M Breil

D Koenig

D Batani

Raffestin

S D Baton

E Le Bel

S Brygoo

X Ribeyre

C Rousseaux

J Breil

M Koenig

D Raffestin

Shock generation comparison with planar and hemispherical targets in shock ignition relevant experiment

Shock generation comparison with planar and hemispherical targets in shock ignition relevant experiment

INTRODUCTION

Inertial Confinement Fusion (ICF), the classical process leading to deuterium-tritium (DT) ignition, is the conversion of the kinetic energy of the imploding shell into the internal energy of a central hot spot [START_REF] Atzeni | The Physics of Inertial Fusion[END_REF][START_REF] Lindl | Inertial Confinement Fusion[END_REF]. An alternative pathway for achieving central ignition, the shock ignition (SI), based on the separation of the compression and the ignition phases, has been considered [START_REF] Betti | [END_REF]4]. This scheme is likely to provide higher gains than the conventional approach. Preliminary experiments performed on the OMEGA laser facility have been quite encouraging [5,6,7] and a proof-of-principle experiment on the National Ignition Facility has been proposed [8]. The SI option is also under investigation for the Laser MegaJoule [9], and has been studied in the context of the European directdrive HiPER project [10,11,12].

In the SI scheme, a strong converging shock is launched in the target by a high-intensity (≤ 10 16 W/cm 2 ) laser spike at the end of the compression phase and prior to the final stagnation of fuel. The collision between the converging shock and the one, originating from the reflection at target center of the first shock produced by the compression beams, creates the conditions for ignition. This concept has been studied theoretically [13] and experimentally (see [14] for a recent review). Experiments in planar geometry have addressed the capability of creating a strong shock in presence of a large plasma corona, the impact of parametric instabilities in laser-plasma interaction and of the hot electrons they produce [15,16,17]. In spherical geometry [5,18,19,20] it was demonstrated experimentally that a properly timed final shock significantly enhances the neutron yield [5]. Recent encouraging results [19,20] demonstrated a crucial issue for the credibility of SI, i.e. the capability of generating very high pressures (~300 Mbar at the ablation front) for laser intensities in the range of 10 15 to 10 16 W/cm 2 . Shock ignition is a two-step process. First, a uniform illumination compresses the target at low implosion velocity and low isentropic assembly. Second, a high-intensity laser spike launches a strong converging shock to ignite the central hot spot. It is worth noticing that when the strong shock is launched, the critical radius of the target is reduced; hence the focal spot becomes not optimized for the igniting beams. The consequence is a bad overlapping of beams and the pointing on the target radius becomes more critical, which should lead to a decrease of the absorption and consequently of the shock strength.

Nevertheless, even if the compression phase requires a uniform spherical irradiation, the simulations show that the irradiation is less restrictive for the ignition spike as observed from a recent numerical work [11]. However, these simulations do not take into account the hot-electron production or magnetic field generation during the high intensity spike, which can modify the shock uniformity during the compression phase. In this configuration, hydrodynamic simulations show that two planar shocks propagate inside the spherical hot spot, collide and bounce to finally ignite the capsule. The production of a quasi-planar shock has also been experimentally observed in another context, i.e. with shock waves driven by ion beam inside spherical mass targets [21].

In this paper, we present an experiment performed on the "Ligne d'Intégration Laser" (LIL) at CEA/CESTA, in order to study the influence of the target shape, which entails a specific ablation front curvature, for producing a strong planar shock by a laser spike pulse in presence of a large plasma corona with parameters relevant for SI. An original configuration is investigated where a planar shock could be produced by using an "usual" Gaussian shaped laser focal spot impinging on a hemispherical target.

II. EXPERIMENTAL SETUP

In the experiment, the four laser beams of the LIL quad were focused on two different kinds of target: first, on planar targets to validate the diagnostics and the simulations in a classical case; second, on hemispherical targets in order to observe the behavior of the spike shock in this specific geometry. VISAR (Velocity Interferometer System for Any Reflector) and SOP (Streaked Optical Pyrometry) diagnostics allowed following the shock wave propagation in time. In order to infer all shock characteristics, experimental results were compared to detailed numerical simulations performed with the CHIC code [22].

The LIL facility delivered 10 kJ total laser energy on target at 351 nm (3ω) wavelength [START_REF] Julien | Proc. of SPIE[END_REF]. In addition to continuous phase plate, two-phase modulators incorporated in the front-end laser temporally smoothed the beams. Their goals are to mitigate plasma instabilities (and hydrodynamic instabilities in direct drive experiments), as well as to improve the propagation of the beam through large plasmas by reducing laser filamentation [START_REF] Glenzer | [END_REF].

We used a temporally shaped laser pulse (Fig. 1b): a 2 ns square pulse is assimilated to the final ignition spike at peak power ~ 4 TW, yielding a mean intensity on target of 1.5×10 15 W/cm 2 ; it is preceded by a 1 ns pedestal at lower intensity (~ 1 TW). The 2ns square pulse has been chosen in order to sufficiently sustain for a long time, in planar geometry, the shocks and in proportion to the CH ablator thickness (> 200 microns). The quad was focused by a f/8 focusing system with continuous phase plates designed to produce a fairly small focal spot in order to reach the highest intensity as possible for producing a strong shock.

