Sylvia Piotin

Aassif Benassarou

Frédéric Blanchard

Olivier Nocent

Eric Bertin
email: ebertin@chu-reims.fr

Éric Bertin

Abdominal Morphometric Data Acquisition Using Depth Sensors

come

I. INTRODUCTION

Nutrition and eating disorders have become a national public health priority. In 2001, France launched the National Health and Nutrition Programme (PNNS 1 ) which aims to improve the health status of the population by acting on one of its major determinants: nutrition. At the regional level, Champagne-Ardenne particularly suffers from obesity. According to the 2012 study ObEpi [START_REF]ObÉpi -enquête épidémiologique nationale sur le surpoids et l'obésité[END_REF], the Champagne Ardenne region has experienced the highest increase in prevalence of obesity between 1997 and 2012 (+145,9%) and became the second region most affected behind the Nord Pas-de-Calais region, with an obesity rate of 20,9% (the national average is 15%). Within this context, the study of eating behaviors is an important issue for the understanding and prevention of disorders and diseases related to food. Building typologies of patients with eating disorders would help to better understand these diseases, thus allowing their prevention.

We plan to develop a new acquisition pipeline in order to identify new objective variables based on morphological parameters like abdominal diameter, body surface area, etc. While a recent report by the French Academy of Medicine on unnecessarily prescribed tests2 highlights the generalized use of heavy and expensive imaging, the novelty of our Figure 1. Presentation of our mobile system approach lies in the use of a consumer electronics device (the Microsoft R Kinect TM sensor was initially dedicated to the Microsoft R Xbox 360 TM game console). This device has the advantage of being inexpensive, lightweight and less intrusive than conventional medical equipments. Even if these devices have been used in eHealth projects, their use has often been limited to the adaptation of successful video games to the medical environment: physical training programs to action against obesity [START_REF] Nitz | Is the Wii Fit TM a newgeneration tool for improving balance, health and well-being? A pilot study[END_REF], rehabilitation programs [START_REF] Huang | Kinerehab: a kinect-based system for physical rehabilitation: a pilot study for young adults with motor disabilities[END_REF]. Within our project, the device is used as a measurement tool to collect morphological information in order to enrich or to confront the information extracted from surveys filled in by patients. Beyond its cost, this device can also be easily deployed in patient's homes or in medical practitioners offices, allowing monitoring on a regular basis (Figure 1)

This paper presents the acquisition and analysis methodology that we have implemented. The results presented were obtained on a sample of seventeen healthy patients without any medical context. The main objective was to establish a proof of concept.

After an overview of the uses of Kinect TM like sensors in a medical context, we present our abdominal morphology acquisition system. In the next section, the computation of quantitative indicators from raw data is exposed. Finally, we propose a first statistical analysis before concluding and presenting our future works.

II. RELATED WORK

The Microsoft R Kinect TM peripheral, as well as its recent competitor ASUS R Xtion TM , are made of an RGB camera, an IR emitter and an IR sensor. The latter is able to evaluate a depth map aligned with the video frame. Based on these two sources of data, it is possible to reconstruct people facing the device in 3D [START_REF] Shotton | Real-time human pose recognition in parts from single depth images[END_REF], [START_REF] Weiss | Home 3D body scans from noisy image and range data[END_REF].

the affordable cost of these new peripherals is probably the reason why they are inspiring so many projects. For instance, this technology is growingly involved in healthcare: apart from their use in fall risk assessment in elderly population [START_REF] Stone | Evaluation of an inexpensive depth camera for passive in-home fall risk assessment[END_REF], these cameras are also employed in motor rehabilitation programs [START_REF] Chang | A Kinect-based system for physical rehabilitation: A pilot study for young adults with motor disabilities[END_REF], [START_REF] Da Gama | Poster: Improving motor rehabilitation process through a natural interaction based system using Kinect sensor[END_REF], or for the improvement of workplace ergonomics [START_REF] Dutta | Evaluation of the Kinect TM sensor for 3-D kinematic measurement in the workplace[END_REF].

Not necessarily calling upon motion capture techniques, others use these cameras to quickly collect anthropometric data. Thanks to usual statistical tools such as principal component analysis, it is possible to extrapolate different morphological dimensions from a few measurements [START_REF] Samejima | A body dimensions estimation method of subject from a few measurement items using KINECT[END_REF]. One can also deduce precisely the position of the center of mass of an individual if one combines a Kinect and a Wii Balance Board, popularized by the video game Wii Fit [START_REF] Gonzalez | Estimation of the center of mass with Kinect and Wii balance board[END_REF]. Finally, Velardo and Dugelay have developed an automated system capable of providing nutritional advice depending on the body mass index and basal metabolism rate calculated from parameters either measured or statistically induced from measurements [START_REF] Velardo | What can computer vision tell you about your weight?[END_REF].

III. ACQUISITION ACCURACY

Since the popularization of RGB-D devices (RGB + depth), many metrological studies have dealt with the quality of acquisition [START_REF] Gonzalez-Jorge | Metrological evaluation of Microsoft Kinect and Asus Xtion sensors[END_REF]: accuracy (difference between the measured and the real value) ranges from 5 mm to 15 mm at a distance of 1 m, from 5 mm to 25 mm at a distance of 2 m. Even if these lightweight tracking systems do not really compete with more cumbersome ones (such as Vicon's3 ), they can be a low cost alternative solution in many applications where high accuracy (less than 1 mm) is not a matter of concern.

We also performed our own experiments. We scanned a wood panel with a 700 mm long diagonal at three different distances (800 mm, 1600 mm and 2400 mm). The values shown in tables I and II represent the mean value of ten consecutive measures. The improved accuracy observed, in comparison with the aforementioned results, can be explained by the use of the mean value of several measurements. According to a framerate of 30 frames per second, a measure can be performed in a third of a second following our method. This is not really an issue since we are measuring static poses of patients. In our study, we use the 3D reconstruction and pose detection capabilities of RGB-D devices to measure in real time several morphological characteristics of a patient (size, bust waist and hip measurements, shape of the abdomen, . . . ). These data produced by the acquisition system can be compared to the patient's responses provided during the Stunkard's test: this test is to ask the patient about his perception (often subjective) of his morphology and ask him to lie within a range of silhouettes [START_REF] Stunkard | Use of the Danish Adoption Register for the study of obesity and thinness[END_REF].

With the OpenNI library [START_REF] Openni | OpenNI framework[END_REF] and the NiTE middleware [START_REF] Nite | NiTE middleware[END_REF], we are able to identify the pixels corresponding to each user located in the sensor field. With this information, we reconstruct the visible body surface in 3D space. Since NiTE also allows us to track the skeleton of each individual, we can position planes on the abdomen: the sagittal plane and the transverse plane. By calculating the intersection between the surface and each plane we get a sagittal profile and a transverse profile (Figure 2). We can record these profiles in the patient's medical file to establish a follow-up. These profiles are also stored in an anonymized database to conduct a statistical analysis.

V. PARAMETER EXTRACTION

The previous step of acquisition provides two profiles from the intersection between the body surface and the sagittal and transverse planes. These profiles, composed of segments joining points from the reconstruction, are relatively noisy. A first step of smoothing using spline interpolation can adjust these geometric data and make them more "understandable" and exploitable (Figure 3).

After the first treatment we already have, for each individual, a first visual "signature" of the abdominal morphology (Figure 4). The interest of these visual signatures is to provide a simplified graphical representation to enable a fast and synthetic visualization. On the other hand, these visual descriptions facilitate description and interpretation of clusters when creating typologies by clustering. Finally, they have a major interest in monitoring the evolution of the patient by the doctor.

The profiles obtained during the acquisition are also used to extract features and more "conventional" measurements. It is possible, for example, from the smoothed cross-section, to calculate various lengths that are good estimations of the abdominal dimensions of the subject. In this study, we have limited ourselves, for example, to calculate the diameter (d H and d V ), the height (h H and h V ) and the length (l H and l V ) of each profile (transverse and sagittal) (Figure 5). Other calculations of lengths or surfaces are possible [START_REF] Lu | Automated anthropometric data collection using 3d whole body scanners[END_REF], [START_REF] Yu | The 3d scanner for measuring body surface area: a simplified calculation in the chinese adult[END_REF], [START_REF] Lin | Comparison of three-dimensional anthropometric body surface scanning to waist-hip ratio and body mass index in correlation with metabolic risk factors[END_REF]. The choice of these numerical descriptors of the abdominal morphology defines a parameter space in which subjects are represented. In this study, individuals who have lent to the experience are represented in a space of dimension 6 defined by the variables

d H , d V , h H , h V , l H et l V (Table III). id d H h H l H d V h V l V 1

VI. TOWARDS A TYPOLOGY OF ABDOMINAL

MORPHOLOGIES

After describing individuals in this parameter space, our goal is to automatically extract groups of abdominal morpholo- gies by clustering.

We first proceed to a principal component analysis to reduce the dimensionality of the problem and to project people in a subspace whose components are uncorrelated. The representation of subjects in the factorial subspace (related to the first two principal components) allows us to visualize the similarities between individuals (Figure 6).

First, clustering is achieved, using the k-medoids algorithm. The algorithm is used in the space of the first 3 factors (consideration of the eigenvalues shows that the first 3 principal components explain 94% of the inertia) and it is set to search for 3 clusters (after reading the dendrogram obtained by hierarchical clustering).

The statistical description of obtained clusters is presented in the table IV. The medoids produced by the clustering algorithm provide representative individuals of each cluster (Table V). For a more detailed extraction of representative individuals (or exemplars) we can use the method described in [START_REF] Blanchard | Data representativeness based on fuzzy set theory[END_REF].

VII. CONCLUSION

In this paper, we presented a software prototype able to acquire abdominal morphological parameters using a consumer electronics depth sensor. This prototype is lightweight and inexpensive, and the acquisition is quite insensitive to the capture conditions (this type of sensor is designed to operate in most environments, in private homes).

We also proposed an algorithmic solution to analyze collected data. The results presented in this paper allowed us to validate the principle and the calculation methods of our tool. The aim was not to draw medical conclusions but to establish a proof of concept. Our prototype will now be used in a clinical context dealing with obesity and eating behaviors. The perspectives of this work are multiple. In order to provide medical practitioners the ability to make a diagnosis, we started to combine this tool with the web framework that we developed [START_REF] Nocent | Toward an immersion platform for the World Wide Web using 3D displays and tracking devices[END_REF]. It already supports RGB-D cameras and can transmit the skeletons of users to a distant browser. Provided we also stream the acquired profiles and the descriptors extracted from the analysis, the practitioner may remotely have all the information he needs. So far, our system performs various measurements from a single capture. Using 3D scanning techniques [START_REF] Curless | A volumetric method for building complex models from range images[END_REF], we could capture the entire body surface of a subject, either by moving the camera around, like KinectFusion [START_REF] Izadi | KinectFusion: real-time 3D reconstruction and interaction using a moving depth camera[END_REF], or by combining several sources simultaneously, as is the case with OmniKinect [START_REF] Kainz | Om-niKinect: real-time dense volumetric data acquisition and applications[END_REF]. Finally, the data analysis should be extended to enable the construction of typologies adapted to the studied diseases, and a much larger amount of data.

