Mathieu Avanzi
email: mathieu.avanzi@unine.ch

Cédric Gendrot
email: cgendrot@univ-paris3.fr

Anne Lacheret-Dujour

Is there a prosodic difference between left-dislocated and heavy subjects? Evidence from spontaneous French

Keywords: clitic left dislocation, heavy subject, prominence, spontaneous French

This paper investigates the prosodic difference between two types of subject-NPs in spontaneous French: left-dislocated NPs (NPs followed by a coreferent pronoun in the subsequent verbal clause, as in mon mari i , il i est instituteur, my husband i he i is a teacher) and heavy NPs (NPs which are not followed by a coreferent pronoun, as in mon mari est instituteur, my husband is a teacher). In order to verify a potential prosodic difference between these two kinds of subject in spontaneous French, two instrumental analyses were conducted: (i) an automatic prominence detection determined whether a boundary tone ends the NP; (ii) a comparison of several acoustic features allowed for the acoustic estimation of the degree of prominence of the NPs of each class. We show that both types of subject-NPs cannot be statistically distinguished according to the measurements on the subject-NPs' final syllable, even if a tendency towards higher prominence values (detection and degree) for dislocated subjects clearly appears. A great variability within and in-between utterances is observed and is suggested to account for the non-significant differences.

Introduction

The aim of this study is to provide a clear diagnosis for prosody of left-dislocation in French. In concrete terms, we will try to answer to the following question: is there a prosodic difference between mon mari est instituteur and mon mari i il i est instiutteur? In the first section, we briefly present the literature on this topic. In the second section, we present our corpus and the computational tools for signal processing and prosodic tagging we used. The third and final section, before discussion and conclusion, is devoted to the presentation of the analyses and their results: the prosodic difference between left-dislocated and heavy subject is approached in two steps. First, we focus on the presence or the absence of a boundary tone at the end of the NP. Second, we detail the acoustic correlates of the break's degree of prominence between the NP and the adjacent clause.

Previous work

Two ways to diagnose the difference between left-dislocated and heavy subjects in spoken French have been proposed in the literature.

The difference lies in the (non-)obligatory presence of a boundary tone at the end of the NP

In the Prosodic Theory framework (Selkirk [START_REF] Selkirk | Phonology and syntax. The relations between sound and structure[END_REF]; Delais-Roussarie et al. [START_REF] Delais-Roussarie | Dislocation[END_REF] [START_REF] Delais-Roussarie | Unités prosodiques et grammaire de l'intonation: vers une nouvelle approche[END_REF]), the two kinds of NPs do not manifest the same structural properties. While heavy subjects can be ended by a boundary tone or not (the realization of such a tone depending in this case on the speech rate and on the NP's metrical structure: the longer the NP is and the slower the rate of speech is, more the NP will be susceptible to carry a final prominence), left-dislocated subjects are always followed by a major prosodic boundary, that is to say they obligatorily form an Intonational Phrase (IP) on their own. Compare:

(1) (mon mari est instituteur) IP (2) (mon mari) IP (est instituteur) IP (3) (mon mari) IP (il est instituteur) IP (4) *(mon mari il est instituteur) IP According to this framework, the first three examples (1)-(3) are possible, while the last one ( 4) is ungrammatical: obligatory prosodic boundary realization is due to the leftdislocated NP syntactic status (they are adjuncts, Rizzi [START_REF] Rizzi | The fine structure of left periphery[END_REF]) and assume a specific pragmatic function (they play the role of topic, Lambrecht [START_REF] Lambrecht | Topic, antitopic, and verb agreement in nonstandard French[END_REF]). We find the same idea, formulated in different terms, in other frameworks, such as in Martin [START_REF] Martin | L'intonation du français[END_REF]'s phono-syntactic model or Mertens [START_REF] Mertens | A Predictive Approach to the Analysis of Intonation in Discourse in French[END_REF]'s morphological approach.

The difference lies in the prominence degree of the boundary tone at the end of the NP

Another analysis has been done by Bally [START_REF] Bally | Linguistique générale et Linguistique française[END_REF], one of the first scholars interested in the prosody of spontaneous French. According to this scholar, the difference between the two kinds of NPs lies in the degree of the boundary tone: the break between the subject and the adjacent verbal clause is weak in the first case, strong in the second one. He tried to characterize this shade by saying that in the dislocated utterance the pitch profile is "segmented" because interrupted by a silent pause and by slope inversion, while in the heavy subject utterance the profile is "continuous".

