Jung-Hoon Lee

Hyeong-Pil Kim

Jong-Hoon Won
email: jh.won@inha.ac.kr

GNSS cloud-data processing technique for jamming detection and localization

In this paper, assuming that a jammer is located in an area where a large number of low-cost GNSS receivers are densely distributed over and their received signals are gathered into a central processing unit like a cloud-data, we propose a 2-dimensional time-frequency correlation method for the cloud-data which can determine the type of jamming signal and estimate the jammer's position. This method is applied for examining the similarity between arbitrary two signals acquired from different receivers, thereby discriminating the presence and the type of a jamming signal in the time-frequency domain and also estimating the jammer's position. The availability of the proposed method is examined by a numerical simulation, where 15 GNSS receivers are randomly placed at the certain distances that range from 100 [m] to 300 [m] from the jammer, so that the jamming to signal ratio(J/S) at the receiver varies as distance increases. Finally, we analyze the results of the time-frequency correlation method to confirm that the jamming signal can be detected and the position of the jammer can be estimated. 1

INTRODUCTION

For the modern human life, the Global Navigation Satellite System (GNSS) is an essential element to provide accurate location-based services with worldwide coverage. However, the satellite signal used in GNSS is vulnerable to jamming because intrinsically it has a very low signal power level on the surface of the earth by being transmitted from satellites at an altitude of about 20,000 [km]. In order to solve this problem, a lot of research has been performed, and recently, as the price of the GNSS receiver gradually decreases due to the development of technology, methods using data acquired from a large number of GNSS receivers have been proposed.

In [START_REF] Lindstrom | GNSS Interference Detection and Localization using a Network of Low-cost Front-End Modules[END_REF] and [START_REF] Isoz | Interference Detection and Localization in the GPS L1 Band[END_REF], researchers proposed an idea for detecting the interference signal and estimating the location of the interference source through a number of independent monitoring stations consisting of low-cost antennas and front-end modules distributed over a certain region. Here, the interference signal was detected based on the automatic gain control (AGC) measurement, and the position of the interference source was estimated using the AGC output and the time difference of arrival (TDOA) between the signals of the monitoring stations. Experiments using 4 monitoring stations showed that the result of the TDOA based localization had an about 80% reduced estimation error than that of the AGC output-only based method.

In [START_REF] Scott | J911: Fast Jammer Detection and Location Using Cell-Phone Crowd-Sourcings[END_REF] and [START_REF] Scott | J911: The Case for Fast Jammer Detection and Location Using Crowdsourcing Approaches[END_REF], the author proposed so-called J911 system that refers to the U.S. E911 system that connects 911 calls to the appropriate public safety answering point (PSAP) including location information. The J911 system was described to identify the constant envelope (CE) type jamming signal and estimate the jammer's location by using jamming-to-noise (J/N) observations and location infor-mation acquired from a large number of smartphones. Here, through the simulation of cloud situation with 1,000 smartphones distributed per 1km 2 area, it was shown that the higher the density of the smartphone that provides the measurements in a certain space, the higher the accuracy of the jammer localization increased.

International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France

In [START_REF] Strizic | Crowdsourcing GNSS Jammer Detection and Localization[END_REF], authors proposed cloud sourcing based method that used carrier-to-noise density (C N 0 ⁄ ) and AGC information obtained from many smartphones to distinguish between the jamming and the spoofing signals and estimate the jammer position. In the experiment to verify the method, satellite information from 15 smartphones was acquired using the self-developed android application software, and the AGC level was measured via the SiGe GN3S Sampler. Then they estimated the moving jammer position through the TDOAs.

In this paper, we propose a cloud-data based jamming detection and localization method. This method uses a 2dimensional time-frequency correlation technique that is in between 2 of cloud-data. It is assumed that we can gather and utilize the intermediate frequency (IF) signal data from many low-cost GNSS receivers distributed in a certain region. Discrimination of jamming signal type uses the fact that the result of the correlation between the cloud-data shows different characteristics with respect to power, TDOA, and frequency variation. Then the position of the jammer is estimated using the obtained TDOA information. The availability of the proposed method is verified by a numerical simulation. It is noted that in this paper the AGC effect that may induces the saturation for high power jamming signal at the RF front-end is assumed to be not existed.

JAMMING SCENARIO

2.1 Schematic depiction of jamming scenario Figure 1 is a schematic depiction of a scenario in which a jammer transmitting weak jamming signals is located between a number of low-cost GNSS receivers distributed in a dense density over a large area, and a processing procedure of the receiver data. The area where the jammer interferes with the receiver is set within a radius of 300 [m], so that in the jamming area, we assume that the amplitude of the jamming signal is larger than the amplitude of the satellite signal. All receivers distributed over a large area continuously transmit the data of the received signal to the cloud server and the server monitors the jamming signal via the time-frequency correlation between the specific data. If the jamming signal is detected, the type of jamming signal, received jamming power, and TDOA information between the two signals are obtained by analyzing the result of the correlation method in the time-frequency domain. As a result, we estimate the position of the jammer by applying a position estimation method suitable for the type of specific jamming signal.

Types of jamming signals applied to the scenario

The jamming signal blocks the reception of the GNSS signal by deliberately emitting electromagnetic radiation to disrupt user receivers by reducing the signal-to-noise level at the user's receiver [START_REF] Dovis | GNSS Interference Threats and Countermeasures[END_REF]. Therefore, various types of signals can be classified as jamming signals.

Chirp √2𝑃 𝑖 𝑐𝑜𝑠[2𝜋 (𝑓 0 + 𝑘 2 𝑡 𝑐ℎ𝑖𝑟𝑝 ) 𝑡]
In this section, we explain 5 types of jamming signals dealt with in this paper and present the modeled equations in Table 1.

• Types of jamming signals

(1) Continuous Wave Interference (CWI) -It is a sinusoidal wave, single-tone frequency signal within the GNSS frequency band.

(2) Matched Spectrum Interference (MSI) -It uses the same spreading code as the GNSS signal and shows the same frequency characteristics.

(3) Band-Limited White Gaussian Noise Interference (BLWI) -It has the white gaussian noise characteristic within the limited band, ideally the power spectral density is constant in frequency domain.

(4) Pulse Interference (Pulse) -It has a large power in a very short time, and the form of pulse signal is generated according to the duty cycle, the pulse repetition frequency, and the pulse width.

(5) Chirp Interference (Chirp) -As a continuous waveform signal with time-varying frequency modulation, it is technically easy to International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France implement and has good jamming characteristics, so it is mainly used in In-car Jammer.

JAMMING DETECTION AND LOCALIZA-TION USING TIME-FREQUENCY CORRELA-TION METHOD

Correlation is generally applied between two data acquired from different receivers. Therefore, in this section, we explain the time-frequency correlation method for two receivers located at different location as shown in Figure 2. where 𝑟 𝐴 and 𝑟 𝐵 are received signal for receiver A and B, respectively, 𝜏 𝑆,𝐴 and 𝜏 𝐽,𝐴 are time delay of satellite and jamming signal for receiver A, respectively, 𝜏 𝑆,𝐵 and 𝜏 𝐽,𝐵 are time delay of satellite and jamming signal for receiver B, respectively, and 𝜂 𝐴 and 𝜂 𝐵 are noise component for receiver A and B, respectively.

Jamming detection using time-frequency correlation method

If the center frequency of the jamming signal differs from the satellite signal or continuously changes like the chirp signal, the frequency value with the largest amplitude will be different from the center frequency of the satellite signal in the time-frequency domain 2-D image. Frequency variation can also occur for fast-moving jammers, so the It is also possible to detect the jamming signal by using signal correlation in the time domain. When the two IF signals in which the satellite signal and the jamming signal have different time delays are correlated in the time domain, the two correlation peaks exist at the index having 𝜏 𝑆,𝐵 -𝜏 𝑆,𝐴 and 𝜏 𝐽,𝐵 -𝜏 𝐽,𝐴 . Although the correlation form of the satellite signal is well known, the correlation form of the jamming signal shows different characteristics depending on the jamming signal type. By analyzing this feature, it is possible to discriminate the presence and the type of jamming signal. Jamming signal detection using correlation method is verified for availability through the simulation in Section 4.2.

Jamming source localization using timefrequency correlation method

In the jamming scenario covered in this paper, the correlation peak value of the jamming signal is generally larger than that of the satellite signal for receivers placed in or near the jamming region. It is possible to obtain the TDOA information of the jamming signal between the receivers by analyzing the correlation peak. If we acquire this TDOA information from a large number of receivers, the position of the jammer can be estimated. It is also International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France possible to estimate the jammer position roughly by using the correlation of time-frequency domain 2-D images obtained from a lot of receivers. This exploits the fact that the 2-D image obtained from a receiver close to the jammer has a higher amplitude than a far-away receiver. For example, if a large number of cloud-data obtained from region 'A' shows a 2-D image having a higher amplitude than that of region 'B', it can be seen that the jammer is located in the vicinity of region 'A'. Jamming source localization using correlation method is verified for availability through simulation in Section 4.3.

SIMULATION RESULTS ANALYSIS

Jamming scenario implementation

In the simulator, we assume that jammer can generate the aforementioned five jamming signals. In order to implement cloud-data, multiple receivers were arbitrarily placed at the certain distances that range from 100 [m] to 300 [m] from the jammer. Considering the free space propagation loss, we set the J/N to 0 [dB] at 100 [m] from the jammer. Thus, all receivers in the jamming region indicate the certain values of the J/N that are smaller than 0, while J/S of receivers are positive values. In order to process IF signal data, the sampling frequency is set to 28.57 [MHz] and the integration time of signal correlation is set to 1 [ms] for receivers. In addition, the center frequency of IF signal is 1.4 [MHz]. Besides this for simplicity, we assume that the setting and time of all receivers are synchronized.

Simulation result analysis of jamming detection

Figure 4 shows the plots of the generated 2-D images of received signals for the five jamming signal types. As mentioned in Section 3.2, jamming can be detected using the characteristics of the FFT result. In order to generate each vertical slice of the 2-D image, we use FFT result of 285 samples of IF jamming signal. The FFT result is filtered by the hamming-window and there is 10 samples delay between two adjacent vertical slices.

In Figure 4(a), the yellow point which is the peak amplitude stays on the 1.4 MHz because the CWI jamming signal is a continuous sinusoidal signal of constant frequency.

In Figure 4(b), since the MSI jamming signal is the signal that uses the same spreading code as the existing GNSS signals, the FFT result is also spread. Therefore, in order to detect jamming signal through the 2-D image of received signal, we should note that the amplitude of combined signal containing jamming signal and satellite signal with noise is slightly greater than that of satellite signal with noise. Thus, the 2-D image of combined signal would be colored more brightly than that of satellite signal. However, it cannot be easy to sense the change of 2-D image color.

In Figure 4(c), the 2-D image show random color with constant bandwidth because the BLWI jamming signal is white noise having a limited bandwidth. This makes the color of 2-D image discontinuous, which is different from the 2-D image with no jamming signal.

In Figure 4(d), since the pulse jamming signal radiates a signal of high power during a short period of time, the FFT The chirp jamming signal is a continuous sinusoidal signal like CWI, but frequency changes according to time elapse. Therefore, in Figure 4(e), the peak amplitude of the FFT result also varies with time. Fortunately, this remarkable characteristics of 2-D image of chirp help us to detect jamming signal easily.

As described in Section 3.2, we can also detect jamming signal by using signal correlation in the time domain.