Figure 2 .Figure 3 . 1 Patient 2 Figure 4 .

 23124 Figure 2. Acquisition of the sagittal and transverse profiles. The individual is highlighted and the two profiles are overprinted on his abdomen.

Figure 5 .

 5 Figure 5. Examples of morphometric indicators calculated from the sagittal and transverse profiles

Figure 6 .

 6 Figure 6. Representation, in the factorial design, of individuals and clusters obtained by the k-medoids algorithm.

Table V MEDOIDS

 V : REPRESENTATIVE INDIVIDUALS OF OBTAINED CLUSTERS (INDIVIDUALS WHOSE IDENTIFIERS ARE 14, 8 AND 16 ARE RESPECTIVELY THE MEDOIDS OF CLUSTERS 1, 2 AND 3)

	param. cluster mean	sd median	min	max
	d H	1	324.76 16.72	323.99 304.31 348.24
	d H	2	275.24 23.53	272.49 253.21 300.02
	d H	3	280.81 25.16	289.61 229.57 308.40
	h H	1	84.39	7.55	82.27	75.24	95.75
	h H	2	56.43	7.32	54.46	50.29	64.53
	h H	3	68.30	14.86	66.71	53.23 100.68
	l H	1	382.20 19.19	383.02 362.60 412.56
	l H	2	301.91 23.29	308.42 276.06 321.26
	l H	3	323.83 18.97	331.72 283.72 344.00
	d V	1	259.31 64.15	268.89 156.98 330.10
	d V	2	222.77 13.79	224.42 208.23 235.66
	d V	3	320.27 37.07	320.30 274.25 384.82
	h V	1	20.91	15.58	15.62	9.29	46.73
	h V	2	7.75	4.89	9.56	2.21	11.49
	h V	3	26.36	18.62	16.55	11.87	64.84
	l V	1	266.88 65.20	273.02 159.15 333.59
	l V	2	226.12 10.57	227.48 214.94 235.95
	l V	3	329.85 38.60	340.40 278.51 386.62
				Table IV
			STATISTICAL DESCRIPTION OF CLUSTERS
	Cluster	id	d H	h H	l H	d V	h V	l V
	1	14 332.15 82.27 383.46 288.23 23.58 296.57
	2	8 272.49 64.53 308.42 208.23 11.49 214.94
	3	16 302.74 62.13 331.72 278.60 30.04 289.58

http://www.mangerbouger.fr/pnns/

http://www.academie-medecine.fr

http://www.vicon.com

ACKNOWLEDGEMENTS

The authors would like to warmly thank the volunteers who helped them test and improve their prototype.


Olivier Nocent
email: olivier.nocent@univ-reims.fr

Crestic Sic

Sylvia Piotin

Jaisson Maxime

Laurent Grespi

Crestic Lucas

Sic

Toward an immersion platform for the World Wide Web using autostereoscopic displays and tracking devices

Keywords: I.3.1 [Computer Graphics]: Hardware Architecture-Three-dimensional displays, I.3.2 [Graphics Systems]: Distributed/network graphics-, I.3.3 [Computer Graphics]: Picture/Image Generation-Display algorithms, I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Virtual reality, J.3 [Life and Medical Sciences]: Medical information systems-, Web graphics, 3D perception, autostereoscopic displays, natural interaction

Figure 1: Autostereoscopic technology allows 3D images popping out of the screen while natural interaction provides a seamless manipulation of 3D contents.

Introduction

A recurrent and key issue for 3D technologies resides in immersion. 3D web technologies try to reach the same goal in order to enhance the user experience. Interaction and depth perception are two factors that significantly improve the feeling of immersion. But these factors rely on dedicated hardware that can not be addressed through JavaScript for security reasons. In this paper, we present an original way to interact with hardware via a web browser, using web protocols by providing an easy-to-use immersion platform for the World Wide Web. This plugin-free solution leveraging the brand new features of HTML5 (WebGL, WebSockets) allows to handle autostereoscopic displays for immersion and different type of tracking devices for natural interaction (Figure 2). Because We-bGL is a low-level API, we decided to develop the WebGLUT (We-bGL Utility Toolkit) API on top of WebGL. WebGLUT enhances WebGL by providing extra features like linear algebra data structures (vectors, matrices, quaternions), triangular meshes, materials, multiview cameras to handle autosteroscopic displays, controllers to address different tracking devices, etc. WebGLUT was created at the same time (and with the same philosophy) as WebGLU [START_REF] Delillo | WebGLU development library for WebGL[END_REF]SpiderGL [Di Benedetto et al. 2010].

Our contribution is written as follows: Section 2 presents related work to 3D content on the web and on 3D displays. Section 3 describes the autostereoscopic display technology by providing equations and algorithms to generate multiple views. Section 4 is dedicated to our network-based tracking system. Finally, we mention very shortly in Section 5 a case study related to medical imaging using our immersion platform.

Related work

Web browsers have acquired over the past years the ability to efficiently incorporate and deliver different kinds of media. 3D content can be considered as the next evolution to these additions although requirements of 3D graphics in terms of computational power and unifying standard are more restrictive than still images or videos.

Several technologies have been developed to achieve this integration. The Virtual Reality Markup Language (VRML) [START_REF] Raggett | Extending WWW to support platform independent virtual reality[END_REF]] replaced afterward by X3D [START_REF] Brutzman | X3D: Extensible 3D Graphics for Web Authors[END_REF] was proposed as a text-based format for specifying 3D scenes in terms of geometry and material properties and for the definition of basic user interaction. Even if the format itself is a standard, the rendering within the web browser usually relies on proprietary plugins. But the promising X3DOM [START_REF] Behr | X3DOM: a DOM-based HTML5/X3D integration model[END_REF]] initiative aims to include X3D elements as part of the HTML5 DOM tree. More recently, the WebGL [Khronos Group 2009] API was introduced to provide imperative programming mechanisms to display 3D contents in a more flexible way. As its name suggests, WebGL is the JavaScript analogous to the OpenGL ES 2.0 API for C/C++. It provides capabilities for displaying 3D content within a web browser which was previously the exclusive domain of desktop environments. Leung and Salga [START_REF] Leung | Enabling WebGL[END_REF] emphasize the fact that WebGL gives the chance to not just replicate desktop 3D contents and applications, but rather to exploit other web features to develop richer content and applications. In this context, web browsers could become the default visualization interface [START_REF] Mouton | Collaborative visualization: current systems and future trends[END_REF]].

Even if real-time 3D rendering has become a common feature in many applications, the resulting images are still bidimensional. Nowadays, this limitation can be partly overcome by the use of 3D displays that significantly improve depth perception and the ability to estimate distances between objects. Therefore, the content creation process needs to be reconsidered For computer generated imagery, the rendering system can seamlessly render one or more related views depending on the application [START_REF] Abildgaard | An autostereoscopic 3D display can improve visualization of 3D models from intracranial MR angiography[END_REF][START_REF] Benassarou | Autostereoscopic visualization of 3D time-varying complex objects in volumetric image sequences[END_REF]]. 3D contents are obviously clearer and more usable than 2D images because they doe not involve any inference step. Finally, 3D contents can also address new emerging devices like 3D smartphones [START_REF] Harrold | Autostereoscopic display technology for mobile 3DTV applications[END_REF] and mobile 3DTV, offering new viable platforms for developing 3D web applications.

Autostereoscopic technology

The term stereoscopy denotes techniques where a separate view is presented to the right and left eye, these separate views inducing a better depth perception. Different solutions exist for the production of these images as well as for their restitution. For image restitution with non time-based techniques, one can use anaglyph and colored filters, polarizing sheets with polarized glasses or autostereoscopic displays [START_REF] Halle | Autostereoscopic displays and computer graphics[END_REF]]. The technology of autostereoscopic displays presents the great advantage to allow multiscopic rendering without the use of glasses. Therefore, the spectator can benefit from a stereoscopic rendering more naturally, and this is especially true for 3D applications in multimedia.

Multiple view computation

The geometry of a single camera is usually defined by its position, orientation and viewing frustum as shown in Figure 3. But in stereo rendering environments, we need two virtual cameras, one for each left/right eye. And for multiview autostereoscopic displays [Prévoteau et al. 2010], we need up to N virtual cameras.

Each virtual camera has a given offset position and its own off-axis asymmetric sheared viewing frustum, the view direction remaining unchanged. The near/far planes are preserved, and a focus plane has to be manually defined where the viewing zones converge. The choice of the focus distance will determine if the objects appear either behind or in front of the screen, providing or not a pop-out effect. Given a perspective projection matrix P, the following calculations allow to identify six parameters l, r, b, t, n, f as defined in the OpenGL glFrustum command: l and r are the left and right coordinates of the vertical clipping planes, b and t are the bottom and top coordinates of the horizontal clipping planes, n and f are the distances to the near and far depth clipping planes. First, the distances n and f (refer to Figure 3 for the geometric signification of these terms) are given by Equation 1.

n = 1 -k 2k P34 f = nk where k = P33 -1 P33 + 1 (1)
In order to compute l, r, b and t, we need to define the half-width wF (respectively wn) of the image at the focus distance F (respectively at the near distance n) from the horizontal field of view α:

wF = tan(α/2)F wn = tan(α/2)n (2)
where tan(α/2) = P -1 11 according to the definition of the projection matrix P.

The viewing frustum shift for the camera j for j ∈ {1, . . . , N } in the near plane, denoted as s j n is given by Equation 3where d is the interocular distance and ws the physical screen width.

s j n = dwn ws (j -1) - N -1 2 (3)
Finally,

l = -wn + s j n r = wn + s j n t = ρwn b = -t (4)
where ρ is the viewport aspect ratio. The camera position offset along the horizontal axis is given by s j F , the viewing frustum shift in the focus plane.

s j F = dwF ws (j -1) - N -1 2 (5)

Autostereoscopic image rendering

Thanks to the computations presented in the previous section we are able to produce N separate perspective views from N different virtual cameras. The use of these images depends on the chosen stereoscopic technology. One of the simplest cases relies on quadbuffering, where the images are rendered into left and right buffers independently, the stereo images being then swapped in sync with shutter glasses. Other techniques, which are not time-based, need the different images to be combined in one single image.

Let I j for j ∈ {1, . . . , N } be the image generated by the virtual camera j, the color components I f inal c (x, y) for c ∈ {R, G, B} of the pixel (x, y) in the final image are given by:

I f inal c (x, y) = I M(x,y,c) c (x, y) c ∈ {R, G, B} (6)
where M is a mask function and R, G, B stand for red, green and blue channels. As Lenticular sheet displays consist of long cylindrical lenses that focus on the underlying image plane so that each vertical pixel stripe corresponds to a given viewing zone, the function M does not depend on the color component c and is simply given by Equation 7.