The target, either plane or hemispherical, consisted of three layers (Fig. 1a): (i) a 270 µm plastic (CH) ablator to absorb the incident laser light and to reduce the X-ray production; (ii) a 30 µm Mo layer to block X-rays from the coronal plasma; (iii) the rear part of the target was a 250 µm α-quartz window for VISAR and SOP diagnostics. The first CH layer was either planar (270 µm thickness) or an hemisphere of 500 µm diameter attached onto 20 µm CH planar layer. Hence, the CH thickness at the top of hemispherical target was the same than the planar one. The diameter of the hemispherical part of the target has been chosen to match the LIL focal spot, measured to be Gaussian with a radius of 180 µm (at 1/e). The laser beam is perpendicular to the target and is focused on the center of the hemisphere (cf. Fig. 1c).

A set of two VISARs [25], one at λ = 1064 nm and one at λ = 532 nm, and a SOP at 650 nm were used to record as a function of time the shock velocity and the self-emission of the target rear side.

The self-emission was also recorded with a GOI (Gated Optical Imager) at 650 nm giving an instantaneous 2D image with a 400 ps gate width. Laser plasma instabilities have also been

III. EXPERIMENTAL RESULTS

The energy fraction of the total backscattered light (SRS, stimulated Raman scattering, and SBS) through the f/8 focusing cone has been measured around 5% with respect to the total incident laser energy and less than 8% with respect to the laser energy encircled in a 500-µm diameter (which is the diameter of the hemispherical target). The energy fraction of SBS collected by the NBI diagnostics was always low, less or equal to ≈ 1.5% (we do not have this measurement for Raman). About 2/3 of the reflected energy is due to SBS and 1/3 to SRS. As an example, Fig. 2 shows SBS and SRS powers measured in f/8 for the two shots discussed below, Fig. 2a for the planar target and Fig. 2b for the hemispherical one. Backscattered SBS predominantly develops during the rising part of the laser spike, where the ion acoustic wave damping is expected to be low (because of low ion temperature) as well as the electron temperature for which the SBS inhomogeneous threshold is proportional. The peak SBS reflectivity reaches 6.5% for the planar target (the time-averaged SBS reflectivity is 2.5% on this shot). On the contrary, SRS develops more smoothly during the high intensity part of the laser pulse, but the peak SRS reflectivity is lower, 3% on this shot (and 1.2% for the time-averaged value). Generally, the SBS and SRS measurements are observed to be approximately twice weaker for hemispherical targets compared to the planar targets. These relatively modest reflectivity values for laser intensities as high as 1.5x10 15 W/cm 2 have already been observed in previous SI experiments [15,17,18]. Low reflectivity is a consequence of the combination between the interaction with laser beams using improved temporal smoothing technique and inhomogeneous plasmas. In the experiment, the inhomogeneity affects both the electron density profile (detuning SRS and SBS) and the expansion velocity of the plasma into the vacuum, altering the resonance conditions of SBS. This is illustrated in Fig. 3 showing the SRS and SBS thresholds [START_REF] Kruer | The physics of laser plasma interactions[END_REF] as calculated for plasma conditions given by our 2D hydrodynamic CHIC code [22]. It is shown that the SRS threshold occurs around 2-4x10 15 W/cm 2 , so that the intensity threshold might be marginally reached in our experiment. For SBS, Fig. 3 shows that the effect of the velocity detuning is more significant than the density profile one. However, the SBS threshold is below the SRS one, around 2 to 5x10 14 W/cm 2 , that qualitatively explain the SBS prevalence over SRS, in agreement with the data. Comparing now these thresholds in plasma profiles expected from targets designed for the direct-drive scheme for LMJ [START_REF] Colaïtis | [END_REF], we note that the electron density scalelength is typically one-third smaller in our experiments, thus the SRS threshold is expected to be one third lower as well for LMJ. On the contrary, both the electron temperature T e and the expansion velocity scale-length L V = [1/V ΔV/Δz] -1 increase by a factor of two in the LMJ design, as calculated by the CHIC code. However, as the SBS threshold is proportional to the ratio T e /L V , the threshold for LMJ is expected to be comparable to the current experiment one. FIG. 3 (color online): convective intensity thresholds as a function of the electron density in the corona, expected in our experiments, for SRS (red) and SBS (blue) (from [START_REF] Kruer | The physics of laser plasma interactions[END_REF]). For SBS, the detuning caused by the plasma expansion is more severe than by the electron density profile one.