Figure 5 shows the correlation results of two received signals from different receivers for the five jamming signal types respectively. The peak point means the highest value of the correlation result.

In Figure 5(a), the correlation result according to time comes out in the form of period, because the CWI jamming signal is a continuous sinusoidal signal. The result can help us to distinguish the CWI jamming signal from the various types of jamming signals. According to this characteristic, the estimation result of the jammer's position obtained by using the correlation of CWI signals can be easily wrong compared to the true jammer's position. It will be explained in Section 4.3.

In Figure 5(b), since MSI is very similar to the existing GNSS signal, it is difficult to detect jamming signal through the correlation in the time domain for MSI. However, assuming that the difference of doppler effect between two receivers is large for the satellite signals and also the amplitude of jamming signal is greater than that of satellite signal, the correlation result between the satellite signals may have small peak. Under these assumption, we can only obtain the correlation peak of MSI, and then detect it.

In Figure 5(c), the correlation result of the BLWI signals has a sharp peak compared to the result of spreading code correlation of GNSS signal.

In Figure 5(d), as mentioned earlier, since pulses radiate a signal of high power during a short period of time, the correlation value of the time domain has a large peak at a constant period.

In Figure 5(e), the correlation result of the chirp signals shows similar pattern as that of the MSI signals. However, if the time used for correlation is longer than the sweep period, the correlation result has one main lobe with side lobes that has smaller peak than main lobe. This can be similar to the result of the pulse signals in which peaks of the same size are periodically displayed.

As can be seen, plots in Figure 5 do not present much information relative to the type of jamming signal. We can only find that CWI and pulse signal have specific correlation results. However, since the correlation peak of jamming signals exists at certain time index that is different from that of correlation peak of satellite signals, we can recognize not only the presence of jamming signal, but also TDOA information. This information will be explained in Section 4.3 for jammer localization.

It is determined whether or not a jamming signal is present through correlation between IF signals collected from a large number of receivers. Signal correlation can be performed regardless of the presence and type of jamming signal because it uses two received signals. We use the TDOA information that is estimated from the correlation result of two received signals to estimate the jammer position. This information is presented in Figure 6 by plotting Figure 5 with limited time scale (7 microseconds) for the five jamming signal types respectively. The limited time scale is really short because the sampling frequency of receiver is high enough to obtain high resolution of TDOA.

In Figure 6, every jamming signal except CWI has one main peak that contains TDOA information on the time axis. However, in case of CWI, the sinusoidal wave characteristic prevent the correlation result from containing the true main peak with true TDOA information. Thus, it can be hard to estimate the jammer position with signal correlation result of CWI signals, while other types can be used for jammer localization.

In simulation, the true jammer position was set to be located at the coordinates (0, 0), and the 15 receivers were randomly placed at the certain distances that range from 100 [m] to 300 [m] from the jammer. With the TDOA information, the position was estimated by using the nonlinear least squares estimation (NLSE) method. As a result of the simulation that is summarized in Figure 7, the estimated position of the jammer had about 10.2 [m] error from the true jammer location.

Figure 7 -The results of jammer position estimation by using proposed method

We proved that it is also possible to estimate the position of the jammer by using the proposed method. If we generate more receivers and the amount of cloud-data increases, the accuracy of jammer position estimation will increase.

CONCLUSIONS

In this paper, assuming that a jammer with a weak jamming signal power is located in between a large number of lowcost GNSS receivers distributed at dense density in a wide area, we proposed a 2-dimensional time-frequency correlation method that is to ideally applied to detect and localize the jamming source by using data obtained from multiple receivers through cloud-data computing. The availability of the proposed method was examined by simulation. By applying the time-frequency correlation method composed of two correlations, it is proved that different characteristics can be seen according to the type of jamming signals, therefore five jamming signals can be detected and classified.

Through the jamming scenario where 15 receivers were arranged, it was proved that the jammer position estimation is possible by using the TDOA information of the jamming signals obtained as the result of the signal correlation between the data of receivers.

As a result, by using the time-frequency correlation method which was proposed in this paper, we can discriminate the presence and the type of jamming signal, and estimate the position of the jamming signal source. Furthermore, if we could monitor a huge amount of the cloud-data simultaneously, it will be helpful for the jamming signal monitoring for a wide area.

Figure 1 -

 1 Figure 1 -Schematic depiction of jamming scenario and processing procedure of the receiver data

3. 1

 1 Time-frequency correlation method In order to detect jamming signals and estimate the jammer's position, the time-frequency correlation method proposed in this paper consists of two correlations as follows -2-D image correlation in the time-frequency domain -Signal correlation in the time domain • 2-D image correlation in the time-frequency domain This correlation is to compare the 2-D images of the timefrequency domain generated with the IF signal data acquired at each receiver. Figure 3 outlines the method of generating a 2-D image in the time-frequency domain for a single receiver. If the current time is 0, the preset signal length is 𝑛 samples, and 10 samples of signal correspond to time 𝑇 𝑠 , the 2-D image can be obtained by applying fast fourier transform (FFT) to the IF signals over time. At this time, the frequency bin is expressed in other colors according to the magnitude of the amplitude. • Signal correlation in the time domain This method means the correlation between the two IF signals received at two different receivers. Referring to Figure 2, the equation representing this correlation is given as follows R(τ) = 𝑟 𝐴 * 𝑟 𝐵 = {𝑆(𝑡 -𝜏 𝑆,𝐴 ) + 𝐽(𝑡 -𝜏 𝐽,𝐴 ) + 𝜂 𝐴 } (1) * {𝑆(𝑡 -𝜏 𝑆,𝐵 ) + 𝐽(𝑡 -𝜏 𝐽,𝐵 ) + 𝜂 𝐵 }

Figure 2 -Figure 3 -

 23 Figure 2 -Jamming scenario for two receivers in another location

Figure 4 -

 4 Figure 4 -Generated 2-D images for jamming signals. (a) CWI (b) MSI (c) BLWI (d) Pulse (e) Chirp

Figure 5 -

 5 Figure 5 -Correlation results of two received signals. (a) CWI (b) MSI (c) BLWI (d) Pulse (e) Chirp

Figure 6 -

 6 Figure 6 -TDOA information of signal correlation results. (a) CWI (b) MSI (c) BLWI (d) Pulse (e) Chirp

Table 1 -Modeling of jamming signals Type Signal Modeling Note

 1

	CWI √2𝑃 𝑖 𝑐𝑜𝑠 (2𝜋𝑓 𝑖 𝑡 + 𝜃 𝑖 )	𝑃 𝑖 : Jamming power 𝑓 𝑖 : Jamming signal
		frequency
		𝜃

𝑖 : Jamming signal phase 𝐶(𝑡) : Spreading code 𝑛(𝑡) : White gaussian noise 𝑟𝑒𝑐𝑡(𝑡, 𝑑, 𝑟): Pulse with duration 𝑑, repetition frequency 𝑟 at time 𝑡 𝑓 0 : initial frequency 𝑘 : Chirp rate 𝑡 𝑐ℎ𝑖𝑟𝑝 : 0 ≤ 𝑡 𝑐ℎ𝑖𝑟𝑝 ≤ 𝑇 𝑠𝑤 (𝑇 𝑠𝑤 : sweep time) MSI √2𝑃 𝑖 𝐶(𝑡)𝑐𝑜𝑠 (2𝜋𝑓 𝑖 𝑡 + 𝜃 𝑖 ) BLWI √2𝑃 𝑖 𝑛(𝑡)𝑐𝑜𝑠 (2𝜋𝑓 𝑖 𝑡 + 𝜃 𝑖 ) Pulse √2𝑃 𝑖 𝑟𝑒𝑐𝑡(𝑡, 𝑑, 𝑟)𝑐𝑜𝑠 (2𝜋𝑓 𝑖 𝑡 + 𝜃 𝑖 )

ACKNOWLEDGMENTS

This research was supported by a grant of Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2017R1D1A1B03031787)

Jong-Hoon Won received the Ph.D. After then, he had worked with the Institute of Space Technology and Space Applications at University Federal Armed Forces (UFAF) He was nominated as Head of GNSS Laboratory in 2011 at the same institute, and involved in lectures on advanced receiver technology at Technical University of Munich (TUM) since 2009.


Xabier Zubizarreta
email: xabier.zubizarreta@iis.fraunhofer.de

J Rossouw Van Der Merwe

Ivana Lukčin

Alexander Rügamer

Wolfgang Felber

Xabier Biographies

Zubizarreta

M Eng

Masters

Receiver Independent Implementation of the Galileo Open Service Navigation Message Authentication (OS-NMA)

University of Erlangen-Nuremberg with a work on the assessment of the Open Service Navigation Message Authentication (OS-NMA

INTRODUCTION

With the rise on location based applications, such as autonomous driving or eCall, a broader audience requires not only accurate and precise positioning solutions, but also secure and reliable ones. Recent studies [START_REF] Marcos | Interference Awareness and Characterization for GNSS Maritime Applications[END_REF] [2] [START_REF] Günther | A Survey of Spoofing and Counter-Measures[END_REF] and incidents [START_REF] Usa Today | Mysterious GPS glitch telling ships they're parked at airport may be anti-drone measure[END_REF] [START_REF] Reuters | Cyber threats prompt return of radio for ship navigation[END_REF] have shown that spoofing and jamming attacks have become commonplace and pose a considerable threat to satellite based positioning solutions. Well protected signals using spreading code encryption such as GPS P(Y), M-Code or Galileo PRS are currently deployed and in use; however, they are mostly under the control of governmental agencies and hardly available to the general public. Therefore, there is a need in the mass market for secure and spoofing-proof receivers. The European Commission (EC) proposes the use of an authentication scheme for the open service (OS) of the Galileo system [START_REF] Fernández-Hernández | A Navigation Message Authentication Proposal for the Galileo Open Service[END_REF]. This new scheme should grant means to authenticate meaningful sections of the Galileo International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France navigation message as well as potential proxy authentication to other satellite navigation systems. Open service navigation message authentication (OS-NMA) is planned to be in full operational capabilities by 2020 [START_REF]A new generation of OS-NMA user terminals[END_REF].

The European satellite based navigation system, Galileo, started offering Early Operational Capability (EOC) on December 2016 and it is planned to reach full operational capabilities (FOC) by 2020. The Galileo system broadcasts four navigation messages: the Freelyaccessible Navigation Message (F/NAV), the Integrity Navigation Message (I/NAV), the Commercial Navigation Message (C/NAV), and the Governmental Navigation Message (G/NAV). I/NAV contains 120 reserved bits, as shown in Figure 1: where 72 are on the E5b-I signal and 48 on the E1B. Due to a redefinition of Galileo's original intended Safety-of-Life (SoL) service towards Galileo integrity via EGNOS / SBAS, a proposal was made to use 40 of the now reserved bits from E1B to implement navigation message authentication (NMA) for the open service (OS) [START_REF]European GNSS (Galileo) open Service Signal-in-Space interface control document[END_REF]: Objectives of the intended authentication scheme are to retain backwards compatibility and to require minimum change to the existing infrastructure. The scheme should also support the low data-rate (125 bps) of the I/NAV and be robust enough to support occasional data-loss, as is the case for GNSS receivers in environments with reduced visibility [START_REF] Fernández-Hernández | A Navigation Message Authentication Proposal for the Galileo Open Service[END_REF].

The paper is structured as follows: In Section 2, the authentication scheme is described with the underlying cryptographic primitives. Section 3 shows the architecture of the developed demonstrator system, focusing on each of the stages. The designed system is put under test by performing several measurements of its speed, operational timing limits, and practical use-cases. These tests are carried out for nominal, limited, and spoofing test cases.