I f inal c (x, y) = I x mod N c (x, y) (7)
It is worth noticing that Equation 7clearly shows that the horizontal resolution of the restituted image is reduced by a factor 1/N compared to the native resolution m of the display. This resolution loss is one of the main drawbacks of autostereoscopic displays, even if it is only limited to 3m/N while using wavelength-selective filters because each pixel's RGB components correspond to three different view zones [Hübner et al. 2006]. Technically speaking, the WebGL implementation of the autostereoscopic image rendering is a twostep rendering process using deferred shading techniques.

Pass #1 (multiple images rendering) renders N low-resolution images by shifting the viewport along the y-axis. The N images are vertically stored in a single texture via a FrameBuffer Object (FBO).

Pass #2 (image post-processing) renders a window-aligned quad. Within the fragment shader, each color component of the output fragment is computed according to Equation 7where the N images are read from an input 2D texture.

These two rendering passes are encapsulated in the method shoot() of the MultiViewCamera object.

Tracking system

Another characteristic of our immersion platform for the World Wide Web resides in the use of tracking devices in order to interact with a 3D scene in a straightforward way. Our aim is to address a large range of tracking devices like mouses, 3D mouses, flysticks or even more recent devices like IR depth sensors (Microsoft R Kinect, ASUS R Xtion Pro). In the same fashion we handle autostereoscopic displays, we propose a plugin-free solution to interact with ad-hoc tracking devices within a web browser by using HTML5 WebSockets [W3C R 2012]. The proprietary ART R DTrack tracking system is an optical tracking system which delivers visual information to a PC in order to be processed. The resulting information, 3D position and orientation of the flystick is then broadcast on every chosen IP address, using the UDP protocol. A PHP server-side script, which can be seen as a WebSocket Server, is running on the web server. The WebSocket server is waiting for UDP datagrams, containing locations and orientations, from the DTrack system. At receipt, the data is parsed and sent via WebSocket to the client using the JSON format. Thanks to this networked architecture, we are able to stream JSON encoded data coming from the tracking system to the web browser. The location and the orientation of the flystick are then used to control the WebGLUT virtual camera.

Using the same approach, we have also imagined a more affordable solution to interact with a 3D scene in an even more straightforward way. Indeed, we use recent IR depth sensors like Microsoft R Kinect and ASUS R Xtion Pro to perform Natural Interaction. Just like the ART R DTrack system, our C++ program acts as a UDP server and, at the same time, collects information about the location and the pose of a user facing an IR depth sensor. Thanks to the OpenNI framework [OpenNI TM 2010], we are able to track the user's body and detect characteristic gestures. This information, streamed over the network using our hardware/software architecture can be used to interact with a 3D scene: move the virtual camera, trigger events, etc. These aspects are exposed within the WebGLUT API through the concept of Controller. A Controller can be attached to any type of Camera. This controller is responsible for updating the properties of the camera (position, orientation, field of view, etc.) depending on its state change. At the time of writing, we manage three types of controllers: MouseController, DTrackController and KinectController.

5 Case study: ModJaw R

The ModJaw R (Modeling the Human Jaw) project has been developed by our research team and Maxime Jaisson who is preparing a PhD thesis in odontology related to the impact of Information Technology on dentistry practice and teaching. ModJaw R [START_REF] Jaisson | ModJaw R : a kind of magic[END_REF] aims to provide 3D interactive educational materials for dentistry tutors for teaching mandibular kinematics. There exist several similarities with a former project called MAJA (Modeling and Animation of JAw movements) [START_REF] Reiberg | MAJA: Modeling and Animating the Human Jaw[END_REF]]. One main difference between ModJaw R and MAJA consists in the data nature. Indeed, we use real-world data obtained by motion capture and CT scanners. In this way, students are able to study a wide variety of mandible motions according to specific diseases or malformations. Among the future works, Jörg Reiber wanted to add an internet connection to MAJA. In this way, ModJaw R can be seen as an enhanced up-to-date version of MAJA relying on real-world data sets and cutting edge web technologies exposed by the brand new features of HTML5 (WebGL, WebSockets, etc.). The choice of web technologies to develop this project was mainly dictated by the following constraints:

Easy-to-use: users just have to open a web browser to access to the software and its user-friendly graphical interface. As it is fully web-based, ModJaw R also incorporates online documentation related to anatomy, mandibular kinematics, etc.

Easy-to-deploy: the framework does not require any install, it can be used from all the computers within the faculty or even from your home computer if you use a HTML5 compatible web browser. Since the software is hosted on a single web server, it is really easy to upgrade it.

Conclusion

In this paper, we have presented an original solution for providing immersion and natural interaction within a web browser. The main benefits of our contribution rely on the seamless interaction between a web browser and autostereoscopic displays and tracking devices through new HTML5 features like WebGL and WebSockets. This plugin-free framework allows to enhance the user experience by leveraging dedicated hardware via JavaScript. Thanks to its network-based approach, this framework can easily be extended to handle other devices in the same fashion. For instance, we began to explore the possibility to manage haptic devices. As our tracking system is completely plugin-free and fully network-based, it could be seamlessly integrated in web-based collaborative environments allowing users to remotely interact with shared 3D contents displayed in a web browser.

Figure 2 :

 2 Figure 2: Global structure of our immersion platform for the World Wide Web

Figure 3 :

 3 Figure 3: Geometry of a single camera (left) and multiple axis-aligned cameras (right).

Acknowledgements

The authors would like to thank Romain Guillemot, research engineer at CReSTIC SIC, for his expertise in autostereoscopic displays and his precious help for porting the source code of multiview cameras from OpenGL to WebGL.


Antoine Jonquet
email: jonquet@leri.univ-reims.fr

Olivier Nocent

Yannick Remion

The art to keep in touch The "good use" of Lagrange multipliers

Keywords: Physically-based animation, constraints, contact simulation

Physically-based modeling for computer animation allows to produce more realistic motions in less time without requiring the expertise of skilled animators. But, a computer animation is not only a numerical simulation based on classical mechanics since it follows a precise story-line. One common way to define aims in an animation is to add geometric constraints. There are several methods to manage these constraints within a physically-based framework. In this paper, we present an algorithm for constraints handling based on Lagrange multipliers. After few remarks on the equations of motion that we use, we present a first algorithm proposed by Platt. We show with a simple example that this method is not reliable. Our contribution consists in improving this algorithm to provide an efficient and robust method to handle simultaneous active constraints.

Introduction

For about two decades, the computer graphics community has investigated the field of physics in order to produce more and more realistic computer animations. In fact, physically-based modeling in animation allows to generate stunning visual effects that would be extremely complex to reproduce manually. On one hand, the addition of physical properties to 3D objects automates the generation of motion just by specifying initial external forces. On the other hand, physicallybased animations are even more realistic than traditional key-framed animations that require the expertise of many skilled animators. As a consequence, the introduction of physically-based methods in modeling and animation significantly reduced the cost and production time of computer generated movies. But, one main drawback of this kind of framework is that it relies on heavy mathematics usually hard to tackle for a computer scientist. A second main disadvantage concerns the input of a physically-based animation: in fact, forces and torques are not really user-friendly since it is really difficult to anticipate a complex motion just by specifying an initial set of external forces.

A computer animation is definitely not a numerical simulation because it follows a story-line. According to Demetri Terzopoulos [TPB + 89], an animation is simulation plus control. One way to ensure that the objects fulfill the goals defined by the animator is to use geometric constraints. A constraint is an equality or an inequality that gathers different parameters of the animation like the total time elapsed, the positions or the orientations of the moving objects. In a less general way, mechanical simulations also benefit from the use of constraints in order to prevent interpenetration between physical objects for example.

There are several methods to handle constraints, summarized in a survey paper by Baraff [Bar93]. But, since our research work is mostly devoted to mechanical simulation, we decided to focus on the use of Lagrange multipliers to manage geometric constraints. In fact, numerical simulations require robust and reliable techniques to ensure that the constraints are never violated. Moreover, with this method we are also able to measure the amount of strain that is necessary to fulfill a given constraint. In this paper, we present a novel algorithm to manage efficiently several simultaneous active geometric constraints. We begin by detailing the physical equations that we use before presenting Platt's algorithm [START_REF] Platt | A Generalization of Dynamic Constraints[END_REF] that is the only algorithm of this type based on Lagrange multipliers. With a simple example, we demonstrate that this algorithm is not suitable for handling simultaneous active constraints. We then introduce our own contribution in order to show how to improve Platt's algorithm to make it reliable, robust and efficient.

Lagrange equations of motion

Lagragian dynamics consist in an extension of newtonian dynamics allowing to generate a wide range of animations in a more efficient way. In fact, Lagrange equations of motion rely on a set of unknowns, denoted as a state vector x of generalized coordinates, that identifies the real degrees of freedom (DOF) of the mechanical systems involved. Within this formalism, the DOF are not only restricted to rotations or translations. For example, a parameter u ∈ [0, 1] which gives the relative position of a point along a 3D parametric curve can be considered as a single generalized coordinate.

Unconstrained motion

The evolution of a free mechanical system only subject to a set of external forces is ruled by the Lagrange equations of motion (1).

M ẍ = f

(1)

M is the mass matrix. ẍ is the second time derivative of the state vector. Finally, the vector f corresponds to the sum of external forces. For more details concerning this formalism, we suggest to read [START_REF] Goldstein | Classical Mechanics[END_REF] and [START_REF] Arnold | Mathematical Methods of Classical Mechanics[END_REF].

Constrained motion

By convention, an equality constraint will always be defined as in equation ( 2) where E is the set of indices of all the equality constraints.

g k (x) = 0 ∀k ∈ E (2)
Constraints restrict the set of reachable configurations to a subspace of R n where n is the total number of degrees of freedom. As mentioned before, there exists three main methods to integrate constraints in equation (1)

The projection method consists in modifying the state vector x and its first time derivative ẋ in order to fulfill the constraint. This modification can be performed with an iterative method like the Newton-Raphson method [START_REF] Vetterling | The Art to keep in touch: The "Good Use of Lagrange Multipliers[END_REF]. Even if this method is very simple and seems to ensure an instantaneous constraint fulfillment, it is not robust enough: indeed it can not guarantee that the process converges in the case of simultaneous active constraints.

The penalty method adds new external forces, acting like virtual springs, in order to minimize the square of the constraint equation, considered as a positive energy function. The main advantage of this method is its compatibility with any dynamic engine since it only relies on forces. But this method leads to inexact constraint fulfillment, allowing interpenetration between the physical objects. In order to diminish this interpenetration, the stiffness of the virtual springs must be significantly increased, making the numerical system unstable.

