France et al., 2016a) and conducted component-level
characterization (e.g., groove efficiency of the diffraction gratings, trade
study of Al vs. Al+MgF2 grating coatings). Instrument assembly and
characterization, integration into the spacecraft, and pre-delivery
environmental testing (e.g., vibration testing, comprehensive performance
testing, thermal vacuum testing, etc.) were completed in 2020 and 2021. The
duration from the start of $CUTE$ funding to delivery of the completed
observatory was almost exactly four years, although approximately 10 months of
schedule were lost to the COVID-19 pandemic.

$CUTE$ proposed to NASA’s CubeSat Launch Initiative (CSLI) for launch support
in fall 2017 and was selected for flight. The proposed spacecraft orbit,
including the initial $CUTE$ mission requirement documentation submitted to
CSLI, requested a dawn-dusk (terminator), sun-synchronous orbit to enable
uninterrupted orbital phase coverage of transiting planets and to minimize
day/night thermal variations. Orbital altitude (450 - 600km) was a secondary
consideration driven by desired mission lifetime. CSLI was unable to
accommodate the requested sun-synchronous orbit within the time-period covered
by the mission funding and $CUTE$ was instead manifested in November 2019 as a
secondary payload on NASA’s Landsat 9 mission. The Landsat 9 launch was
scheduled 8 months after $CUTE$’s targeted launch window. As a result, NASA
provided a 6 month, $0.5M extension to the $CUTE$ program in Fall 2019.
Starting in Spring 2020, COVID delayed the Landsat 9 launch by another 9
months. COVID also impeded $CUTE$’s development timescale owing to supply-
chain delays and the challenges of getting students, scientists, and engineers
into $CUTE$’s labs for continued testing and development. The $CUTE$ mission
submitted a follow-on, competed APRA proposal in December 2020 to conduct
mission operations and carry out the science program (bringing $CUTE$’s cost
to complete the full science mission to approximately $5M). Of the 18 CubeSat
missions originally manifested with Landsat 9, only four (including $CUTE$)
would ultimately deliver and be flown on the mission.

The integrated and tested observatory was delivered to NASA at Vandenberg
Space Force Base (VSFB) in July 2021, and the $CUTE$ team supported
installation into the CubeSat dispenser on the ESPA ring. The mission was
launched on September 27 2021 into a sun-synchronous orbit ($\approx$98°, 10am
Local Time of Ascending Node; LTAN) with a 560km apogee. $CUTE$ deployed from
the dispenser approximately two hours after launch; solar arrays deployed and
the communication beacon started approximately 30 minutes later. $CUTE$’s
beacons were identified by the amateur RF community on the first orbit and
communications were established with the ground station at the University of
Colorado on September 28 2021. We refer the reader to Section 4 and Suresh et
al. (in preparation) for a description of the $CUTE$ ground segment and
commissioning program. We refer the reader to the $SmallSat$ Conference
proceeding by Egan et al. 2022 for a discussion of lessons learned during
$CUTE$ development and early on-orbit operations.

$CUTE$ is part of NASA’s suborbital program, where student training and early-
career mentorship are key ingredients to the definition of mission success.
$CUTE$’s approach was built off of the framework of the NASA Sounding Rocket
Program, which has a long history in the professional development of NASA’s
space scientist workforce. The core science and instrument team (defined as
those working with $CUTE$ for more than 2 years of the implementation phase)
included two Ph.D. candidates (in astrophysics and aerospace engineering),
four undergraduates, two postdoctoral researchers, two early career engineers
($CUTE$ was the first job post-bachelors degree for the mission’s lead
mechanical and electrical engineers), and the early-career project scientist
(Dr. Brian Fleming, who became the PI of a NASA sounding rocket program and
the SPRITE CubeSat (Fleming, 2022) mission during the course of the $CUTE$
development phase). Over the course of $CUTE$’s component-level and instrument
test phase, the project employed another six undergraduate students in various
laboratory and science program development tasks (e.g., target field checking
for crowded fields). In addition to this, the operations team for $CUTE$ (see
Section 5) included an additional two undergraduate students, two graduate
students, and one flight software undergraduate student. Taken as a whole,
$CUTE$ supported the mentoring and training for over 20 early-career
scientists and engineers through the completion of the on-orbit commissioning
phase.

Figure 2: $CUTE$ calibration observations of the O4 supergiant $\zeta$ Puppis.
The top plot shows the full-frame (2048 $\times$ 515 pixel) calibration image
and the bottom plot shows the extracted one-dimensional spectrum (flux
calibrated against archival $HST$ and $IUE$ spectra). $\zeta$ Puppis was
selected as a calibration target because of the wealth of archival NUV spectra
and the high photospheric temperature ensures that iron and magnesium lines in
the spectrum are narrow, interstellar features.

##  4 Implementation: The $CUTE$ Science Payload

The $CUTE$ payload is a magnifying NUV spectrograph fed by a rectangular
Cassegrain telescope. The spectrogram is recorded on a back-illuminated, UV-
enhanced e2v CCD42-10 that is maintained at a nominal operating temperature
($-$15 – $-$5° C) by passive cooling through a radiator panel. $CUTE$ employs
the BCT XB1 bus to provide critical subsystems including power, command and
data handling, communications, and attitude control (ADCS). Figure 1 shows an
instrument schematic, optical testing of the telescope, and an in-band
calibration spectrum from the flight instrument prior to instrument
integration into the BCT chassis.

The $CUTE$ instrument is housed in 4U of the 6U spacecraft. The $CUTE$
aperture is a 206 $\times$ 84 mm, f/0.75 (in the cross-dispersion axis)
primary mirror that is part of the f/2.6 Cassegrain telescope. The rectangular
shape of the primary is matched to the long axis of the 6U CubeSat chassis and
allows for 3 times more throughput than a 1U circular aperture (Fleming et
al., 2018). The hyperbolic secondary mirror is cantilevered off of the primary
mirror, which serves as the bench for the optical instrument, by means of an
Invar tower (see Figure 1). A 15 $\times$ 6 mm fold mirror redirects the beam
90° through a 141 $\mu$m $\times$ 3.5 mm (60′′ $\times$ 1400′′ projected) slit
at the Cassegrain focus. The slit, manufactured by OSH Stencils, was polished
on the incident side and angled 45° about the slit axis to redirect the field
to an aspect camera for use in telescope performance testing and alignment
with the BCT spacecraft during integration. The rectangular telescope design
optimizes collecting area within the mass-volume constraints of the cubesat
form factor, while the large sky field-of-view, increased cost, and mechanical
stress at the primary mirror-secondary tower interface add design
complications.