The results are sketched and discussed in Section 4. Finally, in Section 5 a summary of the obtained results is shown and hints for future development are suggested.

OPEN-SERVICE NAVIGATION MESSAGE AUTHENTICATION (OS-NMA)

The main complexity of GNSS authentication lies on the broadcast nature of the signals. While the signal transmitted by the satellites can reach a large number of users, there is no feedback channel from the users to the satellites. Authentication can also be provided on a signal level or a navigation message level. NMA provides means to verify that the origin of the navigation message lies on a licit sender by means of signature. One of the means to authenticate messages is symmetric cryptography, i.e. cryptographic algorithms that rely on keeping a common secret key accessible only to the transmitter and receiver intended to partake in the communication exchange. This approach requires a careful custody of the key used.

Asymmetric cryptography provides an alternative in which a pair of keys is used. A key is kept secret by one of the parties while the other one is made public. The receiver possessing the public key can decode any message that was encrypted using the secret key, and this way authenticate that the messages originates on the correct party.

Timed Efficient Stream Loss-tolerant Authentication (TESLA)

The Timed Efficient Stream Loss-tolerant Authentication (TESLA) was originally designed to provide authentication for multimedia broadcast applications [START_REF] Perrig | Timed Efficient Stream Loss-Tolerant Authentication (TESLA): Multicast Source Authentication Transform Introduction[END_REF]. This environment is analogous to the multicast (one-tomany) nature of GNSS. It is based on a hybrid symmetric/asymmetric cryptography solution to sign the data to authenticate.

TESLA has a low computational complexity and can authenticate a high downstream data throughput with minimum up-stream transmission by means of delayed disclosure of keys. First, a Message Authentication Code (MAC) is generated to authenticate the plaintext message (M i ). The MACs are generated using a key (K i ). MACs are transmitted first, followed by the key (K i ). This key is extracted from a key-chain, which is generated by applying a one-way hashing function (H) subsequently starting from a seed key (K N ). This transmission and generation sequence is shown in Figure 2.

Figure 2 -Key-chain and message coding used in TESLA

The keys are disclosed only after the plaintext message they encrypt and the produced MAC are transmitted. Further, the keys are disclosed in the reverse order of the generation. This way, messages can neither be verified before their corresponding key is received, nor rogue MACs generated. Future keys can also not be derived from previous ones [START_REF] Caparra | Evaluating the Security of One-way Key Chains in TESLA based GNSS Navigation Message Authentication Schemes[END_REF].

The first disclosed key is called the root-key (K 0 ) and through its signature (sign(K 0 )) it allows to authenticate all subsequent keys. This signature, as it needs to be verified only once, can be performed by means of International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France computationally more expensive asymmetric cryptography. Asymmetric cryptography relies on the usage of a pair of keys from which one of the keys is used to produce the signature (private-key) and the other one is used to authenticate it (public-key). This way, any user knowing the public-key will be able to trust a given message, as only a licit sender possessing the private-key will be able to produce an authenticable signature. Elliptic Curve Digital Signature Algorithm (ECDSA) is a suggested asymmetric method for OS-NMA. The signature produced by this algorithm is roughly twice the size of the public-key required to authenticate it.

OS-NMA Field

Together with the MACs, several other parameters and values need to be transmitted in order to use TESLA for Galileo navigation message authentication. The overview of an OS-NMA subframe is shown in Figure 3.

Figure 3 -OS-NMA subframe structure

Each subframe has 600 available bits (40 bits from the reserved field times 15 odd pages per subframe) which are split in two main sections: The Header and Root-key (HKROOT) section and the MAC and key (MACK) section.

The HKROOT section contains all system parameters (current operational status, length of the keys, length of the MACs, the hashing algorithms used to produce them, and root-key identification parameters), the root-key, and the signature of the root-key, known as Digital Signature Message (DSM). The first 8 bits of the 40 bits are allocated to OS-NMA.

The HKROOT section is split over several I/NAV subframes, as one DSM Block is insufficient to accommodate the root-key together with the signature.

The MACK section contains the most meaningful bits to perform the authentication: the MAC tags and the keys. The amount of MACs per section depends on the length of the MACs themselves and the length of the keys. This section occupies the remaining 32 bits of the field [START_REF]European Union Satellite Navigation Programmes[END_REF].

SYSTEM DESCRIPTION

In order to evaluate the operation of OS-NMA under controlled laboratory conditions, experimental and commercial receivers were selected. As OS-NMA processing takes place on the latest stages of message decoding, a receiver independent solution was chosen. Additionally, since no OS-NMA signal is currently broadcast by the Galileo satellites, a message generator was developed based on the openly available latest version of the OS-NMA ICD.

The OS-NMA demonstrator consists of three main parts: a software module responsible for key generation, message generation, and message parsing; a signal generator; and a receiver.

Software module

The OS-NMA based authentication is performed during the last stages of navigation message decoding. Therefore, it can be completely detached from the receiver itself. To achieve this purpose a file-system based approach was selected. This way, only a basic interface needs to be developed to read the raw navigation message bits from the receiver and write them with the correct format in a file. Additionally, by using a network based interface, remote authentication can be performed. Finally, the parser scans continuously for changes on the file-system and reacts accordingly in case that a meaningful amount of information is received. Figure 4 shows said folder structure, as well as the content of each subframe data.

Figure 4 -Folder structure of the producer-consumer file-system

In addition to allowing receiver independent operation this architecture provides seamless logging and storage capabilities. Moreover, as there are currently no testing signals available to replicate an OS-NMA containing Galileo system, a message generator is developed together with the parser. The generated OS-NMA message is then preloaded in the signal generator.

Signal generator

A Spirent GSS9000 GNSS simulator is used to produce the signal broadcast from the Galileo satellites. The settings of the simulator, as well as the triggering of its start and stop, can be controlled remotely through a International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France TCP/IP link in real time or by uploading pre-generated user command files. This control link allows setting up each individual bit of the navigation message for all the systems. Therefore, the required bits for OS-NMA can be set accordingly to contain the pre-computed values.

Receiver

As all the processing of OS-NMA takes place on a navigation message level, the authentication scheme can be implemented in virtually any current Galileo E1B compatible GNSS receiver which provides access to the raw I/NAV navigation message bits. Table 1 shows some of the receivers that are used to check the implementation:

Manufacturer Receiver

Raw bits Septentrio PolaRx5 Accessible as GalRawINAV u-blox M8T Accessible in UBX-RXM-SFRBX Fraunhofer IIS GOOSE [START_REF] Overbeck | GOOSE -GNSS Receiver with an Open Software Interface[END_REF] Custom direct write to file Table 1

-Table summarizing receivers compatible with the suggested approach

Not only high-end or specialized receivers offer access to the raw navigation message bits; but, since the application programming interface (API) version 24 from Android, GNSS raw measurements are also provided by this mobile-phone operating system. Raw navigation message bits are among the values returned by the API. As a matter of fact, there are already some phone manufacturers supporting the Galileo signal and raw measurement output, opening the door to a plethora of new applications profiting from NMA [START_REF] Android | Raw GNSS Measurements[END_REF].

MEASUREMENTS

First of all, the correct operation of the receiver was evaluated by using the end-to-end demonstrator under a nominal scenario, that is, a scenario under commonly found conditions. A general overview of the system's operation time was obtained at this stage. The Time to the First Authenticated Fix (TTFAF) was measured and evaluated for several parameter combinations. The authentication data throughput, i.e. the amount of navigation bits that could be authenticated through the scheme, was also evaluated. Next, analysis was performed to check the resilience of the authentication scheme in case of data-loss.

Finally, the main functionality of OS-NMA was put under test, namely the possibility to avoid data forgery based spoofing attacks. For that purpose the licit self-generated OS-NMA signal was spoofed using a forged licit-looking Galileo E1B signal. The mismatch of the public-key, resulted in the authentication scheme to serve its purpose to identify the rogue signal. The receiver alerted with a minimum delay about the ongoing attack.

Nominal use-cases

A nominal scenario lasting 3 minutes and with clear sky view was used. The OS-NMA signal was transmitted only by a single satellite (SVID = 3) authenticating its own navigation message. The OS-NMA bits were precalculated and pre-loaded into the signal generator. The receiver was directly connected to the RF output of the signal generator.

The TTFAF limits the end-user applications of the authentication scheme. While for the case of freight tracking it could be permissible to obtain an authenticated position fix after a long delay, and at a reduced rate, in the case of the automotive industry, more precisely for autonomous driving applications, authentication delay may play a vital role.

In the OS-NMA scheme the receiver must undergo three steps until the navigation data bits are authenticated. Figure 5 summarizes these steps as a time-sequence. In order to analyze the lower and upper boundaries of the TTFAF, and the authenticated data-volume, it was iterated through several system parameters. The TTFAF mostly depends on the key-size and the signature size.

The fastest TTFAF occurs for the combination of the shortest possible key-size (80 bits) and the shortest possible signature size (448 bits, rendered by using the P-224 curve for the ECDSA signature). It takes 6 subframes for the complete reception of the parameters, root-key, and its signature for this combination. Therefore, the first navigation data bits can be first authenticated after 180 seconds. Figure 6 shows the evolution of the authentication process versus time.

International Technical Symposium on Navigation and Timing (ITSNT) 2018 13-16 Nov 2018 ENAC, Toulouse, France The slowest possible authentication occurs for the selection of the longest possible key (256 bits, untruncated SHA-256 hash) together with the longest possible signature (1042 bits, produced from using the P-521 curve with the ECDSA). In this case, 14 sub-frames are necessary to fully receive the necessary fields which correspond to 420 seconds. Figure 7 shows the timesequence of the slowest TTFAF. The authentication data size is an important factor. There is a trade-off between the key-lengths and MAC lengths and the security level they provide. While short lengths permit to fit more MAC tags in each subframe, and consequently can authenticate more data, they reduce the cryptographic robustness of the system.

Data-loss resilience

One of the requirements for a GNSS authentication scheme is the ability to still function properly in challenging environments. Bits of the navigation message can be lost due to signal shadowing or interference. Although the low bit-rate of the navigation message transmission together with the used Forward Error Correction (FEC) may compensate any short-time signalloss events, longer signal-loss time may impair the correct operation of OS-NMA.

As far as data-loss is concerned, two main stages can be distinguished on the functioning of OS-NMA. On a first stage in which the receiver is unaware of any of the system parameters and values of the authentication scheme, every single bit carries meaningful and required information. However, once the root-key, system parameters, and the verifying signature are received the whole HKROOT section is dispensable, at least until a new key-chain with a different root-key is introduced. This makes only 80% of the OS-NMA field susceptible to disruption.

Not only is data-loss more critical on different operational stages, but also on different parts of the messages. While MACs missing a few arbitrary bits can still be authenticated, provided that the key is received and the loss on security due to incomplete MACs is permissible, missing parts of the root-key signature voids the signature fully and requires to receive the missing bits again. ECDSA requires the complete error-less signature, and it is not compatible with any type of truncation. Moreover, ECDSA produces unique, non-deterministic, signatures which do not permit to amend lost chunks of a signature with bits received on the future.