The Lagrange method consists in calculating the exact amount of strain, denoted as the Lagrange multiplier, needed to fulfill the constraint. This method guarantees that constraints are always exactly fulfilled. Since the use of Lagrange multipliers introduces a set of new unknowns, equation (1) must be completed by a set of new equations, increasing the size of the initial linear system to solve. But we consider that this method is most suitable for efficiently managing geometric constraints.

For all the reasons mentioned above, we chose the Lagrange method to manage our geometric constraints. According to the principle of virtual work, each constraint g k adds a new force perpendicular to the tangent space of the surface g k (x) = 0. The Lagrange multiplier λ k corresponds to the intensity of the force related to the constraint g k . With these new forces, equation ( 1) is modified as follows:

M ẍ = f + k∈E λ k ∂g k ∂x (3)
We add new equations to our system by calculating the second time derivative of equation ( 2), leading to equation (4).

n i=1 ∂g k ∂x i ẍi = - n i,j=1 ∂ 2 g k ∂x i ∂x j ẋi ẋj ∀k ∈ E (4)
In order to correct the numerical deviation due to round-off errors, Baumgarte proposed in [START_REF] Baumgarte | Stabilization of constraints and integrals of motion in dynamical systems[END_REF] a constraint stabilization scheme illustrated by equation ( 5). The parameter τ -1 can be seen as the speed of constraint fulfillment.

n i=1 ∂g k ∂x i ẍi = - n i,j=1 ∂ 2 g k ∂x i ∂x j ẋi ẋj - 2 τ n i=1 ∂g k ∂x i ẋi - 1 τ 2 g k (5)
When we mix equations (1) and (5), we obtain a linear system where the second time derivative of the state vector x and the vector of Lagrange multipliers Λ are the unknowns.

M -J T -J 0 ẍ Λ = f -d (6)
J is the jacobian matrix of all the geometric constraints and d corresponds to the right term of equation (5).

Inequality constraints management

By convention, an inequality constraint will always be defined as in equation ( 7) where F is the set of indices of all the inequality constraints.

g k (x) ≥ 0 ∀k ∈ F (7)
For a given state vector x, we recall the following definitions:

• the constraint is said to be violated by x when g k (x) < 0. This means that the state vector x corresponds to a non allowed configuration.

• the constraint is said to be satisfied by x when g k (x) ≥ 0.

• the constraint is said to be active when g k (x) = 0. In this case, the state vector x belongs to the boundary of the subspace defined by the inequality constraint g k .

The management of inequality constraints is more difficult than the management of equality constraints. An inequality constraint must be handled only if it is violated or active. In fact, the algorithm is a little more complicated as we explain in the next sections.

That is why we define two subsets within F: F + is the set of indices of all handled inequality constraints and F -is the set of indices of ignored inequality constraints. Finally, we have F = F -∪F + . The jacobian matrix of constraints J of equation ( 6) is built from all the constraints g k where k ∈ E ∪ F + .

Previous work

Within the computer graphics community, the main published method devoted to inequality constraints management using Lagrange multipliers, known as "Generalized Dynamic Constraints", was proposed by Platt in [START_REF] Platt | A Generalization of Dynamic Constraints[END_REF]. In his paper, he describes how to use Lagrange multipliers to assemble and simulate collisions between numerical models. This method is an extension of the work of Barzel and Barr [START_REF] Barzel | A modeling system based on dynamic constraints[END_REF] that specifies how constraints must be satisfied. Moreover, Platt proposes a method to update F + (the set of handled inequality constraints) during the animation. This algorithm can be compared to classical active set methods [START_REF] Björck | Numerical Methods for Least Squares Problems[END_REF][START_REF] Nocedal | Numerical Optimization[END_REF].

We do not focus on collision detection that is a problem by itself. We are aware that this difficult problem can be solved in many ways, we encourage the reader to refer to the survey paper by Teschner et al.

[TKZ + 05]. During the collision detection stage, we assume that the dynamic engine may rewind time until the first constraint activation is detected. This assumption can produce an important computational overhead that can restrict our method to off-line animations production depending on the complexity of the scene simulated. In any case, this stage ensures that constraints urn:nbn:de:0009-6-12767, ISSN 1860-2037 are never violated. But, it is possible that several constraints are activated simultaneously. The main topic of this paper is to provide a reliable algorithm to handle these multiple active constraints in an efficient way.

At the beginning of the animation, Platt populates the set F + with all the active constraints.

Algorithm 1-Platt's algorithm Solve equation (6) to get ẍ and Λ 1 Update x and ẋ (numerical integration)

2 for each k ∈ F do 3 if k ∈ F + then 4 if λ k ≤ 0 then 5 k is moved to F - 6 else 7 if g k (x) < 0 then 8 g k is moved to F + 9
For each time step, according to algorithm 1, we solve equation ( 6) and update the state vector in order to retrieve new positions and velocities at the end of the current time step. We then check the status of each inequality constraint. If a constraint g k is active, it is still handled until its Lagrange multiplier is negative or null, that is to say that the Lagrange multiplier corresponds to a force that prevents from deactivation. According to the new values of the state vector x, if the previously inactive constraint g k is now violated (g k (x) < 0), the constraint must be added to F + in order to prevent the system to enter in such a configuration.

Figure 1: A simple example with two simultaneous active constraints Even if this algorithm seems to give a reliable solution for inequality constraints handling, some prob-lems remain. We set up a simple scene as in figure 1 to illustrate the insufficiencies of Platt's method. A particle of mass m is constrained to slide on a 2D plane. It starts from an acute-angle corner modeled by two linear inequality constraints g 1 (x) ≥ 0 and g 2 (x) ≥ 0 where g 1 (x) = x -y and g 2 (x) = y. Finally, this particle is subject to a single external force f = (2, -1). In this particular case, the state vector x is composed of the 2D coordinates (x, y) of the particle. According to equation (1), the generalized mass matrix for this system is defined by:

M = m 0 0 m (8)
As the geometric constraints g 1 (x) and g 2 (x) are linear, their first and second time derivative do not produce any deviation term defined in equation ( 5):

d = 0 0 (9)
According to the initial value of the state vector x = (0, 0), the two constraints g 1 (x) and g 2 (x) are active, so their indices are inserted in F + and J, the jacobian matrix of constraints, is defined as follows:

J = 1 -1 0 1 (10)
From equations (6) (8) (9) and (10), we obtain a linear system whose unknowns are the second time derivative of the state vector x and the two Lagrange multipliers λ 1 and λ 2 associated with g 1 and g 2 :

    m 0 -1 0 0 m 1 -1 -1 1 0 0 0 -1 0 0         ẍ ÿ λ 1 λ 2     =     2 -1 0 0     (11)
The solutions are ẍ = (0, 0) and Λ = (-2, -1). The particle does not move during this time step because ẍ and ÿ are null. But, since λ 1 and λ 2 are both negative, their corresponding constraints are moved to F -. This means that, for the next time step, the system will be free of any constraints. As the force remains constant, the next value of ẍ will be equal to (2m -1 , -m -1 ). These values will lead to an illegal position of the particle, under the line y = 0. These computations are illustrated by the figure 2.

The amount of violation of the constraint g 2 (x) = y mainly depends on the ratio between the mass m of the particle and the intensity of the external force f . urn:nbn:de:0009-6-12767, ISSN 1860-2037 Section 5 of this paper presents the different results and comparisons.

Our contribution 4.1 A first approach

The problem of Platt's method relies on the fact that it keeps some inequality constraints in F + that should be ignored. In fact, the condition g k (x) < 0 used to populate F + with inequality constraints is not well suited and an alternative approach is proposed. A solution would be to replace the condition g k (x) < 0 by a violation tendency condition expressed as J k ẍ < d k . An active constraint that does not fulfill the violation tendency condition will be satisfied but inactive during the next time step and does not have to be handled.

At the beginning of the animation, we solve equation (1) to get ẍ and we then populate the set F + with the active constraints that fulfill the violation tendency condition. It is clear that we handle less constraints than Platt because our criteria is more restrictive.

We briefly verify that this algorithm gives a correct solution to our example illustrated in figure 1. According to equation (1), ẍ = (2m -1 , -m -1 ). The two constraints g 1 and g 2 are active because x = (0, 0) but only g 2 fulfills the violation tendency condition as mentioned in equation (12).

J 1 ẍ = 3m -1 ⇒ 1 ∈ F - J 2 ẍ = -m -1 ⇒ 2 ∈ F + (12)
In this special case, equation ( 6) becomes:

Algorithm 2-Platt's improved algorithm Solve equation ( 6) to get ẍ and Λ 1 Update x and ẋ (numerical integration)

2 for each k ∈ F do 3 if k ∈ F + then 4 if λ k ≤ 0 then 5 k is moved to F - 6 else 7 if J k ẍ < d k then 8 k is moved to F + 9   m 0 0 0 m -1 0 -1 0     ẍ ÿ λ 2   =   2 -1 0   ( 13
)
The solutions of the linear system (13) are ẍ = (2m -1 , 0) and λ 2 = 1. Finally, the particle will slide along the x-axis without crossing the line y = 0 because the constraint g 1 that was not handled did not introduce a false response.

This new algorithm seems to manage multiple inequality constraints in a good way, but we could highlight a problem with this method by using the same example illustrated in figure 1 with a new external force f = (-1, -2).

At the beginning, since x = (0, 0), the constraints g 1 and g 2 are active. From equation (1), we obtain that ẍ = (-m -1 , -2m -1 ), and from equation ( 14) that only the constraint g 2 is handled. According to equation ( 14), we build the linear system (15).

J 1 ẍ = m -1 ⇒ 1 ∈ F - J 2 ẍ = -2m -1 ⇒ 2 ∈ F + (14) (a) (b) (c)
  m 0 0 0 m -1 0 -1 0     ẍ ÿ λ 2   =   -1 -2 0   (15)
The solutions are ẍ = (-m -1 , 0) and λ 2 = 2. After the update of x and ẋ, the particle slides through the plane defined by the constraint g 1 and reaches an illegal state. This is due to the fact that the Lagrange multiplier λ 2 pushes the system in an illegal state according to the constraint g 1 , which was not previously inserted in equation (6) as it did not satisfy the violation tendency criterion. These computations are again illustrated by the figure 3.

The "right" algorithm

The use of the violation tendency condition J k ẍ < d k improves simultaneous active constraints management, since only the appropriate inequality constraints are handled by equation ( 6). But we have seen, from the second example, that it is not sufficient to produce a consistent configuration. In fact, the constraints from F + that fulfill the violation tendency condition will produce a vector Λ of Lagrange multipliers that prevent the system from being in an illegal configuration according to these handled constraints. In the meantime, the constrained accelerations ẍ of the system could lead to an illegal configuration according to some constraints in F -. The only way to deal with this problem is to use the newly computed constrained accelerations to test if the active inequality constraints g k (where k ∈ F -) fulfill the violation tendency condition and have to be handled. We then need to introduce an iterative process that computes the accelerations and checks if a previously ignored constrained must be handled or not according to the violation tendency condition evaluated with the newly computed constrained accelerations. This process is repeated until the sytem reaches the appropriate state.