Once through the slit, the starlight is diffracted, redirected, and magnified
by a spherical, R = 86.1mm radius, 1714 gr mm-1 aberration correcting, ion-
etched holographic grating fabricated by Horiba Jobin-Yvon (Horiba J-Y). The
holographic grating design was adopted to minimize scattered light in the
system. A second fold mirror with an Rx = 300 mm radius of curvature about the
cross-dispersion dimension provides additional aberration corrections before
the beam reaches the CCD. The final beam focal ratio is f/5.5 in the cross-
dispersion axis, with a detector plate scale of 186′′ mm-1. The detector and
custom avionics were tested and flight ruggedized by the $CUTE$ team in their
on-campus laboratories at the University of Colorado (Nell et al., 2021).

The telescope was delivered fully assembled to CU by Nu-Tek Precision Optical.
All mirrors are coated with MgF2 \+ Al to prevent the formation of an oxide
layer (AlO3). We elected to receive flight and flight-backup gratings coated
in bare Al and MgF2 \+ Al, respectively (coated by Horiba J-Y), to control for
a potential efficiency anomaly similar to that seen on the COS NUV gratings
(Wilkinson, 2002). Detailed pre-flight efficiency and environmental testing
showed better performance with the bare Al grating, without a measurable loss
in efficiency over time. As a result, the instrument team elected to fly the
bare Al-coated mirror on the flight instrument. The design-prediction flight
instrument performance curves are presented in Fleming et al. (2018) and the
on-orbit instrument performance of the $CUTE$ payload is presented in Egan et
al. (2022 – this volume); we provide a brief summary of the key performance
metrics in the following subsection.

Table 1: $CUTE$ Instrument Specifications

Instrument Metric | On-orbit Value
---|---
Bandpass | 2479 – 3306 Å
Spectral Resolutiona | 3.9 Å
Cross-Dispersion Resolutionb |  $\approx$ 30′′
Peak Aeff | 27.5 cm2 at 2500 Å
Background Flux Limit | 5 $\times$ 10-14 erg cm-2 s-1 Å-1
in 300sb |

aafootnotetext: Average resolution over the bandpass, including spacecraft
jitter.

bbfootnotetext: Evaluated at 3000 Å.

Figure 3: Schematic description of $CUTE$ science and calibration
observations.

###  4.1 Instrument Specifications

The final bandpass recorded by the CCD detector is 2479 – 3306 Å (see Table
1), which is a slight change from the pre-flight projection owing to shifts in
the optical system during ascent. The exact bandpass also varies by several Å
depending on the alignment of the stellar point spread function (PSF) in the
spectrograph slit. The spectral resolving power of the instrument is $\approx$
750 ($\Delta\lambda$ $\approx$ 3.3 – 4.5 Å across the bandpass, including the
effects of spacecraft pointing jitter). Figure 2 shows a representative
calibration spectrum from the $CUTE$ on-orbit commissioning program.

The system effective area is a function of the reflection efficiency of the
optics ($R$), efficiency of the grating ($\epsilon_{g}$), and quantum
efficiency of the detector (DQE), multiplied by the geometric collecting area
of the telescope (Ageo):

$A_{eff}(\lambda)=A_{geo}R^{5}(\lambda)\epsilon_{g}(\lambda)DQE(\lambda).$ (1)

The on-orbit effective area was measured by comparing $CUTE$’s observations
(in units of electrons s-1 Å-1) with flux-calibrated observations from the
$IUE$ and $HST$ archives. We measured $A_{eff}$ = 27.5 – 19.0 cm2 across the
$CUTE$ spectral range with a peak at approximately 2500 Å. Component-level
efficiencies were measured prior to instrument assembly in the UV calibration
facilities at the University of Colorado (France et al., 2016a; Egan et al.,
2020). The component-based, pre-flight $A_{eff}$ estimate was about 12% higher
than the median effective area subsequently measured on-orbit (Egan et al. –
this volume). We attribute the loss of sensitivity to two possible causes:
particulate contamination during the failure of $CUTE$’s thermal-electric
cooling system (which occurred during thermal vacuum testing) and
contamination during the $\sim$2 months that $CUTE$ sat in the CubeSat
dispenser at VSFB prior to launch. A dry nitrogen purge was requested in order
to minimize optical degradation following dispenser integration, but was not
made available. The difference in effective area does not have a significant
impact on target selection and detectability, however, the larger and variable
thermal environment resulting from the loss of the active cooling system
removes most of the stars fainter than the nominal target list.

Combining the $CUTE$ effective area with the on-orbit instrumental background
level and the nominal 300 second exposure time for $CUTE$’s exoplanet surveys,
we calculate the typical dispersion in the residual flux following background
subtraction. This sets the minimum flux level that can be detected above the
noise in a 300 second spectrum, which we refer to as the background flux
limit. We measure a background flux limit of $\approx$ 5 $\times$ 10-14 erg
cm-2 s-1 Å-1 at 3000 Å.

##  5 $CUTE$ Mission Operations

The $CUTE$ spacecraft includes a UHF (437.25 MHz) antenna with both
transmission and receiving capabilities. The UHF link is used for uploading
commands to the spacecraft and monitoring real-time telemetry during ground
passes. $CUTE$ also has an S-band (2402 MHz) downlink-only mode for primary
science data transmission. The mission operations and ground station for
$CUTE$ are located at the Laboratory for Atmospheric and Space Physics in
Boulder, Colorado. $CUTE$ typically has 1 – 2 high-elevation ($>$ 50°) passes
and 1 – 2 low-elevation passes per day over the Boulder ground station,
resulting in approximately 10 minutes per day of optimal downlink time. Figure
3 presents an illustration of the $CUTE$ science operations observing mode,
including science data acquisition, approximately monthly calibration
activities, and data downlinks over the Boulder ground station.

The CubeSat Operations Center at LASP utilizes the LASP ground station
initially built for CSSWE and MinXSS, and the recently completed CSIM CubeSat
mission for NASA’s heliophysics division (Mason et al., 2016), using a
combination of HYDRA and OASIS-CC (Flynn et al., 2021) for command and
control. The mission operations are conducted by a team of professionals with
experience from larger NASA flight missions (e.g., $Kepler$, $IXPE$, and
several heliophysics missions) and a dedicated student operations group. The
undergraduate and graduate student operators perform mission planning,
operations, and health and status monitoring for the spacecraft; the science
team has developed a graphical user interface tool to determine optimal target
visibility and viewing conditions. The output of the science planning tool is
processed into $CUTE$’s weekly operations plan to define the observing,
charging, and communication activities for the week.