In Fig. 4 are shown the shock-related data for a planar target. They are plotted as a function of time, t = 0 ns corresponding to the onset of the laser pulse. Fig. 4(a) shows the 2D self-emission image obtained from the GOI taken at t ~ 8 ns. The signal is circular with a diameter matching the SOP profile at the same time. The bright signal in SOP (Fig. 4b) seen at t ~ 1.6 ns coincides with the laser spike hitting the target and generating X-ray radiation from the ablation plasma. X-ray, as well as hot electrons generated by SRS or Two-Plasma Decay instabilities, propagate through the target and can ionize the material ahead of the shock wave changing of its refractive index. This produces a perturbation in VISAR image (Fig. 4c) with a slight drop of the signal at t ~1.6 ns and a complete blanking where the intensity is the strongest. This phenomenon was previously observed in other experiments using VISAR optical probe beam [29,30]. At t ~ 4.3 ns, the shock breakout into the quartz is accompanied by light emission recorded by the SOP diagnostic and a strong fringe shift is observed in the VISAR image. Due to the curvature of the shock, only one fringe, superimposed to weaker ghost fringes, is visible: the non-planar shock is strongest and fastest in the center, corresponding to the maximum laser intensity, and decreases strongly on the edges. The strong curvature of the shock front reflects most of the probe light outside of the collecting aperture of the diagnostic. At ~10.8 ns, the shock breaks out into vacuum, as evidenced by the loss of signal in both images. Calibration shots were performed prior to the experiment in order to calibrate the counts from SOP as a function of the velocity given by the VISAR. Consequently, when the fringes are absent due to the shock curvature, we can extract velocity information directly from the SOP. As the thermal emission is isotropic, SOP signal is indeed less sensitive to the shock curvature than the VISAR signal.

IV. 2D HYDRODYNAMIC SIMULATIONS AND DISCUSSION

The numerical simulations presented in this paper have been performed with CHIC, a 2D radiative ALE (Arbitrary Lagrangian Eulerian) hydrodynamic code [22]. For the interface reconstruction, we used a multi-material treatment of the cells based on the Moment of Fluid method [31,32], in order to correctly track the strong deformations of hemispheric targets during the experiment. The Fig. 4(d Given the error bars, numerical simulations are in fairly good agreement with experimental results. At early times, the calculated velocity seems slightly faster than what was observed in the experiment. This slight disagreement could be explained by the blanking of quartz just after shock breakout, making data analysis more difficult. The shock breakout time into vacuum at the rear side of the quartz window (t ≈ 11 ns), the spatial shape of shock breakout on the VISAR image (plain lines in Fig. 4c) as well as the shape of the self-emission on the GOI image (dashed line in Fig. 4a) are properly reproduced by the simulations. This means that, using the actual intensity distribution in the focal spot, the code correctly predicts the shock strength over the focal spot area as well as its 2D propagation. The experimental/numerical agreement in the shock dynamics and the velocity for planar target is better than 3 %, indicating a good numerical treatment of laser-target interaction and ablation physics. This gives a good confidence in the capability of reproducing shock propagation in a more complex geometry. This also indicates that the laser spot was well centered on the hemisphere. Table 5(c) summarizes the times of these four events, from the experiment and the simulation. Despite the complex geometry and target design, we observe a good agreement (better than 3%), not only for the absolute times, but also for the shock shape. This is highlighted by the red (green respectively) contour line shown in Fig. 5 It is also important to notice the shock emerges with a flat front at the centre of the hemisphere after its propagation. This is the result of the balance between the laser intensity and the target geometry. The laser intensity, thus the shock velocity [33], is maximum at the pole of the CH hemisphere while it decreases across the diameter where the laser interacts both at an increased incident angle and with a reduced CH thickness. The combination of these effects, which compensate themselves, straightens the shock front along its propagation into the hemisphere leading to a flat shock front near the Mo/SiO 2 interface. This is clearly observed on Fig. 6(A) and 6(B), which represent the density maps at the times corresponding to the events cited in Fig. 5 As the shock dynamics is reproduced in details, we can use our numerical simulations to infer the ablation pressure generated by the laser pulse. Just after the spike beam has reached its maximum, we obtain a value of (90 ± 20) Mbar and (120 ± 20) Mbar with planar and hemispherical targets respectively. Considering the 1D scaling law of the ablation pressure P abl in the regime of classical absorption, P abl ~ 57 (η abs I L /λ) 2/3 [34,35], where η abs is the laser absorption fraction, I L denotes the laser intensity in units of 10 15 W/cm 2 and λ is the laser wavelength in µm, we obtain, in our experimental conditions and assuming 80% absorption, an ablation pressure of 130 Mbar. This value agrees with the one inferred from numerical simulations in hemispherical geometry where the planar shock outlasts in the planar geometry. In the last case, the energy losses due to lateral conduction play an important role, reducing the actual ablation pressure. Our experimental results confirm that we are able to launch very strong shocks in the presence of a large plasma corona in realistic ICF conditions and that hemispherical targets generate higher ablation pressures. This is the consequence of the lateral energy flow, spreading away from the focal spot region, in the case of planar target, thus leading to a non-sustained shock. While, due to the geometry counterbalancing such lateral flow, hemispherical targets provide higher ablation pressures.

V. CONCLUSION

In summary, we have investigated the behavior of a shock produced by a laser spike at intensity around 1.5×10 15 W/cm 2 in the presence of a pre-plasma within conditions relevant to shock ignition, in planar and hemispherical geometries. The backscattered light (SBS and SRS) is relatively low, limited to ~ 5% in the focusing cone. 2D hydro-simulations reproduce the shock dynamics in the two cases, and correctly predict the timing and the shape of the shocks allowing to estimate an ablation pressure of 90 Mbar and 120 Mbar in planar and spherical geometry respectively. The increase of the shock velocity in hemispherical CH targets is due to reduced lateral conduction and energy losses, which become important in planar target. With hemispherical targets, the interplay between the target geometry and the spatial intensity distribution entails a final planar shock across the diameter, in spite of using a small focal spot generating an initially curved shock. This could be exploited in the polar direct drive geometry (i.e. with spherical capsule) in order to produce two strong flat shocks colliding at target center to possibly reaching the conditions for fuel ignition [11,36,37].

  measured: the light backscattered through the f/8 focusing cone was collected by the FABS (Full Aperture Backscattered) diagnostic. The 3ω light, backscattered around the focusing cone, was gathered by NBI (Near-Backscattered Imaging) diagnostics, providing the amount of stimulated Brillouin scattering (SBS) energy scattered between f/4 and f/2 apertures [26].