We propose a simple and efficient solution to the inequality constraints handling problem. At the beginning of each time step, all active inequality constraints g k are detected, and F + is emptied. We then begin an iterative process that runs until there is no new insertion in F + . The constrained accelerations ẍ are computed from equation ( 6) and the violation tendency condition J k ẍ < d k is tested on each active inequality constraint. For any inequality constraint g k that fulfills the condition, we insert its index k in F + and start another iterative step. In a recent communication [START_REF] Raghupathi | QP-Collide: A New Approach to Collision Treatment[END_REF], Raghupathi presented a method also based on Lagrange multipliers. For realtime considerations, they do not allow the dynamic engine to rewind time to get back to the first constraint activation. They have to manage constraints at the end of the time step, trying to find the right accelerations to ensure constraints fulfillement. They also confess that this process is not guaranteed to converge for a given situation.

Results and Comparisons

We will now compare the results obtained with Platt's algorithm and our method, using the example illustrated in figure 1. Figure 4 and 5 illustrate a compari-son of the positions and accelerations along y-axis of a particle of mass m = 2 and m = 3. We recall that the inequality constraint g 2 forbids negative values for y and that the constant force f applied to the particle is equal to (2, -1).

As shown on figure 4, Platt's algorithm holds the particle in the corner at the first time step, and releases it at the next time step. As a consequence, the particle evolves in an illegal state during the following steps. With a mass m = 2, the error related to the position is less than 10 -5 with an oscillating acceleration (right column). But if we set the mass m to 3, as shown in the figure 5, errors are much more important, and the particle crosses the line y = 0 modeled by the constraint g 2 .

As illustated, our algorithm keeps the particle along the x-axis within a controlled numerical error value, that is less than 10 -8 in these examples. To illustrate multiple contact constraints, we have set a billiard scene composed of 10 fixed balls placed in a corner and a moving ball that slides towards them. For each ball, we define two inequality constraints according to the corner and one inequality constraint for each pair of balls based on their in-between distance. This example is finally composed of 11 balls and 77 inequality constraints (figure 6).

It is rather difficult to compare the computation times of Platt's algorithm and ours since the simulations made of simultaneous active constraints are not well handled by Platt's algorithm and produce corrupted numerical values that can lead to infinite loops. But it is quite clear that in the worst case, our method may solve n linear systems of increasing size where n is the total number of inequality constraints. The complexity of our solution is then higher than Platt's algorithm. But we recall that our main contribution is not to speed up an existing method but to propose a reliable algorithm mainly dedicated to off-line simulations.

Conclusion

In this paper, we presented a novel algorithm to manage simultaneous active inequality constraints. Among all the existing methods to handle constraints within a physically-based animation, we focused on the Lagrange method which provides a reliable way to ensure that constraints are always exactly fulfilled. But, in the special case of several active inequality constraints, we have to take care on how to handle these simultaneous constraints. Platt proposed an algorithm based on Lagrange multipliers but we showed that this method is unable to solve even simple examples. We then explained how to improve this algorithm in order to propose a new reliable and efficient method for inequality constraints handling. Beyond the example illustrated in figure 1, we produced a short movie simulating a billiard game. Some snapshots are gathered in figure 6.

Figure 2 :

 2 Figure 2: (a) Since the two constraints are active, they handle by Platt's algorithm (b) The related Lagrange multipliers are negative, the constraints are then ignored (c) The new unconstrained acceleration leads to an illegal position

Algorithm 3 -

 3 Figure 3: (a) The two constraints are handled since they are active (b) According to the violation tendency condition, only the constraint g 2 still handled (c) The newly computed constrained acceleration leads to an illegal position

Figure 4 :

 4 Figure 4: Comparison of Platt's algorithm and our method using the example illustrated in figure 1 with a mass m = 2. The numerical values correspond respectively to position and acceleration along the y-axis


Michel Granger
email: granger@univ-angers.fr

Mathias Schulze
email: mschulze@mathematik.uni-kl.de

DEFORMING MONOMIAL SPACE CURVES INTO SET-THEORETIC COMPLETE INTERSECTION SINGULARITIES

Keywords: 2010 Mathematics Subject Classification. Primary 32S30; Secondary 14H50, 20M25 Set-theoretic complete intersection, space curve, singularity, deformation, lattice ideal, determinantal variety

We deform monomial space curves in order to construct examples of set-theoretical complete intersection space curve singularities. As a by-product we describe an inverse to Herzog's construction of minimal generators of non-complete intersection numerical semigroups with three generators.

Introduction

It is a classical problem in algebraic geometry to determine the minimal number of equations that define a variety. A lower bound for this number is the codimension and it is reached in case of set-theoretic complete intersections. Let I be an ideal in a polynomial ring or a regular analytic algebra over a field K. Then I is called a set-theoretic complete intersection if √ I = √ I for some ideal I admitting height of I many generators. The subscheme or analytic subgerm X defined by I is also called a set-theoretic complete intersection in this case. It is hard to determine whether a given X is a set-theoretic complete intersection. We address this problem in the case I ∈ Spec K{x, y, z} of irreducible analytic space curve singularities X over an algebraically closed (complete non-discretely valued) field K.

Cowsik and Nori (see [START_REF] Cowsik | Affine curves in characteristic p are set theoretic complete intersections[END_REF]) showed that over a perfect field K of positive characteristic any algebroid curve and, if K is infinite, any affine curve is a set-theoretic complete intersection. To our knowledge there is no example of an algebroid curve that is not a set-theoretic complete intersection. Over an algebraically closed field K of characteristic zero, Moh (see [START_REF] Moh | A result on the set-theoretic complete intersection problem[END_REF]) showed that an irreducible algebroid curve

K[[ξ, η, ζ]] ⊂ K[[t]
] is a set-theoretic complete intersection if the valuations , m, n = υ(ξ), υ(η), υ(ζ) satisfy (0.1) gcd( , m) = 1, < m, ( -2)m < n.

We deform monomial space curves in order to find new examples of set-theoretic complete intersection space curve singularities. Our main result in Proposition 3.2 gives sufficient numerical conditions for the deformation to preserve both the value semigroup and the set-theoretic complete intersection property. As a consequence we obtain Corollary 0.1. Let C be the irreducible curve germ defined by O C = K t , t m + t p , t n + t q ⊂ K{t} where gcd( , m) = 1, p > m, q > n and there are a, b ≥ 2 such that = b + 2, m = 2a + 1, n = ab + b + 1.

Let γ be the conductor of the semigroup Γ = , m, n and set d 1 = (a + 1)(b + 2), δ = min {p -m, q -n}. In the setup of Corollary 0.1 Moh's third condition in (0.1) becomes ab < 1 and is trivially false. Corollary 0.1 thus yields an infinite list of new examples of non-monomial set-theoretic complete intersection curve germs.

Let us explain our approach and its context in more detail. Let Γ be a numerical semigroup. Delorme (see [START_REF] Delorme | Sous-monoïdes d'intersection complète de N[END_REF]) characterized the complete intersection property of Γ by a recursive condition. The complete intersection property holds equivalently for Γ and its associated monomial curve Spec(K[Γ]) (see [START_REF] Herzog | Generators and relations of abelian semigroups and semigroup rings[END_REF]Cor. 1.13]) and is preserved under flat deformations. For this reason we deform only non-complete intersection Γ. A curve singularity inherits the complete intersection property from its value semigroup since it is a flat deformation of the corresponding monomial curve (see Proposition 2.3). The converse fails as shown by a counter-example of Herzog and Kunz (see [START_REF] Herzog | Die Wertehalbgruppe eines lokalen Rings der Dimension 1[END_REF]).

In case Γ = , m, n , Herzog (see [START_REF] Herzog | Generators and relations of abelian semigroups and semigroup rings[END_REF]) described minimal relations of the generators , m, n. There are two cases (H1) and (H2) (see §1) with 3 and 2 minimal relations respectively. In the non-complete intersection case (H2) we describe an inverse to Herzog's construction (see Proposition 1.4). Bresinsky (see [Bre79b]) showed (for arbitrary K) by an explicit calculation based on Herzog's case (H2) that any monomial space curve is a complete intersection. Our results are obtained by lifting his equations to a (flat) deformation with constant value semigroup. In section §2 we construct such deformations (see Proposition 2.3) following an approach using Rees algebras described by Teissier (see [Zar06, Appendix, Ch. I, §1]). In §3 we prove Proposition 3.2 by lifting Bresinsky's equations under the given numerical conditions. In §4 we derive Corollary 0.1 and give some explicit examples (see Example 4.2).

It is worth mentioning that Bresinsky (see [Bre79b]) showed (for arbitrary K) that all monomial Gorenstein curves in 4-space are settheoretic complete intersections.

Ideals of monomial space curves

Let , m, n ∈ N generate a semigroup Γ = , m, n ⊂ N.

d = gcd( , m).
We assume that Γ is numerical, that is, gcd( , m, n) = 1.

Let K be a field and consider the map

ϕ : K[x, y, z] → K[t], (x, y, z) → (t , t m , t n ) whose image K[Γ] = K[t , t m , t n ] is the semigroup ring of Γ. Pick a, b, c ∈ N minimal such that a = b 1 m + c 2 n, bm = a 2 + c 1 n, cn = a 1 + b 2 m for some a 1 , a 2 , b 2 , b 2 , c 1 , c 2 ∈ N.
/ ∈ {a 1 , a 2 , b 1 , b 2 , c 1 , c 2 }. Then (1.1) a = a 1 + a 2 , b = b 1 + b 2 , c = c 1 + c 2
and the unique minimal relations of , m, n read

a -b 1 m -c 2 n = 0, (1.2) -a 2 + bm -c 1 n = 0, (1.3) -a 1 -b 2 m + cn = 0. (1.4)
Their coefficients form the rows of the matrix

(1.5)   a -b 1 -c 2 -a 2 b -c 1 -a 1 -b 2 c   . Accordingly the ideal I = f 1 , f 2 , f 3 of maximal minors (1.6) f 1 = x a -y b 1 z c 2 , f 2 = y b -x a 2 z c 1 , f 3 = x a 1 y b 2 -z c of the matrix (1.7) M 0 = z c 1 x a 1 y b 1 y b 2 z c 2 x a 2 .
equals ker ϕ, and the rows of this matrix generate the module of relations between f 1 , f 2 , f 3 . Here K[Γ] is not a complete intersection.

(H2) 0 ∈ {a 1 , a 2 , b 1 , b 2 , c 1 , c 2 }. One of the relations (a, -b, 0), (a, 0, -c), or (0, b, -c) is minimal relation of , m, n and, up to a permutation of the variables, the minimal relations are a = bm, (1.8)

a 1 + b 2 m = cn.
(1.9) Their coefficients form the rows of the matrix

(1.10) a -b 0 -a 1 -b 2 c
.