To maximize operational simplicity, $CUTE$ conducts single-target campaigns:
we schedule multiple transit observations in a single command block (typically
lasting 3 - 7 days) that is uploaded to the spacecraft. Each command block
includes calibration exposures, science exposures, and data downlink periods.
We repeat this exercise until 6 – 10 transits of a given planet have been
executed. Interleaved with the transit observing blocks are dedicated downlink
periods, typically between 3 and 5 days, to re-transmit science and
calibration data that experienced low data-completion fractions in the initial
downlink or were lost to spacecraft resets. The need to execute 1 – 3
additional data downlinks per transit campaign, driven by the frequent loss of
fine-pointing control and resets on the spacecraft bus (1 – 2 events per
week), is the limiting factor to $CUTE$’s operational efficiency. Pre-flight
observation planning predicted 3 transits per week could be successfully
executed and downlinked, which would complete 10 transits per target of the 10
target sample in approximately 8 months. The realized mission efficiency is
projected to complete an average of 6 transits per target over a science
mission lifetime of $\sim$ 15 months, or approximately a factor of 3 reduction
in efficiency compared to pre-flight estimates. A summary of the $CUTE$ on-
orbit operations and payload commissioning will be presented in a forthcoming
paper (A. Suresh et al. - in prep.).

Figure 4: $CUTE$ spectral observations of WASP-189 and KELT-20, A-type host
stars of $CUTE$’s first Early Release Science targets. The top plots shows a
”TRIM2D” 2048 $\times$ 100 pixel two-dimensional data product (exposure time =
5 minutes). The bottom plots show the one-dimensional spectral collapse
including wavelength and flux calibration. A 10 pixel boxcar smooth has been
applied to the one-dimensional spectra for display purposes.

##  6 $CUTE$ Science Data Example

In this section we present representative samples of $CUTE$’s individual
science data products and a preliminary reduced transit light curve from the
Early Release Science program. Detailed analyses of the wavelength dependent
transit depths, interpretation and quantification of atmospheric composition
and escape rates, and inter-comparison between different transit visits for
the Early Release Data program will be presented in upcoming works (Egan et
al. and Sreejith et al. 2022 – in prep). The goal here is to present flight
data of exoplanets and their host stars to illustrate the features (and
limitations) of $CUTE$ observations as revealed by the first two targets of
the Early Release Science program.

Figure 4 ($top$) presents a standard spectral data product that $CUTE$
transmits over the S-band downlink. These are “TRIM2D” data products, 2048
$\times$ 100 pixel two-dimensional spectra with a 5 minute exposure time. The
images are trimmed to reduce downlink volume. The 5 minute exposure time is
typical of all $CUTE$ spectra and is a balance of signal-to-noise for our
target brightness, number of exposures possible per orbital night, and
simplicity of operational planning. Each transit visit is buffered by a number
of bias and dark exposures taken at similar celestial pointing, orbital
position (latitude and longitude, and therefore similar temperature and
illumination conditions), and elevation angle of the telescope with respect to
the Earth limb. These calibration files are used to remove thermal and readout
noise effects, as described in Egan et al. (2022).

Data processing beyond the downlink of the TRIM2D data products occurs on the
ground. The two-dimensional data are collapsed along a diagonal extraction
region; the spectra are wavelength and flux calibrated using observations from
the on-orbit commissioning phase. Figure 4 ($bottom$) shows calibrated one-
dimensional spectra of WASP-189 and KELT-20, taken outside of transit. The
spectra are typical of NUV observations from A-type stars in the $IUE$
archive, with the most prominent feature being Mg II absorption in the
photosphere of these intermediate temperature stars. The reader will also
notice the defocus seen in the cross-dispersion direction, manifest as the
double-lobe structure that increases to shorter wavelengths across the band.
This defocus was introduced during an additional payload vibration test that
was not part of the original test specifications but later required by NASA
just prior to the delivery of the spacecraft. This defocus was then
exacerbated during the powered ascent; there is no detectable “breathing” of
the focus with orbital location although the background levels are strongly
driven by the spacecraft thermal and illumination conditions.

$CUTE$’s two-dimensional spectra are calibrated with master bias frames before
cosmic ray correction using LAcosmic algorithm (van Dokkum, 2001). We extract
the one-dimensional spectra from this corrected image as described in Sreejith
et al. (2022 – in prep). The one-dimensional spectra are obtained by
subtracting the background, which are then wavelength calibrated. The spectra
are integrated over the wavelength region of $\approx$2540Å to $\approx$3300Å
to create a light curve point. These light curves can be created down to a
wavelength resolution limit of $\sim$ 4 Å per bin, but for initial
demonstration purposes we display a broad NUV bandpass.

Figure 5: Initial $CUTE$ light curves of WASP-189b, showing three independent
NUV (approximately 2540 – 3300 Å) light curves (black points) and the best-fit
transit models in gray. The plots compare the NUV band light curves with the
optical light curve (in red) from $CHEOPS$ (Lendl et al., 2020). The NUV
transits are significantly deeper than their broadband optical counterparts,
indicating an effective planetary radius increase of RP,NUV $\approx$ 1.5
RP,opt.

Figure 5 presents $CUTE$’s approximately 2540 – 3300 Å light curves for 3
different visits of the ultra-hot Jupiter WASP-189b. The best fit transit
model, taking into account wavelength-dependent stellar limb darkening (shown
in gray), will be presented in detail in Sreejith et al. (2022 – in prep). The
optical transit light curve from $CHEOPS$ (Lendl et al., 2020) is shown for
comparison and suggests excess transit absorption at UV wavelength compared
with the broadband geometric size of the planet. We demonstrate self-
consistent transit depth recoveries of $\approx$ 1.0 – 1.1 % over three
separate transit observations of WASP-189b separated by several weeks.

Excess planetary absorption at NUV wavelengths is consistent with previous
observations of ultra-hot Jupiters observed with $HST$ (Sing et al., 2019;
Cubillos et al., 2020; Lothringer et al., 2022). A transit depth of 1% in
WASP-189b would indicate that the NUV transit observations are probing the
extended upper atmosphere of the planet that is subject to stellar high energy
radiation and escape. This is because the 1 microbar level, used here as a
rough proxy for the base of the thermosphere, has a radius of about 1.1 Rp and
a transit depth of 0.6%, based on an effective temperature of 2410 K and an
atmosphere with solar abundances. In contrast, a transit depth of 1%
corresponds to a larger radius of 1.4 Rp. If we assume a temperature of 8000 K
in the thermosphere, the pressure at 1.4 Rp would be between 0.1 and 1 nbar.
Given that this pressure is too low for significant clouds and hazes, a
pseudo-continuum by these absorbers is unlikely and the broadband NUV transit
depth likely arises from a forest of metal ion lines (e.g., Fossati et al.
2010; Sing et al. 2019). The individual absorption lines responsible would
have to extend to much higher radii than 1.4 Rp in transit to be detectable in
the broad NUV band.

We note that the preliminary lightcurves show significant scatter beyond the
photon noise limit at this stage of the reduction. Work is ongoing to model
the temperature- and orbital position-dependent background to reduce the
observed dispersion in the lightcurves.