Spoofing resilience

The main purpose of the OS-NMA is to provide protection against data spoofing attacks. Spoofing attacks comprise a wide range depending on the complexity of implementation as well as the damage they may cause. By being able to authenticate the contents of the navigation message, protection against signal forgery is provided by OS-NMA. The key-chain has the benefit that the contents of the OS-NMA field cannot be pre-computed in advance by an attacker [START_REF] Humphreys | Assesing the Spoofing Threat: Development of a Portable GPS Civilian Spoofer[END_REF] A spoofing attack was carried out by jamming the receiver and forcing reacquisition on a forged signal as shown in Figure 8 [START_REF] Tippenhauer | On the Requirements for Successful GPS Spoofing Attacks[END_REF]. An attacker would be unable to re-generate future values of the OS-NMA field, therefore the transmitted data is invalid but licit looking. The receiver still did not fully receive the authentic HKROOT field when the jamming occurred. Therefore, it needed to re-start the HKROOT decoding process from the beginning, reading the forged root-key, and signature. Provided that the receiver contains a valid and licit public-key, the alert first comes up at the verification of the root-key's signature. The length of the root-key and the signature ultimately determine this maximum time to first alert. This time coincides with the lower and upper time boundaries of the TTFAF. replay attacks usually have some time delay caused be the rebroadcasting of the signal and the travel time between the attacker and the receiver. If a licit signal is retransmitted at a later time, its contents, including the OS-NMA field will appear to be valid. Therefore, for the OS-NMA scheme to function properly a reliable external time reference is required. The receiver must know the time within an accuracy of at least 2 seconds, which corresponds to knowing which page is currently being received, and thus, also regenerate the key-chain adequately. Receivers that rely solely on the time provided by GNSS are susceptible to signal replay attacks.

CONCLUSIONS AND FUTURE WORK

With the growth of new GNSS based applications the need of increased security proved to be necessary. By introducing OS-NMA on the navigation message, Galileo will be the first GNSS containing an in-built cryptographic authentication mechanism intended for the use of the general public. The scheme has a minimum development overhead and could be implemented in most receivers by only updating their firmware.

Additionally, spoofing attacks relying on signal forgery could be successfully detected. With a minimal technical overhead, the least complex way to carry out attacks could be rendered ineffective. As far as more complex threats are concerned, more reliable and instant authentication mechanisms, such as the spreading code encryption e.g. provided by the Public Regulated Service (PRS) or also intended for the Commercial Service authentication have to be utilized but with further introduced complexity.

Issues such as public-key dissemination and storage, timesynchronization, and inter-system compatibility, still need to be addressed and further defined on new issues of the OS-NMA ICD. When the OS-NMA test transmission commences, the assumptions and conclusions of this exemplary implementation can be tested and validated by future research.

Figure 1 -

 1 Figure 1 -Galileo I/NAV message bit structure, from [8]

Figure 5 -

 5 Figure 5 -TTFAF time-sequence. KS = 80 bits; Signature size = 512 bits First, the receiver must obtain the public key corresponding to the root-key which is currently in use. Although the Interface Control Document (ICD) of the OS-NMA considers Over-the-air (OTA) transmissions of the public-key, it is not yet defined on the current issue of the ICD. Another option would be pre-programming the receivers with the required public keys or providing said keys through third-party channels such as internet, cellular network, or broadcast radio. Next, the receiver must fully receive the DSM message, which extends through several 30 second subframes depending on the signature and key lengths chosen. Finally, the receiver can start processing the MACs.

Figure 6 -

 6 Figure 6 -Time-sequence of signature, root-key, and MAC reception for the fastest possible TTFAF.

Figure 7 -

 7 Figure 7 -Time-sequence of signature, root-key, and MAC reception for the slowest possible TTFAF.

Figure 8 -

 8 Figure 8 -Schematic representation of a jam and spoof attack.OS-NMA does not provide protection against attacks based on signal recording and rebroadcasting. These

ACKNOWLEDGMENTS

The work for this paper has been conducted under the PRoPART project, which has received funding from the European GNSS Agency under the European Union's Horizon 2020 research and innovation program under grant agreement No 776307.

Alexander Rügamer received his Dipl.-Ing.


Angélica Benito
email: angelica.benito@uam.es.

Olivier Piltant
email: piltant@math.uvsq.fr.

Ana J Reguera
email: areguera@agt.uva.es.

Small irreducible components of arc spaces in positive characteristic

may come

Small irreducible components of arc spaces in positive characteristic

Introduction

In 1968, J. Nash initiated the study of the space of arcs X ∞ of a (singular) algebraic variety X with the purpose of understanding the structure of the various resolutions of singularities of X. His work [START_REF] Nash | Arc structure of singularities[END_REF] was done shortly after Hironaka's proof of Resolution of Singularities in characteristic zero [START_REF] Hironaka | Resolution of singularities of an algebraic variety over a field of characteristic zero[END_REF].

Nash's starting point was the following: let X be a variety over a field k of characteristic zero with a given resolution of singularities π : Y → X. For every irreducible component E of the exceptional locus of π, the Nash family of arcs N E ⊂ X ∞ is defined to be the Zariski closure of the image of the set of arcs on Y which are centered at some point of E. He observed that each N E is irreducible and, moreover, N E only depends on the divisorial valuation computing the order along E. Due to the properness of π, every arc in X ∞ \ (SingX) ∞ which is centered at some point of the singular locus of X belongs to some of the N E 's since it lifts to Y . That is, the space of arcs X Sing ∞ centered in SingX decomposes as

X Sing ∞ = ∪ E N E ∪ (SingX) ∞ .
(1.1)

From this, and arguing by induction on dim X, one deduces that the number of irreducible components of X Sing ∞ is finite (see [START_REF] Nash | Arc structure of singularities[END_REF] or [START_REF] Ishii | The Nash problem on arc families of singularities[END_REF], [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF]). It is in general not easy to deduce the decomposition of X Sing ∞ into its irreducible components from (1.1); the Nash problem consists precisely in characterizing these irreducible components. This Nash program extends, with some important differences, to perfect ground fields k of characteristic p > 0. A first obvious difference is that Resolution of Singularities is still an open problem if chark = p > 0 and dim X ≥ 4. Although Nash families N E can be defined only in terms of divisorial valuations, it is not known that the indexing set in (1.1) can be chosen to be a finite set. In particular, it is unknown if the number of irreducible components of X Sing ∞ is always finite [START_REF] Piltant | Local uniformization and arc spaces[END_REF]. Another difference is that, in contrast with characteristic zero, the right hand side term (SingX) ∞ in (1.1) may contain some of the irreducible components of X Sing ∞ . Understanding these "small" components (called small because they consist of families of arcs concentrated inside the singular locus) is the main purpose of this article.

In the spirit of the Nash program, we propose characterizing all small irreducible components of X Sing ∞ in terms of birational morphisms Y → X, or in terms of the various resolutions of singularities of X whenever they exist. We obtain a satisfactory criterion for small components mapping in X onto an irreducible component of Sing X with codimension one in X (corollary 5.8). In particular we identify all small components of X Sing ∞ when X is any surface (corollary 5.9). In higher dimensions the question remains open except in special cases. Among them, an example which is extensively studied is that of the total space of a family of hypersurface cones

X B := Spec O B [y 0 , y 1 , . . . , y n ] (f ) , f := y p 0 + b 1 y p 1 + • • • + b n y p n , b 1 , . . . , b n ∈ O B , (1.
2) where chark = p > 0 and the base k-variety B is regular and irreducible. This subtle example is considered here because it conveys interesting difficulties showing up in the Resolution of Singularities Conjecture [START_REF] Hironaka | Additive groups associated with points of a projective space[END_REF].

Underlying the existence of small components when chark = p > 0 is the fact that Kolchin's irreducibility theorem is not valid in positive characteristic. If X = SpecR is an irreducible affine variety of characteristic zero, then R ∞ := O X∞ is isomorphic to the differential algebra associated with R which Kolchin proved to be irreducible, i.e. the reduced ring R red is a domain. It was known to Kolchin that this statement does not hold in general when chark = p > 0; the most simple counterexample is the irreducible surface

X := V (y p + zx p ) ⊂ A 3 k = Speck[x, y, z]
where SingX = V (x, y) is the z-axis and X ∞ has two irreducible components, (SingX) ∞ and the Zariski closure of its complement in X ∞ . Obviously, any irreducible component of

X ∞ contained in (SingX) ∞ is also an irreducible component of X Sing ∞ .
This article is organized as follows. In section two, we state and reprove Local Uniformization of varieties along formal arcs (proposition 2.3) which plays an important role in this article. It allows us to give a simple new proof of the fact that the arc space X ∞ of a k-variety X has finitely many irreducible components (proposition 2.1) in any characteristic. Every irreducible component C of X ∞ maps to an irreducible subvariety Z C and its generic point is the generic arc in RegZ C . Section 3 introduces basic valuation theory and the notion of Local Uniformization of valuations.

In section 4, we introduce the new terminology arc-sharp/arc-blunt. Given a scheme-theoretic point ζ ∈ X, X is said to be arc-sharp at ζ if ζ is a generic point of X, or if the generic arc in RegZ, Z := {ζ}, is not a specialization of a generic arc in RegX; otherwise, X is said to be arc-blunt at ζ. Using Local Uniformization along formal arcs, we give a simple necessary condition for arc-sharpness in Theorem 4.4: if X admits a resolution of singularities π : Y → X, then X is arc-sharp at z only if π -1 (ζ) has no separable point over k(ζ). Since we work without the Resolution assumption, our criterion is formulated in terms of prime divisors centered at z, see Proposition 4.3 and Theorem 4.4 for a precise formulation. An interesting corollary is that X is arc-sharp at z only if its multiplicity at ζ is divisible by p (Corollary 4.5).

Section 5 provides a sufficient condition for arc-sharpness, stated as Theorem 5.5. For a point ζ of codimension one in X, both necessary and sufficient conditions coincide and this proves our main result characterizing small irreducible components of X ∞ whose generic point is the generic arc in RegZ, Z := {ζ}. Our result is more general and is phrased in terms of radicial morphisms which are relevant here. The proof uses Frobenius techniques which are original in the context of arc spaces. Section 6 contains several questions which came out of this work. Question 6.1 asks whether any small irreducible component of X Sing ∞ is necessarily an irreducible component of the whole space X ∞ . Question 6.2 asks whether our characterization of arc-sharpness for points ζ of codimension one in X is also valid in higher codimension, at least for p > dim O X,ζ . Finally question 6.6 is a mild, but apparently challenging problem of Resolution of Singularities type: can small irreducible components be eliminated by blowing up? More precisely, we ask whether every irreducible proper variety X is birationally equivalent to a proper variety Y (resp. admits a proper birational morphism Y → X) such that Y ∞ is irreducible. Using a classical theorem of Albanese [START_REF] Lipman | Introduction to resolution of singularities[END_REF], we answer the "birationally equivalent" version in the affirmative for X of arbitrary dimension and all big enough characteristics p > (dimX)!. These questions are tested on a family of varieties with equation (1.2).

The irreducibility problem on arc spaces.

Let k be a perfect field. By a variety over k, we mean a reduced separated k-scheme of finite type. Given a variety X|k, let X ∞ |k denote the space of arcs of X. It represents the functor on k

-algebras A → X(A[[t]]). If X ⊂ A N k = Spec k[x 1 , . . . , x N ] is affine with ideal I X = (f 1 , . . . , f r ), we pick infinitely many variables X n = (X 1,n , . . . , X N,n ), n ≥ 0. For 1 ≤ j ≤ r, expand f j ( ∑ n X n t n ) = ∞ ∑ n=0 F j,n t n , ( 2.1)
so we get a description of X ∞ as

X ∞ = Spec k[{X n } n≥0 ] ({F 1,n , . . . , F r,n } n≥0 )
.