It is unique up to adding multiples of the first row to the second. Overall there are 3 cases and an overlap case described equivalently by 3 matrices

(1.11) a -b 0 a 0 c , a -b 0 0 -b c , a 0 -c 0 b -c .
Here K[Γ] is a complete intersection. In the following we describe the image of Herzog's construction and give a left inverse: Proof. The first statement holds due to minimality. By Buchberger's criterion the generators 1.6 form a Gröbner basis with respect to the reverse lexicographical ordering on x, y, z. Let g denote a normal form of g = x ˜ -z ñ with respect to 1.6. Then g ∈ I if and only if g = 0. By (1.1) reductions by f 2 can be avoided in the calculation of g. If r 2 and r 1 many reductions by f 1 and f 3 respectively are applied then Proof.

(H1') Given a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ∈ N \ {0}, define a, b, c by (1.1) and set = b 1 c 1 + b 1 c 2 + b 2 c 2 = b 1 c + b 2 c 2 = b 1 c 1 + bc 2 , (1.12) m = a 1 c 1 + a 2 c 1 + a 2 c 2 = ac 1 + a 2 c 2 = a 1 c 1 + a 2 c, (1.13) n = a 1 b 1 + a 1 b 2 + a 2 b 2 = a 1 b + a 2 b 2 = a 1 b 1 + ab 2 , (1.
g = x ñ-a 1 r 1 -ar 2 y b 1 r 2 -r 1 b 2 z r 1 c+r 2 c 2 -z ˜ . and g = 0 is equivalent to ˜ = r 1 c + r 2 c 2 , b 1 r 2 = r 1 b 2 , ñ = a 1 r 1 + ar 2 . Then r i = b i gcd(b 1 ,b 2 ) for i = 1,
(a) Consider ñ, ˜ ∈ N as in Lemma 1.2. Then x ñ -z ˜ ∈ I = ker ϕ means that (t ) ñ = (t n ) ˜ and hence ñ = ˜ n. So the pair ( , n) is pro- portional to ( ˜ , ñ) which in turn is propotional to ( , n ) by Lemma 1.2.
Then the two triples ( , m, n) and ( , m , n ) are proportional by symmetry. Since gcd( , m, n) = 1 by hypothesis ( , m , n ) = q •( , m, n) for some q ∈ N. By Lemma 1.2 q divides gcd(b 1 , b 2 ) and by symmetry also gcd(a 1 , a 2 ) and gcd(c 1 , c 2 ). By minimality of the relations

(1.2)-(1.4) gcd(a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ) =
( , m , n ) is in the corresponding subcase of (H2), gcd(a, b) = 1, (1.18) ∀q ∈ ∩[-b 2 /b, a 1 /a] ∩ N : gcd(-a 1 + qa, -b 2 -qb, c) = 1. (1.19) In this case, ( , m, n) = ( , m , n ).
Proof.

(a) By Lemma 1.3.(a) e = 1 is a necessary condition. Conversely let e = 1. By definition (1.5) is a matrix of relations of ( , m , n ). Assume that ( , m , n ) is in case (H2). By symmetry we may assume that ( , m , n ) admits a matrix of minimal relations

In particular c ≥ d . Then b 1 ≥ b contradicts (1.12) since = b 1 c + b 2 c 2 ≥ b c + b 2 c 2 > b c ≥ b d = .
We may thus assume that b 1 < b . The difference of first rows of (1.20) and (1.5) is then a relation

a -a b 1 -b c 2 of ( , m , n ) with a -a < 0, b 1 -b < 0 and c 2 > 0. Then c 2 ≥ c ≥ d by choice of c . This contradicts (1.12) since = b 1 c 1 + bc 2 ≥ b 1 c 1 + b d > b d = .
We may thus assume that ( , m , n ) is in case (H1) with a matrix of unique minimal relations

(1.21)   a -b 1 -c 2 -a 2 b -c 1 -a 1 -b 2 c   of type (1.5) where a = a 1 + a 2 , b = b 1 + b 2 , c = c 1 + c 2 .
as in (1.1). Then (a, b, c) ≥ (a , b , c ) by choice of the latter and

= b 1 c + b 2 c 2 = b 1 c 1 + b c 2 by Lemma 1.3.(a). If (a i , b i , c i ) ≥ (a i , b i , c i ) for i = 1, 2, then = b 1 c + b 2 c 2 ≥ b 1 c + b 2 c 2 = implies c = c and hence (a, b, c) = (a , b , c ) by symmetry. By unique- ness of (1.21) then (a 1 , a 2 , b 1 , b 2 , c 1 , c 2 ) = (a 1 , a 2 , b 1 , b 2 , c 1 , c 2 )
and hence the claim. By symmetry it remains to exclude the case c 2 > c 2 . The difference of first rows of (1.21) and (1.5) is then a relation

a -a b 1 -b 1 c 2 -c 2 of ( , m , n ) with a -a ≤ 0, c 2 -c 2 < 0 and hence b 1 -b 1 ≥ b by choice of the latter. This leads to the contradiction = b 2 c 2 + b 1 c > b 1 c ≥ b c + b 1 c > b 2 c 2 + b 1 c = .
= d = b , a = m d = a .
Writing the second row of (1.10) as a linear combination of (1.20) yields

-a 1 + qa -b 2 -qb c = p -a 1 -b 2 c .
with p ∈ N and q ∩ [-b 2 /b, a 1 /a] ∩ N and hence p = 1 by (1.19). The claim follows.

The following examples show some issues that prevent us from formulating stronger statement in Proposition 1.4.(b).

Example 1.5.

(a) Take (a, -b,

0) = (3, -2, 0) and (-a 1 , -b 2 , c) = (-1, -4, 4). Then ( , m , n ) = (4, 6, 7) which is in case (H2). The second minimal rela- tion is (-2, -1, 2) = 1 2 ((-a 1 , -b 2 , c) -(a, -b, 0)).
The same ( , m , n ) is obtained from (a, 0, -c) = (7, 0, -4) and (-a 2 , b, -c 1 ) = (-1, 3, -2). This latter satisfies (1.18) and (1.19) but (a, 0, -c) is not minimal.

(b) Take (a, -b, 0) = (4, -3, 0) and (-a 1 , -b 2 , c) = (-2, -1, 2). Then ( , m , n ) = (3, 4, 5) but (a, -b, 0) is not a minimal relation. In fact the corresponding complete intersection K[Γ] defined by the ideal x 3 -y 4 , z 2 -x 2 y is the union of two branches x = t 3 , y = t 4 , z = ±t 5 .

Deformation with constant semigroup

Let O = (O, m) be a local K-algebra with O/m ∼ = K. Let F • = {F i | i ∈ Z}
A = i∈Z F i s -i ⊂ O[s ±1 ].

It is a finite type graded O[s]-algebra and flat (torsion free) K[s]-algebra with retraction

A A/A ∩ m[s ±1 ] ∼ = K[s].
For u ∈ O * there are isomorphisms

(2.1) A/(s -u)A ∼ = O, A/sA ∼ = gr F O.
Geometrically A defines a flat morphism with section

Spec(A) π / / A 1 K ι i i
with fibers over K-valued points

π -1 (x) ∼ = Spec(O), ι(x) = m, 0 = x ∈ A 1 K , π -1 (0) ∼ = Spec(gr F O), ι(0) = gr F m.
F • = m • W O W , F • = F •,w = m • = υ -1 [•, ∞] O.
Setting t = t /s and identifying K ∼ = O W /m W this yields a finite extension of finite type graded O W -and flat (torsion free) K[s]-algebras

(2.2) A = i∈Z (F i ∩ O W )s -i ⊂ i∈Z F i s -i = O W [s, t] = B ⊂ O W [s ±1 ]
with retraction defined by K[s] ∼ = B/(B <0 + Bm W ). The stalk at w is

A = A w = i∈Z (F i ∩ O)s -i ⊂ i∈Z F i s -i = O[s, t] = B ⊂ O[s ±1 ].
At w = w ∈ W the filtration F w is trivial and the stalk becomes

A w = O W,w [s ± 1].
The graded sheaves gr F O W ⊂ gr F O W are thus supported at w and the isomorphism gr

F (O W ) w = gr F O ∼ = K[t ] ∼ = K[N] identifies (2.3) (gr F O W ) w = gr F O ∼ = K[Γ ], Γ = υ(O \ {0}) with the semigroup ring K[Γ ] of O,
The analytic spectrum Spec an W (-) → W applied to finite type O Walgebras represents the functor T → Hom O T (-T , O T ) from K-analytic spaces over W to sets (see [START_REF] Henri | Familles d'espaces complexes et fondements de la géométrie analytique[END_REF]Exp. 19]). Note that

Spec an W (K[s]) = Spec an {w} (K[s]) = L is the K-analytic line. The normalization of W is ν : W = Spec an W (O W ) → W and B = ν * B where B = O W [s, t]. Applying Spec an W to (2.2) yields a diagram of K-analytic spaces (see [Zar06, Appendix]) (2.4) X = Spec an W (A) π & & Spec an W (B) = Y ρ o o L ι 8 8
where π is flat with π • ρ • ι = id and

π -1 (x) ∼ = Spec an W (O W ) = W, ι(x) = w, 0 = x ∈ L, π -1 (0) ∼ = Spec an W (gr F O W ), ι(0) ↔ gr F m W .
Remark 2.1. Teissier defines X as the analytic spectrum of A over W × L (see [Zar06, Appendix, Ch. I, §1]). This requires to interpret the O W -algebra A as an O W ×L -algebra.

Remark 2.2. In order to describe (2.4) in explicit terms, embed

L ⊃ W ν / / W ⊂ L n
with coordinates t and x = x 1 , . . . , x n and

X = {(x, s) | (s 1 x 1 , . . . , s n x n ) ∈ W, s = 0} ⊂ L n × L, Y = (t, s) t = st ∈ W ∪ L × {0} ⊂ L × L.
This yields the maps X → W ← Y . The map ρ in (2.4) becomes ρ(t, s) = (x 1 (t )/s 1 , . . . , x n (t )/s n ) for s = 0 and the fiber π -1 (0) is the image of the map

ρ(t, 0) = ((ξ 1 (t), . . . , ξ n (t)), 0), ξ k (t) = lim s→0 x k (st)/s k = σ(x k )(t).
Taking germs in (2.4) this yields the following.

Proposition 2.3. There is a flat morphism with section

S = (X, ι(0)) π / / (L, 0) ι k k with fibers π -1 (x) ∼ = (W, w) = C, ι(x) = w, 0 = x ∈ L, π -1 (0) ∼ = Spec an (K[Γ ]) = C 0 , ι(0) ↔ K[Γ + ].
The structure morphism factorizes through a flat morphism

X = Spec an W (A) f 3 3 f / / (|W |, A) / / W and f # ι(0) : A → O X,ι(0) induces an isomorphism of completions (see [Car62, Exp. 19, §2, Prop. 4]) A ι(0) ∼ = O X,ι(0) .