FIG. 1 (

 1 FIG. 1 (color online). (a) schematic drawing of the planar (up) and hemispherical (bottom) targets; (b) laser temporal pulse shape; (c) schematic of the laser focal spot on the hemisphere

FIG. 2 (

 2 FIG. 2 (color online): Time-resolved powers of SBS (in blue) and SRS (in red) measured in f/8 for a planar target (a) and a hemispherical target (b). The incident laser power (divided by 20) is also shown (dashed gray line).

  FIG. 4 (color online). Experimental shock propagation data obtained with (a) GOI, (b) SOP and (c) VISAR with a planar target. (d) shows extracted shock velocities from VISAR, SOP and calculated from CHIC.

Figure 5 (

 5 Figure 5(a) shows the VISAR image for a hemispherical target. The four specific events indicated on the figure relate to: the shock breakout into the quartz (A) and into vacuum (D) for the edge of the laser focal spot, outside the hemisphere; the shock breakout into the quartz (B) and into vacuum (C) for the focal spot center. Figure 5(b) displays a 2D image of the self-emission taken at t ~ 8 ns. The brightest round signal corresponds to self-emission in the axis of the hemisphere. It is centered on a weaker signal, attributed to the self-emission from the shock generated by the wings of the focal spot.

  FIG. 5 (color online). Experimental shock propagation data obtained with (a) VISAR and (b) GOI at t ~ 8 ns for a hemispherical target. Table (c) displays a comparison of data obtained from experiment and 2D hydrodynamic CHIC code for specific events indicated on the VISAR image in (a).

  (a). These simulations allow following the time evolution of the shock front. Because the wings of the laser spot are larger than the hemisphere, an early shock breakout into quartz (A) is observed at ~2.4 ns, before the breakout of the central flat shock (B) at 3.7 ns. The central flat shock propagates faster in the quartz and reaches the vacuum at ~ 7.1 ns (C). Finally, the annular slow shock front and the fast central shock are catching up at the rear face of the quartz at 12.2 ns (D) leading to a change of the shock front curvature faintly observed on VISAR image (Fig. 5(a) point D).

FIG. 6 (

 6 FIG. 6 (color online). Maps of density (in g/cc) at the four times corresponding to the events indicated on the VISAR image displayed in Fig. 5(a). The dashed lines represent the initial interface positions between the three layers CH/Mo/SiO2. The laser comes from the left.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the technical assistance of all LIL teams, and the CEA-Valduc target laboratory for providing high quality targets. This work was supported by the Institute Laser-Plasma (ILP) and within the framework of the EUROfusion Consortium via the ToIFE project. It has received funding from the Euratom research and training programme 2014-2018 under grant agreement N° 633053. The views and opinions expressed herein do not necessary reflect those of European Commission.

⎯⎯⎯⎯⎯⎯⎯⎯


Cédric Bonnafé
email: cedric.bonnafe@umontpellier.fr

A SURFACE OF DEGREE 24 WITH 1 440 SINGULARITIES OF TYPE D 4 by

Using the invariant algebra of the complex reflection group denoted by G 32 in the Shephard-Todd classification, we construct three irreducible surfaces in P 3 ( ) with many singularities: one of them has degree 24 and contains 1 440 quotient singularities of type D 4 .

Let µ D 4 (d ) denote the maximal number of quotient singularities of type D 4 that an irreducible projective surface of degree d in P 3 ( ) might have. Miyaoka [Miy] proved that µ D 4 (d ) 16 117 d (d -1) 2 . For d = 8, 16 or 24, this reads µ D 4 (8) 53, µ D 4 (16) 492 and µ D 4 (24) 1736. The main results of this note are that (1) µ D 4 (8) 44, µ D 4 (16) 352 and µ D 4 (24) 1440 and that (2) µ D 4 (8k ) 44k 3

for all k 1. For this, let [x 1 , x 2 , x 3 , x 4 ] be the polynomial ring over in 4 indeterminates with its usual grading and let

] is homogeneous, we denote by (f ) the projective hypersurface it defines, and by (f ) sing its reduced singular locus. If k is a natural number, we denote by f [k ] the homogeneous polynomial f (x k 1 , x k 2 , x k 3 , x k 4 ).

For proving the above results, we exhibit a particular homogeneous polynomial g of degree 8 such that the associated projective varieties (g ), (g [2]) and (g [3]) (which have respective degrees 8, 16 and 24) have many quotient singularities of type D 4 . The proof relies essentially on Magma computations that will be detailed in the next sections: we have decided to write the full Magma code in this arXiv version, so that the reader can check by himself the computations, but note that this code will not appear in the published version of this paper.