##  7 Conclusions

The $CUTE$ CubeSat mission was launched in September 2021 and is currently
carrying out its primary science mission to collect NUV spectroscopy of
transiting planets. $CUTE$ has successfully completed spacecraft and
instrument commissioning and has completed initial science observations on a
number of exoplanetary systems. Optimal science targets are short-period,
Jovian-sized planets orbiting bright ($V$ $<$ 8) F and A stars. The science
instrument has demonstrated sensitivity to NUV white light transit depth of
$<$ 1% and wavelength-dependent exoplanet atmosphere opacity increase at
$\lambda$ $\lesssim$ 3300 Å.

We have presented the motivation for, mission design of, and on-orbit
characteristics of the $CUTE$ mission. The companion paper by Egan et al.
present the details of the on-orbit instrument performance of $CUTE$ and
future papers will present mission science results as well as information
about the $CUTE$ ground-segment data pipeline, commissioning, and operations
of the mission. $CUTE$ has supported mentoring and training for over 20 early-
career scientists and engineers. Mission operations are planned to continue
until June 2023 (currently limited by project funding) and the initial release
of $CUTE$ data will be delivered to the NexSci archive in 2023.

Acknowledgments: $CUTE$ was developed and operated with the support to two
NASA/APRA awards to the Laboratory for Atmospheric and Space Physics at the
University of Colorado Boulder, NNX17AI84G and 80NSSC21K1667. A. G. S. was
supported by a Schrödinger Fellowship through the Austrian Science Fund (FWF)
[J 4596-N]. A. G. S. and L.F. acknowledge financial support from the Austrian
Forschungsförderungsgesellschaft FFG project 859718 and 865968. AAV
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 817540, ASTROFLOW). K.F. acknowledges the numerous and invaluable
discussions with colleagues excited about ultraviolet transit science and the
potential to do science with small satellites. The $CUTE$ team wishes to
specifically recognize the amateur radio operator community and, and SatNOGS
network specifically, for hosting numerous telemetry tracking tools that have
improved the mission’s ability to recover from faults and understand long-term
spacecraft trends much more efficiently than would have been otherwise
possible.

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# Multiplier ideals of normal surface singularities

László Koltai  HUN-REN Alfréd Rényi Institute of Mathematics, Reáltanoda utca
13-15, H-1053, Budapest, Hungary
ELTE - Faculty of Science, Dept. of Geometry, Pázmány Péter sétány 1/A, 1117
Budapest, Hungary<EMAIL_ADDRESS>,  Tamás László  Babeş-Bolyai
University, Str. Mihail Kogălniceanu nr. 1, 400084 Cluj-Napoca, Romania
Simion Stoilow Institute of Mathematics of the Romanian Academy, Bucharest,
Romania<EMAIL_ADDRESS>and  András Némethi  HUN-REN Alfréd Rényi
Institute of Mathematics, Reáltanoda utca 13-15, H-1053, Budapest, Hungary
ELTE - Faculty of Science, Dept. of Geometry, Pázmány Péter sétány 1/A, 1117
Budapest, Hungary
BBU - Babeş-Bolyai Univ., Str, M. Kogălniceanu 1, 400084 Cluj-Napoca, Romania
BCAM - Basque Center for Applied Math., Mazarredo, 14 E48009 Bilbao, Basque
Country – Spain<EMAIL_ADDRESS>
(Date: June 20, 2024)

###### Abstract.

We study the multiplier ideals and the corresponding jumping numbers and
multiplicities $\\{m(c)\\}_{c\in\mathbb{R}}$ in the following context: $(X,o)$
is a complex analytic normal surface singularity,
${\mathfrak{a}}\subset{\mathscr{O}}_{X,o}$ is an
${\mathfrak{m}}_{X,o}$–primary ideal, $\phi:\widetilde{X}\to X$ is a log
resolution of $\mathfrak{a}$ such that
$\mathfrak{a}{\mathscr{O}}_{\widetilde{X}}={\mathscr{O}}_{\widetilde{X}}(-F)$,
for some nonzero effective divisor $F$ supported on $\phi^{-1}(0)$.

We show that $\\{m(c)\\}_{c>0}$ is combinatorially computable from $F$ and the
resolution graph $\Gamma$ of $\phi$, and we provide several formulae. We also
extend Budur’s result (valid for $(X,o)=(\mathbb{C}^{2},0)$), which makes an
identification of $\sum_{c\in[0,1]}m(c)t^{c}$ with a certain Hodge spectrum.
In our general case we use Hodge spectrum with coefficients in a mixed Hodge
module.

We show that $\\{m(c)\\}_{c\leq 0}$ usually depends on the analytic type of
$(X,o)$. However, for some distinguished analytic types we determine it
concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated
with the central vertex), we recover $\sum_{c}m(c)t^{c}$ from the Poincaré
series of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover
$\sum_{c}m(c)t^{c}$ from the multivariable topological Poincaré (zeta)
function of $\Gamma$.

###### Key words and phrases:

normal surface singularities, plane curve singularities, multiplier ideals,
jumping numbers, jumping multiplicities, Hodge spectrum, mixed Hodge modules,
canonical covering, weighted homogeneous singularities, splice quotient
singularities

###### 2010 Mathematics Subject Classification:

Primary: 14B05, 14Fxx, 32S05; Secondary: 32S10, 32S25

The authors are partially supported by NKFIH Grant “Élvonal (Frontier)” KKP
144148. The second author acknowledges the support of the ’János Bolyai
Research Scholarship’ of the Hungarian Academy of Sciences, and the support of
the project “Singularities and Applications” - CF 132/31.07.2023 funded by the
European Union - NextGenerationEU - through Romania’s National Recovery and
Resilience Plan.

##  1\. Introduction

Assume that $(X,o)$ is a complex analytic normal surface singularity, let
${\mathscr{O}}_{X,o}$ be its local algebra with maximal ideal
${\mathfrak{m}}_{X,o}$. Let us also fix an ${\mathfrak{m}}_{X,o}$–primary
ideal ${\mathfrak{a}}\subset{\mathscr{O}}_{X,o}$. Then there exists a log
resolution $\phi:\widetilde{X}\to X$ and a nonzero integral cycle $F$
supported on the exceptional curve $\phi^{-1}(0)=E$ such that
${\mathfrak{a}}{\mathscr{O}}_{\widetilde{X}}={\mathscr{O}}_{\widetilde{X}}(-F)$.
In fact, in our study we will start with some $\phi$ and such a cycle $F$.