This yields the finite extension of K-analytic domains

O S = O X,ι(0) ⊂ O Y,ι(0) .
We aim to describe O Y,ι(0) and K-analytic algebra generators of O S . In explicit terms O S is obtained from a presentation

I → O[x] → A → 0 mapping x = x 1 , . . . , x n to ι(0) = A ∩ m[s ±1 ] + As as (2.5) O S = O{x}/O{x}I = O{x} ⊗ O[x] A, O{x} = O ⊗K{x}.
The graded K-algebra A/sA is thus generated by ξ. Extend F • to the graded filtration F

• [s ±1 ] on O[s ±1 ]. For i ≥ j, (A/As) i = gr F i A i •s i-j ∼ = / / gr F i A j .
Thus finitely many monomials in ξ, s generate any A j /F i A j ∼ = F j /F i over K. With γ the conductor of Γ and i = γ + j, F γ ⊂ m ∩ O = m and hence F i = F γ F j ⊂ mF j . Therefore these monomials generate A j as O-module by Nakayama's lemma. It follows

A = O[ξ, s] as graded K-algebra. Using O = K ξ and ξ = ξs then O S = K ξ , ξ, s = K ξ, s (see (2.5)).
We now reverse the above construction to deform generators of a semigroup ring. Let Γ be a numerical semigroup with conductor γ generated by = 1 , . . . , n . Pick corresponding indeterminates x = x 1 , . . . , x n . The weighted degree deg(-) defined by deg(x) = makes K[x] a graded K-algebra and induces on K{x} a weighted order ord(-) and initial part inp(-) . The assignment x i → i defines a presentation of the semigroup ring of Γ (see (2.3))

K[x]/I ∼ = K[Γ] ⊂ K[t ] ⊂ K{t } = O.
The defining ideal I is generated by homogeneous binomials f = f 1 , . . . , f m of weighted degrees deg(f ) = d. Consider elements ξ = ξ 1 , . . . , ξ n defined by (2.7)

ξ j = t j + i≥ j +∆ j ξ j,i t i s i-j ∈ K[t, s] ⊂ O[t, s] = B with ∆ i ∈ N \ {0} ∪ {∞} minimal. Set δ = min {∆ }, ∆ = ∆ 1 , . . . , ∆ n .
With deg(t) = 1 = -deg(s) ξ defines a map of graded K-algebras K[x, s] → K[t, s] and a map of analytically graded K-analytic domains K{x, s} → K{t, s} (see [SW73] for analytic gradings).

Remark 2.6. Converse to (2.6), any homogeneous ξ ∈ K{t, s} of weighted degree can be written as ξ = ξ /s for some ξ ∈ K{t }. It follows that ξ(t, 1) = ξ (t) ∈ K{t}.

Consider the curve germ C with K-analytic ring

(2.8) O = O C = K ξ , ξ = ξ(t,
O S = K ξ, s = K{x, s}/ F , F = f -f s.
Proof. First let Γ = Γ. Then Lemma 2.5 yields the first equality in (2.10). By flatness of π in Proposition 2.3, the relations f of ξ(t, 0) = t lift to relations F ∈ K{x, s} m of ξ. That is, F (x, 0) = f and F (ξ, s) = 0. Since f and ξ have homogeneous components of weighted degrees d and , F can be written as F = f -f s where f ∈ K{x, s} m has homogeneous components of weighted degrees d + 1. This proves in particular the last claim. Since f i (t ) = 0, any term in f i (ξ, s)s = f i (ξ) involves a term of the tail of ξ j for some j. Such a term is divisible by t d i +∆ j which yields the bound for ord(f i (x, 1)).

Conversely let f with homogeneous components satisfy (2.9). Suppose that there is a k ∈ Γ \ Γ. Take h ∈ K{x} of maximal weighted order k such that υ(h(ξ )) = k . In particular, k < k and inp h(t

) = 0. Then inp h ∈ I = f and inp h = m i=1 q i f i for some q ∈ K[x] m . Set h = h - m i=1 q i F i (x, 1) = h -inp h + m i=1 q i f i (x, 1).
Then h (ξ ) = h(ξ ) by (2.9) and hence υ(h (ξ )) = k . With (2.9) and homogeneity of f it follows that ord(h ) > k contradicting the maximality of k.

Remark 2.8. The proof of Proposition 2.7 shows in fact that the condition Γ = Γ is equivalent to the flatness of a homogeneous deformation of the parametrization as in (2.7). These Γ-constant deformations are a particular case of δ-constant deformations of germs of complex analytic curves (see [Tei77, §3, Cor. 1]).

The following numerical condition yields the hypothesis of Proposition 2.7. Lemma 2.9.

If min {d} + δ ≥ γ then Γ = Γ. Proof. Any k ∈ Γ is of the form k = υ(p(ξ )) for some p ∈ K{x} with p 0 = inp(p) ∈ K[x]. If p 0 (t ) = 0, then k ∈ Γ. Otherwise, p 0 ∈ f and hence k ≥ min {d} + min { }.
The second claim follows.

Set-theoretic complete intersections

We return to the special case Γ = , m, n of §1. Recall Bresinsky's method to show that Spec(K[Γ]) is a set-theoretic complete intersection (see [Bre79a]). Starting from the defining equations (1.6) in case (H1) he computes

f c 1 = (x a -y b 1 z c 2 ) c = x a g 1 ± y b 1 c z c 2 c = x a g 1 ± y b 1 c z (c 2 -1)c (x a 1 y b 2 -f 3 ) = x a 1 g 2 ∓ y b 1 c z (c 2 -1)c f 3 ≡ x a 1 g 2 mod f 3
where g 1 ∈ x, z and

g 2 = x a-a 1 g 1 ± y b 1 c+b 2 z (c 2 -1)c .
He shows that, if c 2 ≥ 2, then further reducing g 2 by f 3 yields

g 2 = x a-a 1 g 1 ± y b 1 c+b 2 z (c 2 -2)c (x a 1 y b 2 -f 3 ) ≡ x a-a 1 g 1 ± x a 1 y b 1 c+2b 2 z (c 2 -2)c mod f 3 ≡ x a 1 g1 + y b 1 c+2b 2 z (c 2 -2)c mod f 3 ≡ x a 1 g 3 mod f 3 for some g1 ∈ K[x, y, z]. Iterating c 2 many times yields a relation (3.1) f c 1 = qf 3 + x k g, k = a 1 c 2
, where g ≡ y mod x, z with from (1.12). One computes that

x a 1 f 2 = y b 1 f 3 -z c 1 f 1 , z c 2 f 2 = x a 2 f 3 -y b 2 f 1 . Bresinsky concludes that (3.2) Z(x, z) ⊂ Z(g, f 3 ) ⊂ Z(f 1 , f 3 ) = Z(f 1 , f 2 , f 3 ) ∪ Z(x, z) making Spec(K[Γ]) = Z(g, f 3 ) a set-theoretic complete intersection.
As a particular case of (2.7) consider three elements

ξ = t + i≥ +∆ ξ i s i-t i , (3.3) η = t m + i≥m+∆m η i s i-m t i , ζ = t n + i≥n+∆n ζ i s i-n t i ∈ K[t, s].
(3.5) F c 1 = qF 3 + x k G, G(x, y, z, 0) = g, then C = S ∩ Z(s -1) = Z(G, F 3 , s -1)
is a set-theoretic complete intersection.

Proof. Consider a matrix of indeterminates

M = Z 1 X 1 Y 1 Y 2 Z 2 X 2
and the system of equations defined by its maximal minors

F 1 = X 1 X 2 -Y 1 Z 2 , F 2 = Y 1 Y 2 -X 2 Z 1 , F 3 = X 1 Y 2 -Z 1 Z 2 .
By Schap's theorem (see [START_REF] Schaps | Deformations of Cohen-Macaulay schemes of codimension 2 and non-singular deformations of space curves[END_REF]) there is a solution with coefficients in K{x, y, z} [[s]] that satisfies M (x, y, z, 0) = M 0 . Grauert's approximation theorem (see [Gra72]) coefficients can be taken in K{x, y, z, s}.

Using the fact that M is a matrix of relations, we imitate in Bresinsky's argument in (3.2),

Z(G, F 3 ) ⊂ Z(F 1 , F 3 ) = Z(F 1 , F 2 , F 3 ) ∪ Z(X 1 , Z 2 ).
The K-analytic germs Z(G, F 3 ) and Z(G, X 1 , Z 2 ) are deformations of the complete intersections Z(g, f 3 ) and Z(g, x a 1 , z c 2 ), and are thus of pure dimensions 2 and 1 respectively. It follows that Z(G, F 3 ) does not contain any component of Z(X 1 , Z 2 ) and must hence equal Z(F 1 , F 2 , F 3 ) = S. The claim follows.

Proposition 3.2. Set δ = min(∆ , ∆m, ∆n) and k = a 1 c 2 . Then the curve germ C defined by (3.3) is a set-theoretic complete intersection if

min(d 1 , d 2 , d 3 ) + δ ≥ γ, min(d 1 , d 3 ) + δ ≥ γ + k ,
or, equivalently,

min(d 1 , d 2 + k , d 3 ) + δ ≥ γ + k .
Proof. By Lemma 2.9 the first inequality yields the assumption Γ = Γ on (3.3). The conductor of ξ k O equals γ + k and contains (F if i )(ξ , η , ζ ), i = 1, 3, by the second inequality. This makes F i -f i , i = 1, 3, divisible by x k . Substituting into (3.1) yields (3.5) and by Lemma 3.1 the claim.

Remark 3.3. We can permute the roles of the f i in Bresinsky's method.

If the role of (f 1 , f 3 ) is played by (f 1 , f 2 ), we obtain a formula similar to (3.1), f b 1 = qf 2 + x k g with k = a 2 b 1 . Instead of x k , there is a power of y if we use instead (f 2 , f 1 ) or (f 2 , f 3 ) and a power of z if we use (f 3 , f 1 ) or (f 3 , f 1 ). The calculations are the same. In the examples we favor powers of x in order to minimize the conductor γ + k . We assume that a, b ≥ 2 and b + 2 < 2a + 1 so that < m < n. The maximal minors (1.6) of M 0 are then

Series of examples

f 1 = x a+1 -yz, f 2 = y b+1 -x a z, f 3 = z 2 -xy b
with respective weighted degrees

d 1 = (a + 1)(b + 2), d 2 = (2a + 1)(b + 1), d 3 = 2ab + 2b + 2
where d 1 < d 3 < d 2 . In Bresinsky's method (3.1) with k = 1 reads f 2 1 -y 2 f 3 = xg, g = x 2a+1 -2x a yz + y b+2 . We reduce the inequality in Proposition 3.2 to a condition on d 1 .