Our polynomial g is constructed using polynomial invariants of various finite subgroups of GL 4 ( ). Let W 1 be the subgroup of GL 4 ( ) generated by Finally, let W 3 denote the subgroup of GL 4 ( ) generated by

u 1 =    1 0 0 0 0 1 0 0 0 0 ζ 3 0 0 0 0 1    , u 2 =     ζ 3 +2 3 ζ 3 -1 3 ζ 3 -1 3 0 ζ 3 -1 3
ζ 3 +2 3 ζ 3 -1 3 0 ζ 3 -1 3 ζ 3 -1 3
ζ 3 +2 3 0 0 0 0 1     , u 3 =    1 0 0 0 0 ζ 3 0 0 0 0 1 0 0 0 0 1    and u 4 =     ζ 3 +2 3 1-ζ 3 3 0 1-ζ 3 3 1-ζ 3 3 ζ 3 +2 3 0 ζ 3 -1 3 0 0 1 0 1-ζ 3 3 ζ 3 -1 3 0 ζ 3 +2 3     .
Commentaries. The following facts are checked using [Magma], as explained below. Let Z(W i ) denote the center of W i . In all cases, it is isomorphic to a group of roots of unity acting by scalar multiplication. Then: (a) The group W 1 has order 48 and is isomorphic to the non-trivial double cover S4 of the symmetric group S 4 W 1 /Z(W 1 ). (b) The group W 2 has order 768, contains a normal abelian subgroup H of order 32 and W 2 /H S 4 . The group W 2 /Z(W 2 ) has order 192, but is not isomorphic to a Coxeter group of type D 4 . (c) The group W 3 is the complex reflection group denoted by G 32 in the Shephard-Todd classification [ShTo] (it has order 155 920). Recall that the group W 3 /Z(W 3 ) is a simple group of order 25 920 and is isomorphic to the derived subgroup of the Weyl group of type E 6 (i.e. to the derived subgroup of the special orthogonal group SO 5 ( 3 )).

Note that we have used the form implemented by Michel [Mic] in the Chevie package of Gap3. It contains the group W 1 as a subgroup, as well as a subgroup of diagonal matrices isomorphic to (µ 3 ) 4 , where µ d is the group of d -th roots of unity.

If λ = (λ 1 λ 2 λ 3 λ 4 ) is a partition of 8 of length at most 4, we denote by Ω - λ (resp.

Ω + λ ) be the orbit of the monomial x λ 1 1 x λ 2 2 x λ 3 3 x λ 4
4 under the action of W 1 (resp. the symemtric group S 4 ) and we set

m λ = m ∈Ω λ m for ∈ {+, -}. Then m +
λ is the symmetric function traditionnally denoted by m λ . If all the λ i 's are even, then m - λ = m + λ but note for instance that

m + 611 = m - 611 = x 6 1 x 2 x 3 + x 6 1 x 2 x 4 -x 6 1 x 3 x 4 + x 1 x 6 2 x 3 -x 1 x 6 2 x 4 + x 6 2 x 3 x 4 +x 1 x 2 x 6 3 + x 1 x 6 3 x 4 -x 2 x 6 3 x 4 -x 1 x 2 x 6 4 -x 1 x 3 x 6 4 -x 2 x 3 x 6 4 . Now, let g = m - 8 -6m - 62 -60m - 611 + 2 240m - 521 -14m - 44 + 10 180m - 431 + 40 412m - 422 -23 440m - 4211 + 111 980m - 332 + 154 704m - 2222 .
By construction, m - λ is invariant under the action of W 1 and so g is invariant under the action of W 1 S4 . One can check with Magma the following facts:

Proposition 1. If 1 k 3, then the polynomial g [k ] is invariant under the action of W k .
Theorem 2. The homogeneous polynomial g satisfies the following statements: (a) (g ) is an irreducible surface of degree 8 in P 3 ( ) with exactly 44 singular points which are all quotient singularities of type

D 4 . (b) If k 1, then (g [k ]
) is an irreducible surface of degree 8k , whose singular locus has dimension 0 and contains at least 44k 3 quotient singularities of type D 4 . (c) (g [2]) is an irreducible surface of degree 16 with exactly 472 singular points: 24 quotient singularities of type A 1 , 96 quotient singularities of type A 2 and 352 quotient singularities of type D 4 . (d) (g [3]) is an irreducible surface of degree 24 in P 3 ( ) with exactly 1 440 singular points which are all quotient singularities of type D 4 . The automorphism group of (g [3]) contains at least 25 920 elements and acts transitively on the 1 440 singular points.

Remark 1. It turns out that we did not find the polynomial g directly: we found first g [3] by looking at invariants of degree 24 of W 3 G 32 , following ideas of Barth [Bar] and Sarti [START_REF] Sarti | Pencils of symmetric surfaces in P 3[END_REF], [START_REF] Sarti | Symmetric surfaces with many singularities[END_REF], [START_REF] Sarti | Symmetrische Flächen mit gewöhnlichen Doppelpunkten[END_REF] (for constructing the Barth sextic with 65 nodes and, for instance, the Sarti dodecic with 600 nodes), who used invariants of Coxeter groups of type H 3 and H 4 . See also [Bon] for details about the method used for finding g .