We denote by $Z_{K}$ the anticanonical cycle associated with $\phi$,
$-c_{1}(\Omega^{2}_{\widetilde{X}})=-c_{1}({\mathscr{O}}_{\widetilde{X}}(K_{\widetilde{X}}))\in
H^{2}(\widetilde{X},\mathbb{Z})$, identified with a rational cycle (via the
adjunction relations). We consider for any $c\in\mathbb{R}$ the cycle
$L(c):=\lfloor Z_{K}+cF\rfloor$. Then the tower of multiplier ideals is
defined as
$\mathfrak{I}(c,F):=\phi_{*}{\mathscr{O}}_{\widetilde{X}}(-L(c))\subset{\mathscr{O}}_{X,o}$.
They measure the complexity of the singularity $(X,o)$ compatibly with $F$ (or
with $\mathfrak{a}$). They were introduced in [Ein, Laz2] and since then they
were intensively studied, see [budur, Carrami, Carrami2, Galindo, Galindo2,
Howald, Hyry, Jarvilehto, Montaner, Naie, Tucker] with all their connections
with different aspects of singularities and singular map-germs. In fact, the
definition and study extends to any dimension, see e.g. [Ein, Laz2, Pande] and
the references therein.

It turns out that for any $c\in\mathbb{R}$, the ideal
$\mathfrak{I}(c,F)\subset{\mathscr{O}}_{X,o}$ has finite codimension,
$\mathfrak{I}(c,F)\subset\mathfrak{I}(c^{\prime},F)$ for $c^{\prime}<c$, and
$\mathfrak{I}(c,F)={\mathscr{O}}_{X,o}$ for $c\ll 0$. The number $c$ is called
jumping number if
$m(c):=\dim\,\mathfrak{I}(c-\epsilon,F)/\mathfrak{I}(c,F)\not=0$ for
$0<\epsilon\ll 1$, and $m(c)$ is the corresponding jumping multiplicity. One
verifies that in any compact interval there are only finitely many jumping
numbers and they are all rational.

Compared with the earlier literature in the subject we make two major
extensions:

(A) We do not assume any restriction regarding the surface singularity
$(X,o)$, e.g. we do not assume that the link is a rational homology sphere
(i.e. the dual graph might have loops and the irreducible exceptional divisors
might have higher genera). Also, we do not restrict the analytic type of
$(X,o)$: on the supporting topological type the analytic type might vary
(which in the non-rational or non-minimally elliptic cases might have
considerable effects). We emphasize that in general the classification of such
analytic types (or the description of the corresponding moduli spaces) is
still an open problem. Under this generality the general analytic tools (e.g.
vanishing theorems) are more complex (and the zones what they do not cover are
wider).

(B) In the earlier studies only the cases $c>0$ were treated. Our study of the
cases $c\leq 0$ opens a new perspectives. If $(X,o)$ is smooth or log terminal
then $m(c)=0$ for $c\leq 0$, however, in the general case (even for rational
$(X,o)$) the negative zone appears as a very challenging part.

In fact (and this is one of the main messages of the note), we show that the
study of the Poincaré series of multiplier ideals
$\sum_{c\in\mathbb{R}}m(c)t^{c}$ (introduced first in [Galindo]) breaks into
two major parts with rather different aspects.

(i) (the $c>0$ zone) We also call this the Riemann–Roch zone, since in this
case we determine $\sum_{c>0}m(c)t^{c}$ completely topologically from the
resolution graph $\Gamma$ of $\phi$ and the cycle $F$ (without any restriction
regarding the link or the analytic type of $(X,o)$).

We provide several combinatorial formulae. The first one, Proposition
LABEL:lem:chi, is a generalization of expressions obtained in [Carrami2,
Carrami, Tucker] proved in those articles for rational singularities.

We also compute $m(c+1)-m(c)$ (see Corollary LABEL:cor:m0), thus we recover
$\sum_{c>0}m(c)t^{c}$ from $\sum_{c\in(0,1]}m(c)t^{c}$ and $(D_{c},F)$, where
$D_{c}=L(c)-L(c-\epsilon)$ ($0<\epsilon\ll 1$) is the jumping divisor
(introduced in [Carrami2]), computable combinatorially. (This fact was also
shown for rational $(X,o)$ in [Carrami].)

Hence, the first interval and $\sum_{c\in(0,1]}m(c)t^{c}$ has a distinguished
role.

Regarding this we stress out two facts. Firstly, Budur in [budur] identified
$\sum_{c\in(0,1]}m(c)t^{c}$ with the Hodge spectrum ${\rm Sp}_{(0,1]}(f)$ of a
function germ $f$, where in his case, $(X,o)=(\mathbb{C}^{2},0)$ is smooth,
$f:(\mathbb{C}^{2},0)\to(\mathbb{C},0)$ is a function germ with an isolated
singularity, $\phi$ is a log embedded resolution and $F$ is the divisor ${\rm
div}_{\phi,E}(f)$ of $\phi^{*}(f)$ along $\phi^{-1}(0)=E$. This was reproved
in [Carrami] as well. (For the theory of mixed Hodge structures and of the
spectrum associated with isolated hypersurface singularities see the work of
Steenbrink, e.g. [steenbrink, steenbrink2] or [SSS].)

In this note we succeed to generalize this statement. The first step is that
we rewrite $J_{[0,1]}(t)=\sum_{c\in[0,1]}m(c)t^{c}$ in a different way, in
terms of $\Gamma$ and the multiplicities of $F$, for any $(X,o)$ and any $F$
(cf. Theorem LABEL:th:main_mult). This is the second combinatorial formula of
the jumping multiplicities.

This formula will resonate perfectly with Hodge spectrum computations of a
generic element $f\in\mathfrak{a}$. Indeed, if $(X,o)$ is numerically
Gorenstein (that is, $Z_{K}$ is an integral cycle), and $f$ is an analytic
function gems $(X,o)\to(\mathbb{C},0)$ with isolated singularities, ${\rm
div}_{\phi,E}(f)=F$, we prove the generalization of the statement of Budur:
$J_{[0,1]}(t)={\rm Sp}_{[0,1]}(t)$, cf. Corollary LABEL:cor:NS. However, when
$\Gamma$ is not numerically Gorenstein, $\\{Z_{K}\\}\not=0$, then we need to
use a more general spectrum computation established in [NS]. In [NS] the usual
spectrum is replaced by the spectrum of $f$ with coefficients in a mixed Hodge
module ${\mathscr{M}}$ of $(X,o)$, such that ${\mathscr{M}}$ restricted to
$X\setminus\\{o\\}$ is a variation of Hodge structure whose underlying
representation is abelian. Indeed, under the assumption that the link is a
rational homology sphere, from $Z_{K}$ we construct a character of the first
homology of the link, hence a rank one variation of Hodge structure
${\mathbb{V}}$ on $X\setminus\\{o\\}$, and we prove the identity
$J_{[0,1]}(t)={\rm Sp}_{[0,1]}(t)(f,j_{*}{\mathbb{V}})$ (where
$j:X\setminus\\{o\\}\subset X$ is the inclusion), see Theorem LABEL:th:NS2.
The construction of ${\mathbb{V}}$ can be done using the canonical cyclic
covering of $(X,o)$ (associated with $Z_{K}$, or in $\mathbb{Q}$–Gorenstein
case associated with the canonical divisor $K_{\widetilde{X}}$, as well). We
emphasize again, though a priori both objects $J_{[0,1]}(t)$ and ${\rm
Sp}_{[0,1]}(t)(f,j_{*}{\mathbb{V}})$ are analytical, we provide combinatorial
formulae for them.