There is a natural map j : X ∞ → X. Given a subvariety Z ⊆ X, we denote by X Z ∞ the subscheme j -1 (Z) of X ∞ . As a subset, X Z ∞ consists of all arcs with center inside Z. There is an inclusion

Z ∞ ⊆ X Z ∞ which is strict if Z ̸ = X.
The scheme X ∞ is not of finite type over k if dim X > 0. However it satisfies several finiteness properties. The following result refers to one of them. It is proved in [START_REF] Kolchin | Differential Algebra and algebraic groups[END_REF] chap. IV, prop. 10 if char k = 0 and in [START_REF] Gillet | Differential algebra. A scheme theory approach[END_REF] cor. 1.28, [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF] th. 2.9 for any perfect field k. Proposition 2.1. ([8] and [START_REF] Gillet | Differential algebra. A scheme theory approach[END_REF], [START_REF] Reguera | Towards the singular locus of the space of arcs[END_REF]) Let X be a variety over a perfect field k with irreducible components X 1 , . . . , X c . Let Z ⊆ X be a nowhere dense subvariety such that

c ∪ i=1 SingX i ⊆ Z.
The natural map X ∞ \Z ∞ -→ X induces a bijection on irreducible components. In particular X ∞ has finitely many irreducible components.

The last part of the statement follows immediately from the first one: apply Noetherian induction on X, changing X by its (reduced) singular locus SingX. We will give below an alternate proof of this statement as an application of uniformization along arcs (proposition 2.3 below). Uniformization also plays an important role in the next sections. First we consider the simpler case of a regular variety.

Lemma 2.2. Let Y be a regular k-variety. The following holds:

(1) For every subvariety

Z ⊆ Y , the induced map Y Z ∞ → Z induces a bijection on irreducible components. More precisely, if Z = E 1 ∪• • •∪E r is the decomposition of Z into irreducible components, then Y Z ∞ = Y E 1 ∞ ∪ • • • ∪ Y Er ∞ is the decomposition of Y Z ∞ into irreducible components. (2) If furthermore Z ⊆ Y is nowhere dense, then Y ∞ \ Y Z ∞ = (Y \ Z) ∞ is Zariski dense in Y ∞ .
Proof. We may suppose that Y is affine, let Y ⊆ A m . Picking a finite open covering of Y , we may assume that there exists an étale morphism from Y to a subset of A d k where d = dim Y . Then we have

O Y∞ ∼ = O Y [X 1 , X 2 , . . . , X n , . . .]
where, for n ≥ 0, X n = (X 1,n , . . . , X m,n ) is a m-uple of variables. Then, (1) follows from the induced isomorphisms

O Y Z ∞ ∼ = O Z [X 1 , . . . , X n , . . .] and O Y E i ∞ ∼ = O E i [X 1 , . . . ,
(η) ̸ = h P (0).
Of course, there is a specialization h P (η) h P (0). Specialization in X and X ∞ will play an important role in this article.

Let P ∈ X ∞ , and π : X ′ → X be a blowing up along a subscheme Y ⊂ X such that h P (η) ̸ ∈ Y . There exists a unique lifting

h ′ P : Specκ(P )[[t]] → X ′ .
We have

x ′ := h ′ P (0) ∈ Σ(h ′ P ) and π is an isomorphism at h ′ P (η). Iterating, let X ← X ′ ← • • • ← X (r) ← • • • (2.2)
be the resulting sequence of blowing ups and centers with

x (r) ∈ Σ(h (r) P ), x (r) ∈ Y (r) Σ(h (r) P ). (2.3)
Note that the local ring O X (r) ,h (r) P (η) is independent of r ≥ 0. An important case of such sequences is when taking Y (r) = {x (r) } for every r ≥ 0; then (2.2) is called the quadratic sequence along h P . Proposition 2.3. (Uniformization along arcs) Let P ∈ X ∞ . Assume that h P is a nonconstant arc and h P (η) ̸ ∈ Sing X. Consider the quadratic sequence (2.2) and let π (r) : X (r) → X, and h (r)

P : Specκ(P )[[t]] → X (r)
be the corresponding morphisms for r ≥ 0. Then both Σ(h (r) P ) and X (r) are regular at x (r) for every r >> 0.

Proof. To prove the statement for Σ(h (r) P ), it can be assumed without loss of generality that Σ(h P ) = X. We build up a discrete invariant

i(x) := (a(x), b(x), e(x), δ(x)) ∈ N 4
which decreases for the lexicographical ordering by blowing up along x provided O X,x is not regular.

Let v = v P be the discrete valuation associated to h P and fix an isomorphism v(k(h P (η))\{0}) ≃ Z. Consider the value semigroup

S(x) := {v(f ), 0 ̸ = f ∈ O X,x } ⊆ N. Let a(x) := min{S(x)\{0}}. If S(x) ̸ = N, we denote b(x) := min{S(x)\a(x)N}. Note that we have (a(x ′ ), b(x ′ )) < (a(x), b(x))
for the lexicographical ordering if S(x) ̸ = N. In other terms, it can be assumed that

S(x (r) ) = N, r ≥ 0. (2.4)
We denote: e(x) := emb.dim x X.

Since e(x ′ ) ≤ e(x), we may also assume that e(x (r) ) = e := e(x) for every r ≥ 0. In particular, we have

d(x (r) ) := dimO X (r) ,x (r) = d is constant and k(x (r) )|k(x) is finite algebraic. Let t := tr.deg k k(x) ≥ 0. We write O X,x = R/P,
where (R, M, k(x)) is a regular local ring of dimension e. There is nothing to prove if P = 0, so we assume that P ̸ = 0 from now on. For f ∈ R, we denote by f its image in R/P . Let (u 1 , . . . , u e ) be a r.s.p. of R, with the convention that v(u 1 ) = 1, viz. (2.4). Let

Ω := Ω R|k (log u 1 )
be the module of Kähler differentials over k with logarithmic pole along u 1 .

Then Ω is a free R-module with basis

B = ( du 1 u 1 , {du j } 2≤j≤e , {dλ s } 1≤s≤t ), (2.5)
where the images of λ 1 , . . . ,

λ t ∈ R r in k(x) form a separating transcendence basis of k(x)|k. Let ∆(x) ⊆ Der R|k
be the dual module of Ω and define:

D(x) := {D ∈ ∆(x), D • M ⊆ M }.
The R-module D(x) is generated by the family

< u 1 ∂ ∂u 1 , {M ∂ ∂u j } 2≤j≤e , { ∂ ∂λ s } 1≤s≤t > . (2.6)
We define

δ(x) := min{v(D • f ), D ∈ ∆(x), f ∈ P, D • f ̸ ∈ P }.
Note that since k is perfect, X is generically smooth over k, so there exists

f ∈ P and D ∈ ∆(x) such that D •f ̸ ∈ P . In particular, we have δ(x) < +∞. Pick then D 0 ∈ ∆(x), f ∈ P such that v(D 0 • f ) = δ(x).
W.l.o.g. it can be assumed that D 0 belongs to the dual basis B ∨ . Let m := ord x f ≥ 2. We now compute how δ(x) transforms by blowing up along x. To begin with, let

O X ′ ,x ′ = R ′ /P ′ , R ′ = R[ u 2 u 1 , . . . , u e u 1 ] x ′ .
We have f ′ := u -m 1 f ∈ P ′ . Note that any D ∈ D(x) extends to a derivation D ′ ∈ ∆(x ′ ). On the other hand, we have

(D ′ • f ′ ) = u -m 1 ( (D • f ) -m (D • u 1 ) u 1 f ) and we deduce that v(D ′ • f ′ ) = v(D • f ) -m.
Since u 1 ∆(x) ⊆ D(x), we apply the previous equality to

D := u 1 D 0 to get δ(x ′ ) ≤ v(u 1 D 0 • f ) -m ≤ δ(x) -(m -1) < δ(x)
and the conclusion follows. The proof of the proposition for X (r) is similar.

Proof of proposition 2.1: there is a commutative diagram

j -1 (X\Z) ⊆ X ∞ \Z ∞ ↓ ↓ X\Z ⊆ X .
By lemma 2.2(1), the left hand side arrow induces a bijection on irreducible components because

X\Z ⊆ c ∩ i=1 Reg X i .
It is therefore sufficient to prove that

j -1 (Z)\Z ∞ ⊆ j -1 (X\Z), (2.7)
where bars denote Zariski closure in X ∞ . For P ∈ j -1 (Z)\Z ∞ , we have (h P (0) ∈ Z and h P (η) ̸ ∈ Z). In particular h is not constant. We now apply proposition 2.3. There exists a regular irreducible k-variety Y , and a birational morphism

f : Y → X i , X i an irreducible component of X, such that h P lifts to Y . Furthermore, if W := f -1 (Z), by proposition 2.2(2), we have P ∈ f ∞ (j -1 Y (Y \W )). Therefore (2.7) follows from the self-evident inclusions f ∞ (j -1 Y (Y \W )) = f ∞ (j -1 Y (Y \W )) ⊆ j -1 (X\Z).
3 Reminder on valuations.

Let k ⊂ K be a field extension. By a k-valuation v of K, we mean a valuation of K which is trivial on k. The corresponding valuation ring is denoted by

O v := {f ∈ K : v(f ) ≥ 0} ∪ {0},
and its maximal ideal by

M v := {f ∈ K : v(f ) > 0} ∪ {0}. The residue field k v := O v /M v contains k as a subfield. Given two local subrings (R, M ) and (S, N ) of K containing k, we say that S dominates R if R ⊆ S and M = N ∩ R.
Let A ⊆ K be a k-subalgebra. We say that a k-valuation v has a center in

X := SpecA if A ⊆ O v . The center x ∈ X is the prime ideal P := M v ∩ A, so O v dominates A P .
Suppose a k-valuation v of the residue field k v is given. The ring

R := {g ∈ O v : g modM v ∈ O v }
is the valuation ring of a k-valuation v 0 , called composite of v with v, and denoted by v 0 = v • v. It has the following property: for every k-subalgebra A ⊆ K such that v 0 has a center x 0 ∈ X = SpecA, v has a center x ∈ X and x 0 is a specialization of x. More precisely, the prime ideal P 0 := M v ∩ O v 0 satisfies the properties:

O v 0 /P 0 ≃ O v and (O v 0 ) P 0 = O v .
See [START_REF] Zariski | Commutative Algebra[END_REF] chap. VI, sec. 10 for more details about composition of valuations.

Let X|k be a k-variety with irreducible components X 1 , . . . , X c . By a k-valuation of X, we mean a k-valuation of the fraction field k(X i ) for some i, with a center on X i . Definition 3.1. Let X|k be an affine k-variety and v be a k-valuation of X with center x v and associated component

X i . A local uniformization of v is a birational morphism: Y v = SpecA → X i A = O X i [f 1 , . . . , f r ] ⊆ O v ,
such that A Mv∩A is a regular local ring. We say that Local Uniformization holds on X at some x ∈ X (LU (X, x) holds for short) if every k-valuation of X with center x has a local uniformization. Local Uniformization results ([13] A.III th. U 3 and [1] theorem on p.1839): Let X|k be a k-variety and x ∈ X. Local Uniformization holds at x if we assume that char k = 0 or that (char k = p > 0 and dim O X,x ≤ 3).