Remark 2. Note that g has coefficients in but the singular points of (g ), (g [2]) and (g [3]) have coordinates in various field extensions of , and most of the singular points are not real (at least in this model).

So let us start by defining the polynomial g and the three groups W i in MAGMA and checking the facts stated in Proposition 1 and Commentaries. These data, together with the definition of the fields K and L as well as a function ProjectiveOrbit used for computing orbits of various points under the action ot the groups W i , are all contained in a file g32-article.m, whose content is given in the Appendix. Note that we will first work with the projective space over , and the polynomial g will be defined over . We now turn to the study of the singularities of the varieties (g [i ]) for i ∈ {1, 2, 3}. Note the following fact, that will be used further: Lemma 3. If 1 i < j 4, then the closed subscheme of P 3 ( ) defined by the ideal 〈g ,

∂ g ∂ x i , ∂ g ∂ x j 〉 has dimension 0.
This is checked thanks to the following code:

> dg:=[Derivative(g,i) : i in [1..4]]; > pairs:=[[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]; > time [Dimension(Scheme(P3,[g,dg[i[1]],dg[i[2]]]
)) : > i in pairs]; [ 0, 0, 0, 0, 0, 0 ] Time: 20.510 // about 20 seconds

Degree 8

The Magma computations leading to the proof of the statement (a) of Theorem 2 are detailed in this section. Along these computations, the following facts are obtained (here, denotes the open subset of P 3 ( ) defined by x 1 x 2 x 3 x 4 = 0):

Proposition 4. We have:

(a) dim (g ) sing = 0, so (g ) is irreducible. (b) (g ) sing is contained in . (c)
The group W 1 has 3 orbits in (g ) sing , of respective length 8, 12 and 24.

We first check that (g ) sing has dimension 0 and is contained in . In particular, all the singular points are contained (for instance) in the affine chart (g ) aff defined by "x 4 = 0". We will make all the remaining computations in this affine chart (and extend the scalars to the field K ):

> Zgaff:=AffinePatch(Zg,1); > ZgaffK:=ChangeRing(Zgaff,K); > ZgaffKsing:=SingularSubscheme(ZgaffK); > irr1:=IrreducibleComponents(ZgaffKsing);

> irr1:=[ReducedSubscheme(i) : i in irr1]; > Set([Degree(i) : i in irr1]); { 1 } > ZgK:=ChangeRing(Zg,K); > sings1:=[[Coordinates(i) : i in RationalPoints(j)] : > j in irr1]; > sings1:=&cat sings1; > sings1:=[ZgK ! (i cat [1]) : i in sings1]; > # sings1; 44
The result of the command Set([Degree(i) : i in irrg]) shows that all singular points have coordinate in K , and the last command shows that the number of singular points in (g ) is equal to 44. We then determine the W 1 -orbits in (g ) sing and check that they are all quotient singularities of type D 4 by picking up one point in each orbit. Note that the points in the W 1 -orbit of cardinality 8 are the only real singular points of (g ). Figure 1 shows part of the real locus of (g ).

> orbits

Degree 8k

Let denote the open subset of P 3 ( ) defined by x 1 x 2 x 3 x 4 = 0 and let σ k : P 3 ( ) →

P 3 ( ), [x 1 ; x 2 ; x 3 ; x 4 ] → [x k 1 ; x k 2 ; x k 3 ; x k 4 ]. The restriction of σ k to a morphism → is an étale Galois covering, with group (µ k ) 4 /∆µ k (here, ∆ : µ k → (µ k ) 4 is the diagonal embed- ding). We have (g [k ]) = σ -1 k ( (g )).
Let us first prove that (g [k ]) is irreducible. We may assume that k 2, as the result has been proved for k = 1 in the previous section. Recall that ) is contained in

∂ g [k ] ∂ x i = k x k -1 i ( ∂ g ∂ x i • σ k ),
{p 1 , p 2 , p 3 , p 4 } ∪ i = j σ -1 k ( i , j ) ,
where p i = [δ i 1 ; δ i 2 ; δ i 3 ; δ i 4 ] (and δ i j is the Kronecker symbol) and i , j is the subscheme of P 3 ( ) defined by the ideal 〈g ,

∂ g ∂ x i , ∂ g ∂ x j 〉
(and which has dimension 0 by Lemma 3). Since σ k is finite, this implies that (g [k ]) sing has dimension 0, so (g [k ]) is irreducible. Now, σ k : → is étale and the singular locus of (g ) is contained in (see Proposition 4(b)). Therefore, the 44 singularities of (g ) lift to 44k 3 singularities in (g [k ]) ∩ of the same type, i.e. quotient singularities of type D 4 . This proves the statement (b) of Theorem 2.

Note that, for k = 2, 3 and 4 (and maybe for bigger k ) we will prove in the next sections that (g [k ]) contains singular points outside of .