The second new aspect regarding $\sum_{c\in(0,1]}m(c)t^{c}$ is the following
connection. Assume that there exists again a function
$f:(X,o)\to(\mathbb{C},0)$ with an isolated singularity such that $F={\rm
div}_{\phi,E}(f)$. Set $(C,o):=(f^{-1}(0),o)$ and let $\delta(C,o)$ be its
delta invariant. Then $\sum_{c\in(0,1]}m(c)=\delta(C,o)+(\\{Z_{K}\\},F)$.

(ii) (the $c\leq 0$ zone) We show by examples that the multiplicity $m(0)$
cannot be determined topologically. However, we characterize it completely in
terms of the analytic type of $(X,o)$. The analytic contribution is a term
$\mathfrak{h}^{0}\in\\{0,1\\}$ characterized by an $h^{0}$–sheaf cohomology.

The $c\leq 0$ zone can be called the ‘$h^{0}$–zone’, where the Riemann–Roch
expressions or vanishing theorems cannot be applied. In this zone $m(c)$
usually cannot be computed form $\Gamma$ and $F$. We exemplify this fact by
concrete examples when we fix $\Gamma$ and $F$ and we consider several
different analytic realizations, and we compute the $m(c)$’s. Interestingly
enough, already the numerical invariant $\sum_{c\leq 0}m(c)$ measures the
analytic type and it is connected to the most important analytic invariant of
$(X,o)$, the geometric genus $p_{g}(X,o)$. More precisely, we prove that
$\sum_{c\leq 0}m(c)=p_{g}(X,o)+\chi(\lfloor Z_{K}\rfloor)$. Already when
$(X,o)$ is rational (i.e. $p_{g}(X,o)=0$), $\sum_{c\leq 0}m(c)$ can be
arbitrarily large (see e.g. Example LABEL:ex:sum).

On the other hand, even for this zone, we provide concrete formulae valid for
certain distinguished analytic types.

$\bullet$ In the case of a weighted homogeneous germs $(X,o)$ with local
graded algebra $\oplus_{\ell\geq 0}R_{\ell}$ (and $F$ is associated with the
central vertex), we show the correspondence $\dim(R_{\ell})=m(c)$ with
$c=(\ell-\mathfrak{r})/\alpha$ (for notations and precise statement see
subsection LABEL:ss:we). For $\ell>{\mathfrak{r}}$ (corresponding to $c>0$)
$\dim(R_{\ell})$ is topological computable from the Seifert invariants of the
link, however in general it is not. But, if the link is a rational homology
sphere, then by Dolgachev-Pinkham-Demazure formulae all the terms for any
$\ell$ are topological. (For the theory of weighted homogeneous germs see e.g.
[pinkham].)

$\bullet$ If the link is a rational homology sphere and we fix a resolution
graph $\Gamma$, and we assume that the analytic type of $(X,o)$ is splice
quotient with respect to $\Gamma$ (a family introduced by Neumann and Wahl,
cf. [NW]). Then in Proposition LABEL:PZ we provide a combinatorial formula for
$\sum_{c\in\mathbb{R}}m(c)t^{c}$ in terms of $\\{L(c)\\}_{c}$ and the
multivariable topological Poincaré (zeta) function $Z({\bf t})$ of $\Gamma$.
This includes e.g. all the cases when $(X,o)$ is rational or minimally
elliptic (with $\Gamma$ minimal). (For more on $Z({\bf t})$ see [CDG, CDGEq,
Nkonyv].)

But we emphasize again that in general, the computation of $\\{m(c)\\}_{c\leq
0}$ is of analytic nature, hence it runs in parallel with the classification
of the analytic types of $(X,o)$ supported of a fixed topological type
identified by $\Gamma$. We will return back to this study in the forthcoming
manuscript.

##  2\. Preliminaries

###  2.1. Notations regarding resolutions and resolution graphs

######  2.1.1.

Let us fix a normal surface singularity $(X,o)$ and one of its log (good)
resolutions $\phi:\widetilde{X}\to X$. Let $\Gamma$ be the dual resolution
graph of $\phi$, we denote its vertices by $\mathscr{V}$ and its edges by
$\mathscr{E}$. We write $E$ for the irreducible exceptional curve
$\phi^{-1}(o)$, and let $\\{E_{i}\\}_{i\in\mathscr{V}}$ be its irreducible
components.

The lattice $L:=H_{2}(\widetilde{X},\mathbb{Z})$ is a free $\mathbb{Z}$–module
generated by the classes of $\\{E_{i}\\}_{i\in\mathscr{V}}$ and it is endowed
with the natural negative definite intersection form $(\,,\,)$. The dual
lattice is $L^{\prime}={\rm
Hom}_{\mathbb{Z}}(L,\mathbb{Z})\simeq\\{l^{\prime}\in
L\otimes\mathbb{Q}\,:\,(l^{\prime},L)\in\mathbb{Z}\\}\subset
L\otimes\mathbb{Q}$. We use the same symbol $(\,,\,)$ for the extended
intersection form to $L\otimes\mathbb{Q}$. The overlattice $L^{\prime}$ is
generated over $\mathbb{Z}$ by the (anti)dual classes
$\\{E^{*}_{i}\\}_{i\in\mathscr{V}}$ defined by
$(E^{*}_{i},E_{j})=-\delta_{ij}$ (where $\delta_{ij}$ stays for the Kronecker
symbol). $L^{\prime}$ can also identified with
$H^{2}(\widetilde{X},\mathbb{Z})$ (the target of the first Chern class
$c_{1}:{\rm Pic}(\widetilde{X})\to H^{2}(\widetilde{X},\mathbb{Z})$).

We denote by $M$ the link of $(X,o)$. It is known that $M$ is a rational
homology sphere if and only if each $E_{i}$ is rational and $\Gamma$ is a
tree. It is an integral homology sphere if additionally $\det\,(\,,\,)=\pm 1$.
More generally, if $g_{i}$ denotes the genus of $E_{i}$, $g:=\sum_{i}g_{i}$
and $h$ the first Betti number of the topological realization of $\Gamma$ (the
number of independent cycle in the graph), then the first Betti number of $M$
is $2g+h$, and ${\rm Tors}\,H_{1}(M,\mathbb{Z})\simeq L^{\prime}/L$.

There is a natural partial ordering of $L\otimes\mathbb{Q}$: we write
$l_{1}^{\prime}\geq l_{2}^{\prime}$ if
$l_{1}^{\prime}-l_{2}^{\prime}=\sum_{i}a_{i}E_{i}$ with every $a_{i}\geq 0$.
We set $L_{\geq 0}=\\{l\in L\,:\,l\geq 0\\}$ and $L_{>0}=L_{\geq
0}\setminus\\{0\\}$.