Suppose that X is affine and irreducible. Recall that the Riemann-Zariski space V(X), consisting of all k-valuations of X, is provided with the following topology: given a subring A of k(X) which is a O X -algebra of finite type, let E(A) be the set of all v ∈ V(X) which are nonnegative on A, that is A ⊆ O v . Then the sets E(A), where A runs over all O X -algebras of finite type contained in k(X), define a basis of a topology on V(X). With this topology, V(X) is quasi-compact ( [START_REF] Zariski | Commutative Algebra[END_REF] chap. VI, sec. 17, theorem 40) and the map π X : V(X) → X, sending v to its center on X, is closed and continuous ( [START_REF] Zariski | Commutative Algebra[END_REF] chap. VI, sec. 17, lemma 4).

Small irreducible components of the arc space.

In this section we study the irreducible components of the arc space of a variety. Let X|k be a k-variety, and denote by X 1 , . . . , X c its reduced irreducible components. The corresponding maximal points of X are denoted by ξ 1 , . . . , ξ c .

Let ζ ∈ X and Z := {ζ} ⊆ X. Recall from proposition 2.1 that

Z ∞ \(SingZ) ∞ is irreducible. The Zariski closure Z ∞ \(SingZ) ∞ ⊆ Z ∞ ⊆ X ∞ of Z ∞ \(SingZ) ∞ is denoted by Z • ∞ , its generic point by ζ ∞ . Note that ζ ∞ ∈ (Z \ SingZ) ∞ , hence Z 0 ∞ = (Z\SingZ) ∞ (4.1)
This applies in particular to the maximal points ξ 1 , . . . , ξ c of X and the corresponding irreducible components of X ∞ are denoted by X • 1∞ , . . . , X • c∞ . These may not be all irreducible components of X ∞ and motivates the following definition:

Definition 4.1. Let X|k be a k-variety, ζ ∈ X and Z := {ζ} ⊆ X. We say that X is arc-sharp at ζ if Z • ∞ = X • i∞ for some i (i.e. ζ = ξ i ), or if Z • ∞ c ∪ i=1 X • i∞ .
If X is not arc-sharp at ζ, we say that X is arc-blunt at ζ.

Remark 4.1. The variety X is arc-sharp at ζ if and only if

X i is arc-sharp at ζ for each i such that ζ ∈ X i . Furthermore Z • ∞ is an irreducible component of X ∞ if and only if for every irreducible subvariety F , Z ⊆ F ⊆ X, F is arc-sharp at ζ.
In order to characterize irreducible components of X ∞ , we need to introduce prime divisors ([14] sec. 14). It is well known that the residue field k(v) of a prime divisor v over ζ is a finitely generated field extension of k(ζ).

Definition 4.2. A prime divisor of X over ζ is a discrete valuation v of some k(X i ), ζ ∈ X i , with ring (O v , M v , k v ) such that O X i ,ζ ⊆ O v , M v ∩ O X i ,ζ = ζ, and tr.deg k(ζ) k v = dim O X i ,ζ -1. We say that O v is generically smooth over ζ if dim kv Ω kv|k(ζ) = tr.deg k(ζ) k v .
E := π -1 (Z) red is irreducible and O Y,E = O v .
If v is generically smooth over ζ, we may furthermore take the induced map E → Z smooth.

Proof. Let π 1 : Y 1 → X be a projective birational morphism such that the center E 1 of v in Y 1 is a divisor, Y 1 normal. In particular, the generic point of E 1 is regular. We have π -1 1 (Z) red = E 1 ∪ E 2 , where E 2 does not contain E 1 . Take Y := RegY 1 \E 2 .
Note that we may replace Y with Y ∩π -1 (U ), where U ⊆ X is any Zariski neighborhood of ζ. If v is generically smooth over ζ, then E∩π -1 (U ) → Z ∩U is smooth for suitable U and this concludes the proof. Proposition 4.3. Let X|k be an irreducible k-variety, ζ ∈ X and Z := {ζ} ⊆ X. We consider the following properties:

(1) there exists a generically smooth prime divisor over ζ;

(2) there exists P ∈ X ∞ such that h P (0) = ζ, h P (η) ∈ Reg(X) and κ(P )|k(ζ) is a finite and separable field extension;

( We have equivalences ( 1) ⇔ ( 2) and ( 3) ⇔ (4). Furthermore, we have an implication (2) ⇒ (3); the converse holds if LU (X, ζ) holds.

Proof. We get (1) =⇒ (2) by applying lemma 4.2: when

E → Z is smooth, π -1 (ζ) has points over k(ζ) sep . Let ζ ′ ∈ E be such that k(ζ ′ )|k(ζ) is separable. There exists an arc h : Spec k(ζ ′ )[[t]] → Y centered at ζ ′ and such that h(η) ̸ ∈ E. Then the arc h = π • h satisfies (2).
Conversely, we apply proposition 2.3 to obtain a birational morphism π : X ′ → X and a point Finally let us prove (4) =⇒ (3). Let π α : X α → X an arbitrary proper and birational morphism. The birational correspondence map is further denoted by

ζ ′ := h P ′ (0) ∈ Reg(X ′ ) ∩ π -1 (ζ),
k-varieties Y v → X such that the center y v ∈ Y v of v sits inside Reg(Y v ). Let ζ ′ ∈ Reg(Y v ) ∩ {y v } be such that k(ζ ′ )|k(ζ)
π αβ : X β • • • → X α .
Let

S α := {ζ ′ ∈ π -1 α (ζ) : k(ζ ′ )|k(ζ)
is a finite and separable extension}.

We remark that every maximal point c α of S α corresponds to a separable field extension k(c α )|k(ζ). It follows from these definitions that, whenever π αβ is a morphism, we have

π αβ (S β ) ⊆ S α . (4.2)
We define F α to be the intersection of π αβ (S β ) whenever π αβ is a morphism. Since X α is Noetherian, (4.2) implies that F α ̸ = ∅. Furthermore, we have

π αβ (F β ) = F α (4.3) Let F := lim ← -F α ⊆ π -1 X (ζ) ⊂ lim ← -X α = V(X).
We have that F is a closed set and F α = π Xα (F) where we denote by π Xα : V(X) → X α the projection maps (recall that they are closed maps, [START_REF] Zariski | Commutative Algebra[END_REF] VI sec. 17, lemma 4). Besides, F α ̸ = ∅ for all α, (4.3) and the quasicompacity of the Zariski-Riemann surface imply that F is nonempty. We may therefore pick v ∈ F such that y α := π Xα (v) ∈ X α is the generic point of some irreducible component of I α for every α. Note that k(y α )|k(ζ) is a separable field extension: since π αβ (F β ) = I α whenever π αβ is defined at x β , we have k(y α ) ⊆ k(c β ), where c β is any maximal point of F β specializing to y β . In particular, the residue field k v = lim ← -k(y α ) of v is a separable field extension of k(ζ). This proves the proposition.

The main result of this section is the following theorem. We list below some corollaries. Proof. The argument is the same as in the proof of proposition 2.1. To begin with, it can be assumed that X is irreducible. We use property (2) of proposition 4.3 which provides some h P .

By proposition 2.3, there exists a regular irreducible k-variety X ′ , and a birational morphism f : X ′ → X, such that h P lifts to X ′ . Furthermore, the exceptional locus E of X ′ → X maps to Z.

Let h P lift with center ζ ′ ∈ X ′ . Since κ(P )|k(ζ) is finite and separable, so is k(ζ ′ )|k(ζ). Let Z ′ := {ζ ′ } ⊆ X ′ . Since k(ζ ′ )|k(ζ) s finite and separable, we have f ∞ (ζ ′ ∞ ) = ζ ∞ . Here ζ ∞ (resp. ζ ′ ∞ ) is the generic point of Z 0 ∞ (resp. Z ′ 0 ∞ ). By lemma 2.2(2), we have ζ ∞ ∈ f ∞ (j -1 X ′ (X ′ \E))
and the conclusion follows from the self-evident inclusions

f ∞ (j -1 X ′ (X ′ \E)) = f ∞ (j -1 X ′ (X ′ \E)) ⊆ j -1 (X\SingX) = X • ∞ .
:= X → X is a unique point ζ ∈ X, rational over k(ζ). Then, m 1 = dim k(ζ) ( O X,ζ / M X,ζ O X,ζ )
where 5 Existence of small irreducible components.

The main result characterizing small irreducible components of X ∞ is theorem 5.5 below. We first recall some general facts about radicial morphisms ([3] chap. 1, sec. 3.5). Let f : Y → X be a morphism of schemes having finitely many irreducible components. We say that

f is dominant if f (Y) = X . Equivalently, the induced map O X red → O Y red is injective.
We say that f is strongly dominant if f induces a surjective application from the set of maximal points {y 1 , . . . , y n } of Y into the set of maximal points {x 1 , . . . , x m } of X . If f is strongly dominant, there is an induced inclusion

Tot(X red ) = m ∏ i=1 k(x i ) ⊆ Tot(Y red ) = n ∏ j=1 k(y j )
between total rings of fractions. We say that f is birational if f is strongly dominant, and if the above inclusion is an equality.

A morphism f : X ′ → X of schemes is said to be radicial if every nonempty fiber f -1 (x) of f has only one element {x ′ } and is such that the induced field extension k(x) ⊆ k(x ′ ) is radicial, i.e. algebraic and separably closed ([3] Def. 3.5.4 and Prop. 3.5.8). In other terms, if chark(x) = 0, we have k

(x) = k(x ′ ); if chark(x) = p > 0, every λ ∈ k(x ′ ) satisfies: λ p α ∈ k(x) for some α ≥ 0.
(5.1)

We remark at this point that an integral, radicial and dominant morphism f : Y → X is actually strongly dominant. Namely, let z ∈ Y be a maximal point mapping to a non maximal point of X and let Y 0 := Y\Z, where Z := {z}. The map f 0 : Y 0 → X is again integral, radicial and dominant.

In particular f 0 is surjective by the going up theorem. Since f is radicial, f -1 (f (z)) has only one element: a contradiction. We now state a couple of lemmas.

Lemma 5.1. Let f : X ′ → X be a finite, radicial and dominant morphism of k-varieties, with chark = p > 0. There exists α ≥ 0 and inclusions

O p α X ′ ⊆ O X ⊆ O X ′ . Proof.
It can be assumed that X = SpecR, hence X ′ = SpecR ′ is affine, and that R ′ = R[x] is generated by one element as a R-algebra. The right hand side map is an inclusion R ⊆ R[x] because f is dominant. Let P 1 , . . . , P c be the minimal primes of R. Since f is radicial and surjective (finite and dominant), we may label P ′ 1 , . . . , P ′ c the minimal primes of R ′ , with

P ′ i ∩ R = P i , K i := QF (R/P i ) ⊆ K ′ i := QF (R ′ /P ′ i ) radicial, 1 ≤ i ≤ c. By (5.1), we may pick α ≥ 0 such that Tot(R ′ ) p α ⊆ Tot(R) = c ∏ i=1 K i ⊆ Tot(R ′ ) = c ∏ i=1 K ′ i . ( 5.2)
In particular, we may assume that x ∈ Tot(R) in order to prove the lemma, i.e. f birational.

We now argue by induction on the pair (d = dim R, c). For d = 0, f birational implies R = R ′ so there is nothing more to prove. Assume that d ≥ 1. Using induction on c, we may furthermore assume that x is not a zero divisor in R ′ . Then we may pick h ∈ R such that hR ′ ⊆ R, h not a zero-divisor.