Degree 16

Using the morphism σ 2 defined in the previous section, we get that (g [2]) ∩ has exactly 352 singular points, which are all quotient singularities of type D 4 . We now need to determine the singularities which are not contained in . So let be the complement of in P 3 ( ). We now study the singularity of (g [2]) at the points of the orbit of cardinality 96. It turns out that that the command IsSimpleSurfaceSingularity takes too much time to get a conclusion, so we will investigate properties of the equation of (g [2]) in a neighborhood of the first point p (in Magma list orbits [2]). We work in the affine chart x 3 = 0 (where p = [ξ 1 ; ξ 2 ; ξ 3 ; ξ 4 ] lives), and we denote by (x , y , z ) the coordinates of the affine chart x 3 = 0 equal to (x 1 /x 3 + ξ 1 /ξ 3 , x 2 /x 3 + ξ 2 /ξ 3 , x 4 /x 3 + ξ 4 /ξ 3 ) and we set

f (x , y , z ) = g [2](x , y , 1, z ).
If j 0, we denote by f j the homogeneous component of f of degree j . As p ∈ (g [2]) sing , we have The last command shows that there exists a linear change of the coordinates (x , y , z ) → (x , Y , Z ) such that f 2 might be transformed into Y 2 + Z 2 . By standard arguments, this proves that p is a quotient singularity of type A k , for some k 2, which can be obtained as the Milnor number of f : however, Magma cannot compute this Milnor number in a reasonable amount of time and we need to copy the polynomial f in the software Singular [DGPS] to compute this Milnor number (!): we obtain 2. So p is a quotient singularity of type A 2 . This completes the proof of statement (c) of Theorem 2.

f 0 = f 1 = 0. > p:=orbits2[2][1]; > A3L<x,y,z>:=AffineSpace(L,3); > cop:=Coordinates(p); > f:=Evaluate(g2L,[x+cop[1],y+cop[2],1,z+cop[4]]); > cof:=Coefficients(f); > mof:=Monomials(f); > l:=# mof; > f2:=&+ [cof[i] * mof[i] : i in [1..l] | > Degree(mof[i])
Figure 2 shows part of the real locus of (g ).

Degree 24

Using the morphism σ 3 defined in Section 2, we get that (g [3])∩ has exactly 44×3 3 = 1 188 singular points, which are all quotient singularities of type D 4 . Let us compute (g [3]) sing ∩ : The last command shows that (g [3]) sing ∩ contains 252 points. We now show that they are all defined over K : The proof of statement (d) of Theorem 2 is complete: Figure 3 gives partial views of its real locus.

> Zg3:=Surface(P3,
> Zg3K:=ChangeRing(Zg3,K); > sings3H:=[[Coordinates(i) :

Complements

Remark 3. From Section 2, we deduce that (g [4]) sing has 2 816 quotient singularities of type D 4 lying in the open subset and it can be checked that it has 480 other singular points not lying in , for which we did not determine the type.

Remark 4. After investigations in the invariant rings of several irreducible primitive complex reflection groups (there are 34 such groups, denoted by G i with 4 i 37 in Shephard-Todd classification [ShTo]), we have also been able to construct curves with many singularities. For example:

• Using the reflection group W = G 24 , we have obtained a cuspidal curve 14 of degree 14 in P 2 ( ) with exactly 42 cusps (all lying in a single W -orbit). Note that W /Z(W ) ⊂ Aut( 14) has order 168 and is isomorphic to GL 3 ( 2 ). • Using the reflection group W = G 26 , we have obtained a curve 18 of degree 18 in P 2 ( ) with 72 cusps and 12 nodes (these are the two W -orbits of singular points). Note that W /Z(W ) ⊂ Aut( 18) has order 216.

Also, other singular surfaces have been obtained. For example:

• Using the reflection group W = G 29 (note that W /Z(W ) has order 1 920), we have obtained:

-a surface of degree 8 in P 3 ( ) with 160 nodes, all belonging to the same Worbit. Recall that the Endraß octic [End] has degree 8 and 168 nodes (and its automorphism group has order 16) and the Sarti octic [START_REF] Sarti | Pencils of symmetric surfaces in P 3[END_REF] has 144 nodes. -a surface of degree 8 in P 3 ( ) with 20 singular points of multiplicity 3 and Milnor number 11, all belonging to the same W -orbit. • Using the reflection group W = G 31 , we have obtained a surface 20 of degree 20 in P 3 ( ) with 1 920 nodes, all lying in the same W -orbit. Note that W /Z(W ) ⊂ Aut( 20) has order 11 520) and recall that the Chmutov surface [Chm] of degree 20 has 2 926 nodes. Details will appear in a forthcoming paper [Bon]. [(1-zeta3)/3,(zeta3+2)/3, 0,(zeta3-1)/3 ], [ 0, 0, 1, 0 ], [(1-zeta3)/3,(zeta3-1)/3, 0,(zeta3+2)/3 ]]); W3:=MatrixGroup<4,K | [u1,u2,u3,u4]>;

  Let ζ 3 (resp. ζ 4 ) be a primitive third (resp. fourth) root of unity. Let W 2 be the subgroup of GL 4 ( ) generated by

  =ChangeRing(g,K); > [gK^i eq gK : i in Generators(W1)]; [ true, true, true ] eq g3K : i in Generators(W3)];[ true, true, true, true ]

>

  Zg:=Surface(P3,g); > Zgsing:=SingularSubscheme(Zg); > Dimension(Zgsing); 0 > H:=Scheme(P3,x1 * x2 * x3 * x4); > Dimension(Intersection(Zgsing,H)); -1

FIGURE 1 .