We define the rational Lipman cone as
${\mathscr{S}}_{\mathbb{Q}}:=\\{l^{\prime}\in
L\otimes\mathbb{Q}\,:\,(l^{\prime},E_{i})\leq 0\ \mbox{for all $i$}\\}$, and
we also set ${\mathscr{S}}^{\prime}:={\mathscr{S}}_{\mathbb{Q}}\cap
L^{\prime}$ and ${\mathscr{S}}:={\mathscr{S}}_{\mathbb{Q}}\cap L$. As a monoid
${\mathscr{S}}^{\prime}$ is generated over ${\mathbb{Z}}_{\geq 0}$ by
$\\{E^{*}_{i}\\}_{i}$. If
$s^{\prime}\in{\mathscr{S}}_{\mathbb{Q}}\setminus\\{0\\}$, then all its
$E_{i}$–coordinates are strictly positive (see [Nkonyv, Corollary 2.1.19]).

Let $K_{\widetilde{X}}$ be the canonical divisor  of $\widetilde{X}$
(well–defined up to a linear equivalence), that is,
${\mathscr{O}}_{\widetilde{X}}(K_{\widetilde{X}})\simeq\Omega^{2}_{\widetilde{X}}$,
the sheaf of holomorphic 2–forms. We denote its first Chern class
$c_{1}({\mathscr{O}}_{\widetilde{X}}(K_{\widetilde{X}}))\in
H^{2}(\widetilde{X},\mathbb{Z})\simeq L^{\prime}$ by $-Z_{K}$; $Z_{K}$ is
called the (anti)canonical cycle of $\phi$. The (rational) cycle $Z_{K}\in
L^{\prime}$ can also be determined combinatorially from $(\,,\,)$ by the
adjunction formulae $(Z_{K},E_{i})=(E_{i},E_{i})+2-2g_{i}$ for all
$i\in\mathscr{V}$.

We call $(X,o)$ numerically Gorenstein  if $Z_{K}\in L$, i.e. if all its
$E_{i}$–coefficients are integral (this condition is independent of the choice
of $\phi$). We call $(X,o)$ Gorenstein if $Z_{K}\in L$ and one can choose
$K_{\widetilde{X}}$ as $-Z_{K}$, i.e.
${\mathscr{O}}_{\widetilde{X}}(K_{\widetilde{X}}+Z_{K})\simeq{\mathscr{O}}_{\widetilde{X}}$.
In a different language, $(X,o)$ is numerically Gorenstein (resp. Gorenstein)
exactly when the line bundle $\Omega^{2}_{X\setminus\\{o\\}}$ is topologically
(resp. analytically) trivial. (For a different characterization see also
LABEL:bek:qg.)

Finally, we set for any $l^{\prime}\in L^{\prime}$ the Riemann-Roch expression
$\chi(l^{\prime})=(Z_{K}-l^{\prime},l^{\prime})/2$. If $l\in L_{>0}$, then
$\chi(l)=h^{0}({\mathscr{O}}_{l})-h^{1}({\mathscr{O}}_{l})$. E.g.,
$\chi(E)=1-g-h$. As usual, we also write $\chi({\mathscr{O}}_{l}(D))$ for
$h^{0}({\mathscr{O}}_{l}(D))-h^{1}({\mathscr{O}}_{l}(D))$, which, by Riemann-
Roch, equals $\chi(l)+(D,l)$ ($D\in L$).

For any ${\mathbb{R}}$-divisor $D=\sum_{i}d_{i}E_{i}$ of $\widetilde{X}$
($d_{i}\in{\mathbb{R}}$) we define its integral part $\lfloor D\rfloor$ and
fractional part $\\{D\\}$ by applying the corresponding operation to each
coefficient $d_{i}$.

For more details see e.g. [NOSz, Nkonyv].