(

To see this, write x = f /g, f, g ∈ R, g not a zero-divisor, a relation

x m + f 1 x m-1 + • • • + f m = 0, f 1 , . . . , f m ∈ R
and take h := g m-1 . By (5.3), we are reduced to proving that x p α ∈ hR ′ for some α ≥ 0.

To complete the proof, let

R := R √ (h) , R ′ := R ′ √ (hR ′ ) ,
so the map SpecR ′ → SpecR is again finite, dominant and radicial. Let x ∈ R ′ be the image of x. Applying now induction on d, there exists r ∈ R and α ≥ 0 such that

x p α -r ∈ √ (hR ′ ).
Since there exists n 0 ≥ 1 such that ( √ (hR ′ )) n ⊆ hR ′ for every n ≥ n 0 , we have

x p α ∈ hR ′ for every α ≥ α + log p n 0 .
This concludes the proof.

Lemma 5.2. Let f : X ′ → X be a finite and radicial morphism of kvarieties, with chark = p > 0. Then f ∞ : X ′ ∞ → X ∞ is integral and radicial. More precisely, there exists α ≥ 0 such that

O p α X ′ ∞ ⊆ Im(O X∞ → O X ′ ∞ ). (5.4)
Proof. It can be assumed that X is affine and f is dominant. By lemma 5.1, there exists α ≥ 0 and inclusions

O p α X ′ ⊆ O X ⊆ O X ′ .
Let u 1 , . . . , u r be generators of the O X -algebra O X ′ and,

u p α j -f j = 0, f j ∈ O X , 1 ≤ j ≤ r,
be relations satisfied by the generators. Then O X ′ ∞ is generated as an O X∞algebra by elements U j,n , 1 ≤ j ≤ r and n ≥ 0; there are relations

U p α j,n -F j,np α = 0, F j,np α ∈ O X∞ , 1 ≤ j ≤ r, n ≥ 0 in O X ′ ∞ as required.
Remark 5.3. We do not know if f ∞ is an integral morphism when removing the assumption chark = p > 0. 2) is an immediate application of lemma 5.4 and holds more generally for separable and strongly dominant morphisms of k-varieties with no extra assumption. Assume now that (2) holds. We first reduce to the case where f is a radicial morphism. To begin with, we may replace X by any Zariski neighborhood of ζ, thus assume that X is affine and Z regular. By remark 4.1, we may furthermore assume that X is irreducible. Let e : Y → X be a finite and étale neighborhood of ζ, and consider the diagram:

Lemma 5.4. Let f : X ′ → X be a dominant morphism of irreducible k- varieties. Let ζ ′ ∈ X ′ and ζ := f (ζ ′ ). Assume that both field extensions k(X ′ )|k(X) and k(ζ ′ )|k(ζ) are separable. If X ′ is arc-blunt at ζ ′ , so is X arc-blunt at ζ. Proof. Part (1) =⇒ (
Y g ← -Y ′ := Y × X X ′ e ↓ e ′ ↓ X f ← -X ′ .
By [START_REF] Grothendieck | Éléments de Géométrie Algébrique IV-2. Étude locale des schémas et des morphismes de schémas[END_REF] lemma 6.15.3.1 (ii), g satisfies the extra assumption (5.6) at each τ ∈ e -1 (ζ). Also some τ ′ ∈ e ′ -1 (τ ) induces a separable residue extension

k(τ ′ )|k(τ ) if and only if k(ζ ′ )|k(ζ) is separable, with ζ ′ := e ′ (τ ′ ). Finally, we have Y ∞ ≃ Y × X X ∞ , Y ′ ∞ ≃ Y ′ × X ′ X ′ ∞
because e, e ′ are étale morphisms. Therefore the theorem holds if and only if it holds for every τ ∈ e -1 (ζ), and for each morphism

Y i × X X ′ → Y i , Y i an irreducible component of Y with τ ∈ Y i .
Choosing a suitable étale neighborhood of ζ, it can be assumed the following for each i:

• the integral closure of the local ring O Y i ,τ is local, and

• the residue extension k(τ ′ )|k(τ ) is radicial for each τ ′ ∈ e ′ -1 (τ ).

In view of assumption (5.6), we have now achieved the following reduction:

the map X ′ × X SpecO X,ζ → X × X SpecO X,ζ is radicial. ( 5.7)
Let now g : Y ′ → Y be a finite and strongly dominant morphism of k-varieties, τ ∈ Y . Suppose that the map

Y ′ × Y SpecO Y,τ → Y × Y SpecO Y,τ
is radicial. After possibly replacing Y by some Zariski neighborhood U of ζ, Y ′ by g -1 (U ), there exists a nowhere dense closed subset W ⊂ Y such that τ belongs to each irreducible component of W , and g -1 (Y \W ) → Y \W is a radicial morphism. Note that the map

W ′ × W SpecO W,τ → W × W SpecO W,τ
is also radicial. Arguing by induction on the dimension, we may eventually assume that the map f :

X ′ → X is radicial. Let {ζ ′ } := f -1 (ζ) red , Z ′ := {ζ ′ } ⊆ X ′ .
There are two cases to consider:

Case 1: k(ζ ′ ) = k(ζ). By assumption, X ′ is arc-sharp at ζ ′ . Therefore there exists F ∈ O X ′ ∞ vanishing on X ′ • ∞ but not on Z ′ • ∞ .
By lemma 5.2, we may assume that F ∈ O X∞ by replacing the original F with

F p α . Since the morphisms X ′ • ∞ → X • ∞ and Z ′ • ∞ → Z • ∞ are birational, F ∈ O X∞ vanishes on X • ∞ but not on Z • ∞ . Therefore X is arc-sharp at ζ. Case 2: k(ζ ′ )|k(ζ) is purely inseparable. Let r ′ ∈ O X ′ satisfy r ′ p -r ∈ I Z ′ , r ∈ O X
s := (r ′ p -r) p α ∈ O X .
We deduce that there exists a relation y p α+1 -x p α+1 (r p α + s) = 0 in O X . In particular, there is a relation

X p α+1 0 (R p α 1 + S p α ) = 0 in O X∞ . The rational function X 0 is not a zero-divisor on X • ∞ , so R p α 1 +S p α = 0 in O X • ∞ . Since r ̸ ∈ k(ζ) p ,
r is a differential parameter on some nonempty open set of Z, whence none of functions R l , l ≥ 0, vanishes on Z. On the other hand, we have S l ∈ I Z∞ for every l ≥ 0 because s ∈ I Z ′ ∩ O X = I Z . This proves that the function

R p α 1 + S p α ∈ O X∞ vanishes identically on X • ∞ , but does not on Z • ∞ .
Therefore X is arc-sharp at ζ as stated.

Example 5.7. Let k be a field of characteristic p > 0. For n ≥ 1, let X n be the 2n-dimensional variety given by

(• • • ((y p + z 1 x p 1 ) p + z 2 x p 2 ) p + • • • ) p + z n x p n = 0
6 Some open problems and an example.

This section contains the main open questions about small components. We test and illustrate this material with a family of hypersurfaces with equation of the type (1.2).

The definition of arc-blunt/sharp is for the whole arc space X ∞ . A variant consists in replacing X ∞ by j -1 (SingX) = X SingX

:= {ζ}. Assume that X is arc-blunt at ζ, i.e. Z • ∞ ⊆ X ∞ \(SingX) ∞ . Is it true that Z • ∞ ⊆ j -1 (SingX)\(SingX) ∞ ?
In other terms, we ask whether each small irreducible component of the closed set j -1 (SingX) is an irreducible component of X ∞ . The answer is affirmative when dim O X,ζ = 1: this follows from the criterion for arc-sharpness in corollary 5.8 and theorem 4.4. Question 6.2. Let f : X ′ → X be a proper and birational morphism of k-varieties, with chark = p > 0, and let ζ ∈ X. Under which conditions are the following properties equivalent:

(1) X is arc-sharp at ζ;

(2) X ′ is arc-sharp at each ζ ′ ∈ f -1 (ζ) such that the field extension k(ζ ′ )|k(ζ)
is separable?

Implication (1) =⇒ (2) holds in a more general context for proper and surjective morphisms which are generically smooth (lemma 5.3). Theorem 5.5 gives a special case of the converse implication. Example 6.3 below points out a necessary restriction to big enough characteristics when compared to the dimension (tentatively, p > dim O X,ζ ). It is possible that (1) ⇔ (2) holds without further restriction on f .

We restate this question in the case of an irreducible variety X of dimension 3 in characteristic p ≥ 3: is it true that X ∞ is irreducible if and only if for every resolution of singularities π : Y → X and every ζ ∈ X, π

= {f = 0} ⊂ A 2n+1 k with f = y p + z 1 x p 1 + z 2 x p 2 • • • + z n x p n .
Then Z := Sing X = {y = x 1 = . . . = x n = 0} is regular. If ζ is the generic point of Z and π : X ′ → X the blowing up along Z, then π -1 (ζ) has no separable point. But p ≤ dim O X,ζ = n and we will next show that X ∞ is irreducible. In fact, note that

F 0 = Y p 0 + Z 1,0 X p 1,0 + • • • + Z n,0 X p n,0 , F i = Z 1,i X p 1,0 + • • • + Z n,i X p n,0 for 1 ≤ i ≤ p -1.
Hence there exist polynomials

g i,0 , h 0 ∈ ( k [ {Z i,r } 1≤i≤n 0≤r≤p-1 ]) L [W p-1,0 , . . . , W n,0 ] ,
where

L := Z 1,1 Z 2,2 • • • Z p-1,p-1 and W p-1,0 , . . . , W n,0 are indeterminacies, such that {F i = 0} p-1 i=0 if and only if X p i,0 = g i,0 (X p p-1,0 , . . . , X p n,0 ), 1 ≤ i ≤ p -1, and Y p 0 = h 0 (X p-1 p-1,0 , . . . , X p n,0
). We extend this argument by induction and we obtain that there exist

g i,r , h r ∈ ( k [ {Z i,r } 1≤i≤n r≥0 ]) L [W p-1,0 , . . . , W n,0 , . . . , W p-1,r , . . . , W n,r ] such that the ring (O X∞ ) L is isomorphic to   k [ {Y r , Z i,r , X i,r } 1≤i≤n r≥0 ] /   { Y p r -h r (X p p-1,0 , . . . , X p n,r ), X p i,r -g i,r (X p p-1,0 , . . . , X p n,r ) } 1≤i≤p-1,r≥0     L .

Now, let us consider the field

K := k({Z i,r } 1≤i≤n,r≥0 ) where Z 2 i,r = Z i,r .

From the previous study it follows that we may define an arc in X

∞ ϕ : Spec K[[ξ]] → X ∞ by Z i,r = Z 2 i,r , 1 ≤ i ≤ n, r ≥ 0, X p,0 = ξ, X i,r = 0 for i = p and r ≥ 1 or i ≥ p + 1, r ≥ 0 and X i,r = g i,r (Z i,r , X p-1,0 , . . . , X n,r ), Y r = h r (Z i,r , X p-1,0 , . . . , X n,r )
where g i,r (Z i,r , X p-1,0 , . . . , X n,r ) p = g i,r (X p p-1,0 , . . . , X p n,r ), 1 ≤ i ≤ p -1, r ≥ 0, and h r (Z i,r , X p-1,0 , . . . , X n,r ) p = h r (X p p-1,0 , . . . , X p n,r ). Then ϕ sends the generic point of Spec K where the a i (λ 1 , . . . , λ s ) are regular functions on some regular variety U with function field K, λ 1 , . . . , λ s defines a p-basis of K over the prime field and s ≥ n. For simplicity, write a i (λ) = a i (λ 1 , . . . , λ s ). Assume that rank((d λ i (a j ) i,j ) = n. This condition implies that Z := Sing X = {y = x 1 = . . . = x n = 0}.