Lemma 4.1. The conductor of ξO is bounded by

γ + ≤ d 2 - m < d 3 .
In particular, d 2 ≥ γ + 2 and d 3 > γ + .

Proof. The subsemigroup Γ 1 = , m ⊂ Γ has conductor

γ 1 = ( -1)(m -1) = 2a(b + 1) = n + (a -1) + 1 ≥ γ.
To obtain a sharper upper bound for γ we think of Γ as obtained from Γ 1 by filling gaps of Γ

1 . Since 2n ≥ γ 1 , Γ \ Γ 1 = (n + Γ 1 ) \ Γ 1 .
The smallest elements of Γ 1 are i where i = 0, . . . , m . By symmetry of Γ 1 (see [Kun70]) the largest elements of N \ Γ 1 are

γ 1 -1 -i = n + (a -1 -i) , i = 0, . . . , m ,
and contained in n + Γ 1 since the minimal coefficient a -1 -i is nonnegative by

a -1 - m ≥ a -1 - m = (a -1)b -3 b + 2 > -1.
They are thus the largest elements of Γ \ Γ 1 . Their minimum attained at i = m then bounds

γ ≤ γ 1 -1 - m .
Substituting γ 1 + -1 = d 2 yields the first particular inequality. The second one follows from

d 2 -d 3 = 2a -b -1 = m -< m .
Proof of Corollary 0.1.

(a) This follows from Lemma 2.9. (b) By Lemma 4.1, the inequality in Proposition 3.2 simplifies to d 1 + δ ≥ γ + . The claim follows.

(c) Suppose that d 1 + q -n ≥ γ + for some q > n and a, b ≥ 3. Set p = γ -1 -. Then n > m + and Γ ∩ (m + , m + 2 ) can include at most n and some multiple of . Since ≥ 4 it follows that (m + , m + 2 ) contains a gap of Γ and hence γ -1 > + m and p > m. Moreover (a -1)b ≥ 4 is equivalent to Example 4.2. We discuss a list of special cases of Corollary 0.1.

d 1 + p -m ≥ γ + .
(a) a = b = 2. The monomial curve C 0 defined by (x, y, z) = (t 4 , t 5 , t 7 ) has conductor γ = 7. Its only admissible deformation is (x, y, z) = (t 4 , t 5 + st 6 , t 7 ). However this deformation is trivial and our method does not yield a new example. To see this, we adapt a method of Zariski (see [Zar06, Ch. III, (2.5), (2.6)]). Consider the change of coordinates x = x + 4s 5 y = t 4 + 4s 5 t 5 + 4s 2 5 t 6

and the change of parameters of the form τ = t+O(t 2 ) such that x = τ 4 . Then τ = t + s 5 t 2 + O(t 3 ) and hence y = τ 5 + O(t 7 ) and z = τ 7 + O(t 8 ). Since O(t 7 ) lies in the conductor, it follows that C ∼ = C 0 . In all other cases, Corollary 0.1 yields an infinite list of new examples. (b) a = 3, b = 2. Consider the monomial curve C 0 defined by (x, y, z) = (t 4 , t 7 , t 9 ). By Zariski's method from (a) we reduce to considering the deformation (x, y, z) = (t 4 , t 7 , t 9 + st 10 ). (c) a = b = 3. The monomial curve C 0 defined by (x, y, z) = (t 5 , t 7 , t 13 ) has conductor γ = 17. We want to satisfy p ≥ γ+ -d 1 +m = 9. The most general deformation of y thus reads y = t 7 + s 1 t 9 + s 2 t 11 + s 3 t 16 .

The parameter s 1 can be again eliminated by Zariski's method as in (a). This leaves us with the deformation (x, y, z) = (t 5 , t 7 + s 2 t 11 + s 3 t 16 , t 13 + s 4 t 16 , t 13 ) which is non-trivial due to part (c) of Corollary 0.1 with p = 11.

(d) a = 8, b = 3. The monomial curve C 0 defined by (x, y, z) = (t 5 , t 17 , t 28 ) has conductor γ = 47. The condition in part (b) of Corollary 0.1 requires p ≥ γ -d 1 + m = 19. In fact, the deformation (x, y, z) = (t 5 , t 17 + st 18 , t 28 ) is not flat since C has value semigroup Γ = Γ ∪ {46}. However C is isomorphic to the general fiber of the flat deformation in 4-space (x, y, z, w) = (t 5 , t 17 + st 18 , t 28 , t 46 ).

  (a) If d 1 + δ ≥ γ, then Γ is the value semigroup of C. (b) If d 1 + δ ≥ γ + , then C is a set-theoretic complete intersection. (c) If a, b ≥ 3 and d 1 + q -n ≥ γ + , then C defined by p := γ -1 -> mis a non-monomial set-theoretic complete intersection.

  14) and e = gcd( , m , n ). Note that , m , n are the submaximal minors of the matrix in (1.5). (H2') Given a, b, c ∈ N \ {0} and a 1 , b 2 ∈ N, define , m , n , d by = bd , (1.15) m = ad , (1.16) n d = a 1 b + ab 2 c , gcd(n , d ) = 1. (1.17) Remark 1.1. In the overlap case (1.11) the formulas (1.15)-(1.16) yield ( , m , n ) = (bc, ac, ab). Lemma 1.2. In case (H1), let ñ ∈ N be minimal with x ñ -z ˜ ∈ I for some ˜ ∈ N. Then gcd( ˜ , ñ) = 1 and (ñ, ˜ ) • gcd(b 1 , b 2 ) = (n , ).

  2 and the claim follows. Lemma 1.3. (a) In case (H1), equations (1.12)-(1.14) recover , m, n. (b) In case (H2), equations (1.15)-(1.17) recover , m, n, d.

  1 and hence q = 1. The claim follows. (b) By the minimal relation (1.8) gcd(a, b) = 1 and hence ( , m) = d • (b, a). Substitution into equation (1.9) and comparison with (1.17) gives n d = a 1 b+ab 2 c = n d with gcd(n, d) = gcd( , m, n) = 1 by hypothesis. We deduce that (n, d) = (n , d ) and then ( , m) = ( , m ). Proposition 1.4. (a) In case (H1'), a 1 , a 2 , b 1 , b 2 , c 1 , c 2 arise through (H1) from some numerical semigroup Γ = , m, n if and only if e = 1. In this case, ( , m, n) = ( , m , n ). (b) In case (H2'), a, b, c, a 1 , b 2 arise through (H2) from some from some numerical semigroup Γ = , m, n if and only if

  -b 2 c of type (1.10). By choice of a , b , c it follows that a > a , b > b , c ≥ c . By Lemma 1.3.(b) d is the denominator of a 1 b +a b 2 c and = b d .

  (b) By Lemma 1.3.(b) the conditions are necessary. Conversely assume that the conditions hold true. By definition (1.10) is a matrix of relations of ( , m , n ). By hypothesis (1.20) is a matrix of minimal relations of ( , m , n ). By (1.18) gcd( , m ) = d and hence by Lemma 1.3.(b) b

  be a decreasing filtration by ideals such that F i = O for all i ≤ 0 and F 1 ⊂ m. Consider the Rees ring

Let

  K be an algebraically closed complete non-discretely valued field. Let C be an irreducible K-analytic curve germ. Its ring O = O C is a one-dimensional K-analytic domain. Denote by Γ its value semigroup. Pick a representative W such that C = (W, w). We allow to shrink W suitably without explicit mention. Let O W be the normalization of O W . Then O W,w = (O, m) ∼ = (K{t }, t ) υ / / N ∪ {∞} is a discrete valuation ring. Denote by m W and m W the ideal sheaves corresponding to m and m. There are decreasing filtrations by ideal (sheaves)

  Consider the curve germ C in (2.8) with K-analytic ring(3.4) O = O C = K{ξ , η , ζ }, (ξ , η , ζ ) = (ξ, η, ζ)(t,1), and value semigroup Γ ⊃ Γ. We aim to describe situations where C is a set-theoretic complete intersection under the hypothesis that Γ = Γ. By Proposition 2.7, (ξ, η, ζ) then generate the flat deformation of C 0 = Spec an (K[Γ]) in Proposition 2.3. Let F 1 , F 2 , F 3 be the defining equations from Proposition 2.7. Lemma 3.1. If g in (3.1) deforms to G ∈ K{x, y, z, s} such that

  Redefining a, b suitably, we specialize to the case where the matrix in (1.7) is of the formM 0 = z xy y b z x a . By Proposition 1.4.(a) these define Spec(K[ , m, n ]) if and only if = b+2, m = 2a+1, n = ab+b+1(= (a+1) -m), gcd( , m) = 1.

  By (b), C is a set-theoretic complete intersection.It remains to show that C ∼ = C 0 . This follows from the fact thatΩ 1 C 0 → K{t}dt has valuations Γ \ {0} whereas the 1-form ω = mydx -xdy = (m -p)t p+ -1 dt ∈ Ω 1 C → K{t}dt has valuation p + = γ -1 ∈ Γ.

  While part (c) of Corollary 0.1 does not apply, C ∼ = C 0 remains valid. To see assume that C 0 ∼ = C induced by an automorphism ϕ of C{t}. Then ϕ(x) ∈ O C shows that ϕ has no quadratic term. This however contradicts ϕ(z) ∈ O C .

  The deformation (2.7) satisfies Γ = Γ if and only if there is a f ∈ K{x, s} m with homogeneous components such that

	and ord(f i (x, 1)) ≥ d i + min {∆ }. The flat deformation in Proposi-
	tion 2.3 is then defined by
	(2.10)
		1),
	and value semigroup Γ ⊃ Γ.
	We now describe when (2.7) generate the flat deformation in Propo-
	sition 2.3.
	Proposition 2.7. (2.9)	f (ξ) = f (ξ, s)s

Any O W -module M gives rise to an O X -module

With M = M w , its stalk at ι(0) becomes

Lemma 2.4. Spec an W (B) = Spec an W (B) and hence O Y,ι(0) = K{s, t}. Proof. By finiteness of ν (see [START_REF] Henri | Familles d'espaces complexes et fondements de la géométrie analytique[END_REF]Exp. 19, §3, Prop. 9]),

By the universal property of Spec an it follows that (see [Con06, Thm. 2.2.5.(2)]) Proof. By choice of F • there is a cartesian square

By hypothesis and (2.3) the symbols σ(ξ ) generate the graded Kalgebra gr F O. Then σ(ξ ) = σ(ξ ) generate gr F m/ gr F m 2 = gr F (m/m 2 ) and hence ξ generate m/m 2 over K. Then m = ξ O by Nakayama's lemma and hence O = K ξ by the analytic inverse function theorem.

Under the graded isomorphism (2.1) with ξ as in (2.6) (A/As)