FSA | 57.4 $\pm$ 2.3 | 44.8 $\pm$ 2.8 | 39.3 $\pm$ 1.6 | 34.8 $\pm$ 2.1 | 32.9 $\pm$ 1.9 | 33.2 $\pm$ 2.6 | 33.7 $\pm$ 1.7
FSA-FiLM | 62.7 $\pm$ 2.1 | 50.2 $\pm$ 2.4 | 46.9 $\pm$ 2.3 | 40.1 $\pm$ 2.2 | 37.3 $\pm$ 2.5 | 36.1 $\pm$ 2.2 | 35.7 $\pm$ 2.1
Table 40: Detailed accuracy for each incremental session on dSprites-position under the few-shot CIL setting. GDumb is the only memory-based method used for comparisons; we use a buffer size equal to the first session’s number of images $N_{1}$. The best results across all methods are in bold while the best results across the no-memory methods are underlined. A pre-trained EfficientNet-B0 on Imagenet-1k is used as a backbone for all methods. Method |  Accuracy (%) in each session ($\uparrow$)
---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
NA |  35.5 $\pm$ 0.9 |  28.7 $\pm$ 0.7 |  36.4 $\pm$ 0.8 |  42.8 $\pm$ 0.4 |  40.0 $\pm$ 0.4 |  39.4 $\pm$ 0.6 |  41.4 $\pm$ 0.7 |  40.3 $\pm$ 0.9 |  41.0 $\pm$ 0.7
GDumb |  51.1 $\pm$ 1.7 |  45.4 $\pm$ 1.2 |  46.9 $\pm$ 1.8 |  52.2 $\pm$ 1.0 |  45.4 $\pm$ 1.7 |  42.5 $\pm$ 1.6 |  41.8 $\pm$ 0.9 |  39.5 $\pm$ 1.9 |  38.6 $\pm$ 1.0
FACT |  41.4 $\pm$ 0.6 |  25.9 $\pm$ 0.9 |  19.6 $\pm$ 0.4 |  16.3 $\pm$ 0.4 |  13.7 $\pm$ 0.4 |  11.3 $\pm$ 0.5 |  10.4 $\pm$ 0.5 |  9.4 $\pm$ 0.6 |  8.3 $\pm$ 0.6
FSA |  42.9 $\pm$ 2.6 |  39.5 $\pm$ 2.0 |  45.5 $\pm$ 1.5 |  51.6 $\pm$ 1.7 |  48.2 $\pm$ 1.8 |  47.3 $\pm$ 1.4 |  49.8 $\pm$ 1.5 |  49.1 $\pm$ 1.7 |  50.1 $\pm$ 1.5
FSA-FiLM | 46.6 $\pm$ 1.9 | 44.8 $\pm$ 1.1 | 49.4 $\pm$ 0.8 | 52.9 $\pm$ 1.6 | 54.0 $\pm$ 0.7 | 53.5 $\pm$ 1.0 | 55.8 $\pm$ 0.7 | 55.2 $\pm$ 0.5 | 55.8 $\pm$ 0.6
Table 41: Detailed accuracy for each incremental session on FGVC-Aircraft under the few-shot CIL setting. GDumb is the only memory-based method used for comparisons; we use a buffer size equal to the first session’s number of images $N_{1}$. The best results across all methods are in bold while the best results across the no-memory methods are underlined. A pre-trained EfficientNet-B0 on Imagenet-1k is used as a backbone for all methods. Method |  Accuracy (%) in each session ($\uparrow$)
---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
NA |  82.1 $\pm$ 0.8 |  76.6 $\pm$ 0.9 |  73.0 $\pm$ 1.5 |  69.6 $\pm$ 1.0 |  69.0 $\pm$ 1.3 |  67.6 $\pm$ 1.3 |  65.8 $\pm$ 1.2 |  64.1 $\pm$ 1.0 |  60.4 $\pm$ 0.6 |  59.0 $\pm$ 0.8 |  57.6 $\pm$ 0.8
GDumb |  91.3 $\pm$ 1.6 |  91.8 $\pm$ 1.2 |  80.0 $\pm$ 1.4 |  72.0 $\pm$ 1.9 |  69.2 $\pm$ 3.0 |  63.5 $\pm$ 1.5 |  59.9 $\pm$ 1.1 |  54.5 $\pm$ 0.3 |  48.0 $\pm$ 0.4 |  44.3 $\pm$ 1.9 |  41.2 $\pm$ 1.7
FACT |  84.3 $\pm$ 1.4 |  72.0 $\pm$ 1.2 |  68.0 $\pm$ 2.2 |  63.2 $\pm$ 1.0 |  62.3 $\pm$ 1.0 |  59.5 $\pm$ 1.2 |  58.0 $\pm$ 1.0 |  55.8 $\pm$ 1.0 |  52.8 $\pm$ 0.7 |  51.7 $\pm$ 0.8 |  49.8 $\pm$ 0.8
FSA |  87.0 $\pm$ 1.4 |  79.6 $\pm$ 1.1 |  76.4 $\pm$ 0.9 |  72.7 $\pm$ 0.7 |  73.0 $\pm$ 0.9 |  71.3 $\pm$ 0.4 |  69.7 $\pm$ 0.4 |  68.4 $\pm$ 0.4 |  64.7 $\pm$ 0.5 |  62.9 $\pm$ 0.4 |  62.2 $\pm$ 0.4
FSA-FiLM | 94.3 $\pm$ 0.9 | 90.6 $\pm$ 0.3 | 88.6 $\pm$ 1.0 | 85.1 $\pm$ 0.6 | 84.9 $\pm$ 0.4 | 84.0 $\pm$ 0.4 | 82.5 $\pm$ 0.7 | 81.1 $\pm$ 0.4 | 76.8 $\pm$ 0.4 | 75.0 $\pm$ 0.4 | 73.4 $\pm$ 0.4
Table 42: Detailed accuracy for each incremental session on Letters under the few-shot CIL setting. GDumb is the only memory-based method used for comparisons; we use a buffer size equal to the first session’s number of images $N_{1}$. The best results across all methods are in bold while the best results across the no-memory methods are underlined. A pre-trained EfficientNet-B0 on Imagenet-1k is used as a backbone for all methods. Method |  Accuracy (%) in each session ($\uparrow$)
---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
NA |  83.3 $\pm$ 1.0 |  62.6 $\pm$ 0.5 |  60.9 $\pm$ 1.0 |  61.5 $\pm$ 0.3 |  68.3 $\pm$ 1.2 |  71.1 $\pm$ 0.6 |  72.0 $\pm$ 0.7 |  70.8 $\pm$ 1.0 |  69.0 $\pm$ 0.3
GDumb |  88.3 $\pm$ 0.4 |  63.6 $\pm$ 0.8 |  58.6 $\pm$ 0.6 |  56.4 $\pm$ 1.0 |  66.0 $\pm$ 1.1 |  68.7 $\pm$ 0.4 |  68.9 $\pm$ 0.6 |  65.8 $\pm$ 1.0 |  63.2 $\pm$ 1.1
FACT |  84.3 $\pm$ 0.6 |  53.6 $\pm$ 0.6 |  43.0 $\pm$ 0.5 |  35.9 $\pm$ 0.6 |  28.3 $\pm$ 0.6 |  24.2 $\pm$ 0.3 |  22.6 $\pm$ 0.3 |  22.0 $\pm$ 0.5 |  20.6 $\pm$ 0.2
FSA |  85.2 $\pm$ 0.3 |  63.3 $\pm$ 0.3 |  61.4 $\pm$ 0.7 |  61.6 $\pm$ 0.5 |  68.5 $\pm$ 0.9 |  71.2 $\pm$ 1.1 |  72.2 $\pm$ 0.5 |  71.2 $\pm$ 0.7 |  70.3 $\pm$ 0.4
FSA-FiLM | 87.7 $\pm$ 0.3 | 68.5 $\pm$ 0.6 | 66.9 $\pm$ 0.4 | 66.7 $\pm$ 0.9 | 73.7 $\pm$ 0.6 | 76.0 $\pm$ 0.7 | 76.0 $\pm$ 0.5 | 75.0 $\pm$ 0.6 | 74.0 $\pm$ 0.3
Table 43: Detailed accuracy for each incremental session on DomainNet under the few-shot CIL setting. GDumb is the only memory-based method used for comparisons; we use a buffer size equal to the first session’s number of images $N_{1}$. The best results across all methods are in bold while the best results across the no-memory methods are underlined. A pre-trained EfficientNet-B0 on Imagenet-1k is used as a backbone for all methods. Method | Accuracy (%) in each session ($\uparrow$)
---|---
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
NA | 51.9 | 52.8 | 44.9 | 46.8 | 49.3 | 51.7 | 54.4 | 53.8 | 49.7
GDumb | 56.4 | 50.1 | 36.3 | 47.5 | 44.2 | 44.7 | 46.4 | 40.9 | 40.4
FACT | 54.9 | 29.9 | 24.6 | 23.8 | 20.4 | 17.8 | 15.7 | 16.4 | 14.3
FSA | 52.1 | 53.6 | 40.0 | 47.8 | 49.2 | 51.3 | 55.1 | 55.1 | 51.5
FSA-FiLM | 61.8 | 61.6 | 52.0 | 56.5 | 57.0 | 59.4 | 61.8 | 61.2 | 58.8
Table 44: Detailed accuracy for each incremental session on iNaturalist under
the few-shot CIL setting. GDumb is the only memory-based method used for
comparisons; we use a buffer size equal to the first session’s number of
images $N_{1}$. The best results across all methods are in bold while the best
results across the no-memory methods are underlined. A pre-trained
EfficientNet-B0 on Imagenet-1k is used as a backbone for all methods.