[[ξ]] to X 0 ∞ and the special point (ξ) to the generic point of Z ∞ . Therefore Z ∞ ⊂ X 0 ∞ ,
In particular Z is regular.

Here

F 0 = Y p 0 + A 1,0 X p 1,0 + • • • + A n,0 X p n,0 , F r = A 1,r X p 1,0 + • • • + A n,r X p n,0 for 1 ≤ r ≤ p -1,
where ∑ n A j,n t n is the Taylor development of a j ( ∑ n Λ n t n ) in the same way of (2.1). Note that A j,1 = ∑ s i=1 d λ i (a j )(Λ 0 )Λ i,1 for 1 ≤ j ≤ n. The condition rank((d λ i (a j ) i,j ) = n together with the expression of the A j,1 implies that no linear combination of the A j,1 is equal to 0.

Two different cases occur, first consider that n ≤ p -1. There exists a n × n minor, say,

∆ =    A 1,1 • • • A n,1 . . . . . . A 1,n • • • A n,n    ,
such that L = det(∆) ̸ = 0. In fact, if all minors have determinant equal zero, then there is a linear combination of the columns of the matrix that is equal to zero. Hence, there exists c 1 , . . . , c n ∈ K such that c 1 A 1,1 + • • • + c n A n,1 = 0 which contradicts the previous statement. This will imply that X ∞ has two irreducible components since Z ∞ = {X i,m = 0} ̸ ⊂ {L = 0} and Z is regular.

We have to prove that X 0 ∞ ⊂ {L = 0}. Take h ∈ X ∞ \ Z ∞ . There exists indices (i, m) such that X i,m (h) ̸ = 0 with 1 ≤ i ≤ n and m ≥ 0. Take m minimum with this property and consider F mp (h), F mp+1 (h), • • • , F mp+n (h):

F mp+j (h) = (A 1,j (h)X 1,m (h) p +• • •+A n,j (h)X n,m (h) p ) = 0, 1 ≤ j ≤ n. (6.1)

Here, we make use of the following notation: given G ∈ O X∞ , by G(h) we mean the image of G by the morphism O X∞ -→ K induced by h : Spec K[[t]] -→ X.

Recall now that n ≤ p -1. The statement in (6.1) implies that L(h) = 0, and hence that X 0 ∞ ⊂ {L = 0}. Consider now the case n > p-1. The same argument used in the previous case, implies that there exists a (p -1) × (p -1) minor, say are indeterminacies, such that {F i = 0} p-1 i=0 if and only if X p i,0 = g i,0 (X p p-1,0 , . . . , X p n,0 ), 1 ≤ i ≤ p -1, and Y p 0 = h 0 (X p-1 p-1,0 , . . . , X p n,0 ). The same argument used in Example 6.3 implies that Z ∞ ⊂ X 0 ∞ , and hence X ∞ is irreducible.

∆ =    A 1,1 • • • A p-1,1 . . . . . .
The next question is about existence of a curve selection lemma. Computing a Zariski closure is a non trivial problem in arc spaces and a curve selection lemma is useful. Question 6.6. Let X|k be an irreducible k-variety and v be a valuation of k(X)|k. Denote by z ∈ X the center of v. Does there exist a birational morphism X v → X such that v has a center z v ∈ X v and (X v ) ∞ is irreducible?

Remark 3 . 1 .

 31 The definition of Local Uniformization in 3.1 is precisely the one in Zariski's generalized theorem of Local Uniformization, in its stronger form([13] A.III theorem U 3 , p. 858).

Lemma 4 . 2 .

 42 Let X|k be an irreducible k-variety, ζ ∈ X and Z := {ζ} ⊆ X. Let v be a prime divisor of X over ζ. There exists a birational morphism of varieties π : Y → X such that (1) Y is regular, and[START_REF] Gillet | Differential algebra. A scheme theory approach[END_REF]

  where h P = π • h P ′ . Then k(ζ ′ )|k(ζ) is a finite and separable field extension since k(ζ ′ ) ⊆ κ(P ). Blowing up at ζ ′ produces the required generically smooth prime divisor over ζ. The previous argument also shows that (2) =⇒ (3): the regular local ring O X ′ ,ζ ′ has a valuation v centered at ζ ′ and residue field k(ζ ′ ). Conversely, suppose that LU (X, ζ) holds. There exists a birational morphism of

  is a finite and separable field extension. Blowing up at ζ ′ produces the required generically smooth prime divisor over ζ, hence (3) =⇒ (1) holds. Statement (3) =⇒ (4) is trivial: let y ∈ Y be the center of the valuation v provided by (3). Since k(y)|k(ζ) is separable, there exists ζ ′ ∈ Reg({y}) with k(ζ ′ )|k(ζ) a finite and separable field extension.

Theorem 4 . 4 .

 44 Let X|k be a k-variety, ζ ∈ X and Z := {ζ} ⊆ X. Assume that there exists a generically smooth prime divisor over ζ. Then X is arcblunt at ζ.

Corollary 4 . 5 .

 45 Let X|k be an irreducible k-variety, ζ ∈ X and Z := {ζ} ⊆ X. Let C Z|X → Z be the corresponding normal cone and denote by C 1 , . . . C r those irreducible components of C Z|X which map dominantly to Z. Let[ C Z|X ] = m 1 [C 1 ] + • • • + m r [C r ]be the corresponding fundamental cycle. If there exists i,1 ≤ i ≤ r, such that (p m i and C i is generically smooth over k(ζ)), then X is arc-blunt at ζ.In particular, X is arc-blunt at ζ whenever the multiplicity m(ζ) of X at ζ is prime to p.Proof. It can be assumed that ζ ∈ X is a point of codimension one by cutting locally at ζ. Let X ′ → X be an étale covering over a neighborhood of ζ. Changing X by some irreducible component of X ′ , it can be assumed that r = 1, [C 1 ] is an affine line over k(ζ), i.e. the inverse image of ζ by the normalization map n

  M X,ζ is the maximal ideal of O X,ζ . Hence [k(ζ) : k(ζ)] divides m 1 and, since p m 1 by hypothesis, it follows that p [k(ζ) : k(ζ)]. Thus the extension k(ζ)|k(ζ) is separable. The conclusion follows from theorem 4.4. The last statement follows from the fact m(ζ) = r ∑ i=1 m i degP (C i ), so X satisfies the assumptions of the theorem if p m(ζ).

  and the residue r of r in k(ζ) is not in k(ζ) p . Let x, y ∈ I Z satisfy r ′ = y/x. By lemma 5.1, there exists α ≥ 0 such that

∞.Question 6 . 1 .

 61 Let X|k be an irreducible k-variety, ζ ∈ SingX and Z

A 1 ,

 1 p-1 • • • A p-1,p-1    , such that L = det(∆) ̸ = 0.Following the ideas of Example 6.3, we can find functions g i,0 , h 0 ∈ -1,0 , . . . , W n,0 ] , where W p-1,0 , . . . , W n,0

Question 6 . 5 .

 65 (Curve Selection) Let X|k be an irreducible k-variety, ζ ∈ X and f ∈ O X , f ̸ = 0. Assume that X is arc-blunt at ζ. Does there exist a field extension K ⊇ k(ζ ∞ ) and a wedge ϕ : SpecK[[u]] → X ∞ such that ϕ(0) = ζ ∞ and ϕ(η) ̸ ∈ (V (f )) ∞ ?

  X n , . . .] 1 ≤ i ≤ r and the fact that, for 1 ≤ i ≤ r, E i is irreducible hence O E i is a domain.Analogously, (2) follows from the fact that Y \Z is Zariski dense in Y .

	Definition 2.1. Given P ∈ X ∞ , with residue field κ(P ), we denote by
	h P : Spec κ(P )[[t]] → X
	the induced κ(P )-arc on X. We denote by 0 and η respectively the closed
	point and the generic point of Spec κ(P )[[t]]. The point h P (0) = j(P ) ∈ X
	is called the center of h P . The closure of h P (η) is called the support of h P
	and denote by Σ(h P ). We denote by v P the order function ord t h ♯ P : O X,h P (0) → N ∪ {∞}. The
	arc h P is said to be nonconstant if h P

  3) there exists a k-valuation v of X with center ζ in X and such that k v |k(ζ) is a separable field extension;

(4) for every proper and birational morphism π : Y → X, π -1 (ζ) has a point over the separable closure k(ζ) sep of k(ζ);

  The condition p > dim O X,ζ is necessary. For example, let n ≥ p > 0 and let X

	Example 6.3.
	-1 (ζ) has
	a point over the separable closure of k(ζ)?

  hence X ∞ is irreducible. Fix K a function field of positive transcendence degree over the perfect field k.Let X = {f = 0} ⊂ A n+1 U with f = y p + a 1 (λ 1 , . . . , λ s )x p 1 + a 2 (λ 1 , . . . , λ s )x p 2 + • • • + a n (λ 1 , . . . , λ s )x p n ,

	Example 6.4.

Partially supported by MTM2015-68524-P. Partially supported by MTM2017-87017-P and VA128G18.

Proof. Since k(X ′ )|k(X) is separable and f is dominant, we have:

(5.5)

On the other hand, (4.1)) and the conclusion follows from (5.5), since

Theorem 5.5. Let f : X ′ → X be a finite and birational morphism of k-varieties, with chark = p > 0. Let ζ ∈ X and Z := {ζ} ⊆ X. We assume that the following additional property holds: the map X ′ × X S → X × X S is radicial, where S := SpecO X,ζ \{ζ}. (5.6) The following properties are equivalent:

is separable.

Remark 5.6. We do not know if assumption (5.6) can be avoided. It is satisfied in the following two situations:

(1) dim O X,ζ = 1;

(2) there exists a finite and strongly dominant morphism X → X 0 which is generically purely inseparable with X 0 normal.

Examples of (2) include hypersurfaces with equation of the form

Such singularities have played an important role in Resolution of Singularities and have been pointed out by Zariski as a test case for Resolution, see [START_REF] Zariski | The fundamental ideas of abstract algebraic geometry[END_REF] p.88.

in A 2n+1 k . Applying theorem 5.4 to the normalization f : X n → X n of X n it can be proved that (X n ) ∞ has exactly n + 1 irreducible components. More precisely, fix n and set X := X n , we have

x n .

Since u p = z n , the hypothesis (5.6) in th. 5.4 holds. For 0 ≤ r ≤ n let Y r be the subvariety of X given by

Arguing by induction on n, we conclude that {Y 0 r } n r=0 are the irreducible components of (X n ) ∞ .

We state two immediate corollaries of theorem 5.4:

Proof. It follows from theorems 4.3 and 5.4. Corollary 5.9. Let X|k be a k-variety of dimension two and n : X → X be the normalization morphism. The decomposition into irreducible components of X ∞ is Proof. It follows from corol. 5.5. Note that X is arc-blunt at every closed point since k is a perfect field.

Using a classical theorem of Albanese, we can prove the following: Proposition 6.7. Let k be an algebraically closed field of characteristic p > 0 and K|k be a function field of dimension d ≥ 1. If p > d!, there exists a projective variety X|k such that k(X) = K and X ∞ is irreducible.

Proof. It follows from [START_REF] Lipman | Introduction to resolution of singularities[END_REF], last sentence on p. 202, and corol. 4.4 from this article.