 1 FIGURE 1. Part of the real locus of (g )

FIGURE 2 .

 2 FIGURE 2. Part of the real locus of (g [2])

FIGURE 3 .

 3 FIGURE 3. Part of the real locus of (g [3])

  x1 * x3^4 * x4^3 -x1 * x3^3 * x4^4 + x2^4 * x3 * x4^3 -x2^3 * x3^4 * x4 -x1 * x2^4 * x4^3 + x1 * x2^3 * x3^4 -x1 * x2^3 * x4^4 -x2^3 * x3 * x4^4 -x2 * x3^4 * x4^3 -x2 * x3^3 * x4^4 + x2^4 * x3^3 * x4 -x1^3 * x3 * x4^4 ) + 40412 * (x1^4 * x2^2 * x3^2 + x1^4 * x2^2 * x4^2 + x1^4 * x3^2 * x4^2 + x1^2 * x2^4 * x3^2 + x1^2 * x2^4 * x4^2 + x1^2 * x3^4 * x4^2 + x1^2 * x3^2 * x4^4 + x1^2 * x2^2 * x3^4 + x1^2 * x2^2 * x4^4 + x2^4 * x3^2 * x4^2 + x2^2 * x3^4 * x4^2 + x2^2 * x3^2 * x4^4) -23440 * (x1^4 * x2^2 * x3 * x4 -x1^4 * x2 * x3^2 * x4 -x1^4 * x2 * x3 * x4^2 + x1^2 * x2 * x3^4 * x4 + x1^2 * x2 * x3 * x4^4 -x1^2 * x2^4 * x3 * x4 -x1 * x2^4 * x3 * x4^2 -x1 * x2^2 * x3^4 * x4 + x1 * x2^2 * x3 * x4^4 -x1 * x2 * x3^4 * x4^2 + x1 * x2 * x3^2 * x4^4 + x1 * x2^4 * x3^2 * x4 ) + 111980 * (x1^3 * x2^3 * x3^2 -x1^3 * x2^3 * x4^2 + x1^3 * x2^2 * x3^3 -x1^3 * x2^2 * x4^3 -x1^3 * x3^3 * x4^2 + x1^3 * x3^2 * x4^3 + x1^2 * x2^3 * x3^3 + x1^2 * x2^3 * x4^3 -x1^2 * x3^3 * x4^3 -x2^3 * x3^3 * x4^2 -x2^3 * x3^2 * x4^3 + x2^2 * x3^3 * x4^3) + 154704 * x1^2 * x2^2 * x3^2 * x4^2;g2:=Evaluate(g,[x1^2,x2^2,x3^2,x4^2]); g3:=Evaluate(g,[x1^3,x2^3,x3^3,x4^3]

  We now determine the W 2 -orbits in (g ) sing ∩ : there is one W 2 -orbit of cardinality 24 (and we check that its elements are quotient singularities of type A 1 ) and one of cardinality 96.

	The last command shows that (g ) sing ∩	contains 120 points. We now check that all
	the singular points contained in	have coordinates in L :
	> Zg2L:=ChangeRing(Zg2,L);
	> sings2H:=[[Coordinates(i) :
	>	i in RationalPoints(ChangeRing(j,L))] :
	>	j in irr2H];
	> sings2H:=&cat sings2H;
	> sings2H:=[Zg2L ! i : i in sings2H];
	> # sings2H;
	120
	> orbits2:=[];
	> test:=sings2H;
	> W2L:=ChangeRing(W2,L);
	> while (# test) gt 0 do
	while>	orb:=ProjectiveOrbit(W2L,test[1]);
	while>	orb:=[Zg2L ! Coordinates(i) : i in orb];
	while>	orbits2:=orbits2 cat [orb];
	while>	test:=[i : i in test | (i in orb) eq false];
	while> end while;
	> [# i : i in orbits2];
	[ 24, 96 ]
	> IsSimpleSurfaceSingularity(orbits2[1][1]);
	true A 1
	> Zg2:=Surface(P3,g2);
	> Zg2sing:=SingularSubscheme(Zg2);
	> H:=Scheme(P3,x1 * x2 * x3 * x4);
	> Zg2singH:=Intersection(Zg2sing,H);
	> time Zg2singH:=ReducedSubscheme(Zg2singH);
	Time: 11.300
	> irr2H:=IrreducibleComponents(Zg2singH);
	> # irr2H;
	18
	> &+ [Degree(i) : i in irr2H];
	120

The author is partly supported by the ANR (Project No ANR-16-CE40-0010-01 GeRepMod).

Acknowledgements. This paper is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2018 semester. The hidden computations which led to the discovery of the polynomial g were done using the High Performance Computing facilities of the MSRI.

I wish to thank warmly Alessandra Sarti, Oliver Labs and Duco van Straten for useful comments and references and Gunter Malle for a careful reading of a first version of this note. Figures were realized using the software SURFER [Sur].

Appendix

This appendix gives a copy of the file g32-article.m loaded at the beginning of the computations. It contains the data of the polynomial g , the fields K and L , the three groups W i and the function ProjectiveOrbit which is used throughout the computations (it is certainly not the most efficient code, but it is sufficient for our purpose). Note that W 1 and W 3 are defined over the field K , while W 2 is defined over the field L .