Rossi [START_REF] Rossi | L'intonation, le système du français -description et modélisation[END_REF]'s conclusions confirmed [START_REF] Bally | Linguistique générale et Linguistique française[END_REF]'s first perceptual intuitions. [START_REF] Rossi | L'intonation, le système du français -description et modélisation[END_REF] compared the prosody of left-dislocated and heavy subjects in read aloud sentences. His results showed that the difference between heavy and left-dislocated subjects relies in the degree of prominence of the terminal syllable of the NP. More precisely, he concluded that the last syllable of the NP seemed to be more salient when the subject was dislocated than when it was heavy. Indeed, he showed that classic prosodic parameters implied in the realization of French stress were more important at the end of a dislocated subject than at the end of a heavy subject. Regarding the last syllable of the left-dislocated element, his measures showed that (i) the melodic target was higher (it reached the highest level of the speaker range), (ii) the rise was broader due to a greater duration (this syllable lengthening entailing the perception of a silent pause).

Summary

Scholars agree that heavy and left-dislocated subjects express prosodic differences in the way they are prosodically phrased. The problem is that these conclusions have been obtained from the analysis of read aloud speech, that is to say on punctuated texts which have certainly strongly influenced the prosodic phrasing of the subjects (a comma after the left-dislocated NP suggesting a "segmented" prosodic realization; the absence of a comma after a heavy subject facilitating a "continuous" prosodic realization). As far as we know, specific studies addressing the specific question of the acoustic difference between heavy and left-dislocated subjects in spontaneous French do not yet exist (nevertheless, see De Cat [START_REF] De Cat | French Dislocation. Interpretation, Syntax, Acquisition[END_REF] for an attempt). On the basis of a certain number of utterances, such an analysis should also allow for the estimation of the prosodic characteristics which would differentiate left-dislocated and heavy subjects. In the next section, we present the data and the instrumentation we used to conduct such an analysis.

Material

Corpus

If it is impossible to obtain minimal pairs when we work on spontaneous speech, we can nevertheless control a certain number of variables, and elaborate a corpus as homogenous as possible. For this purpose, we investigated two spoken Parisian French corpora (a subpart of the PFC (Phonologie du Français Contemporain) corpus, 16 speakers, 7 h. of free and semi-directed speech, see Durand et al. [START_REF] Durand | La phonologie du français contemporain: usages, variétés et structure[END_REF]; the whole CFPP2000, 22 speakers, 17 h. of semi-directed speech, see Branca et al. [START_REF] Branca-Rosoff | Discours sur la ville. Corpus de Français Parlé Parisien des années 2000[END_REF]) and manually extracted only subject-NPs which comprised 2 or 3 syllables. We collected altogether 134 utterances, namely 67 left-dislocated subjects and 67 heavy subjects. Each pair was pronounced by the same speaker. Let us note that not all speakers do alternate between dislocated and heavy subjects; moreover, there are speakers who frequently alternate, while others do not. That's why our corpus contains only 18 speakers (13 females and 5 males, from to 24 to 85 years old) out of the 38 speakers the whole corpus contains. These 18 speakers perform at least 1, at most 11 items. To conclude on this point, we would like to highlight that only utterances where the pronominal recovery has been made with a verb pre-posed personal subject pronoun, such as je, tu, il/elle/on, nous, vous, ils/elles, have been considered. This means the utterances where the recovery was done with a demonstrative pronoun, such as those from the /c-/ paradigm (like ce, cela, ça, ceux-ci, etc.), had not been taken into account because they give the NP a particular semantic status (i.e. the NP gains a generic interpretation, and the sentence would in turn be like a pseudo-cleft-sentence, see Carlier [START_REF] Carlier | Les gosses ça se lève tôt le matin': l'interprétation générique du syntagme nominal disloqué au moyen de ce ou ça[END_REF], Pekarek-Doehler & Müller [START_REF] Pekarek-Doehler | Le problème c'est de les distinguer. Disloquée à gauche et pseudo-clivée dans la conversation[END_REF]) and they do not resort to the same syntactic analysis (Avanzi [START_REF] Avanzi | La dislocation à gauche avec reprise anaphorique en français parlé. Etude prosodique[END_REF] [START_REF] Avanzi | La dislocation à gauche en français. Etude instrumentale[END_REF] for the details of the argumentation).

Prosodic preprocessing

All the utterances of the corpus (2 x 67 utterances) were aligned semi-automatically into phones, syllables and orthographic words with the EasyAlign (Goldman [START_REF] Goldman | EasyAlign: a semi-automatic phonetic alignment tool under Praat[END_REF]) Praat script (Boersma & Weenick [18]). Next, we conducted prosodic prominence detection with the ANALOR software. Because of lack of space, we cannot present the entire procedure here (see Avanzi et al. [START_REF] Avanzi | A Tool for Semi-Automatic Annotation of French Prosodic Structure[END_REF] for a first experiment, [START_REF] Avanzi | A corpus-based learning method for prominence detection in spontaneous speech[END_REF] for the last state of description of the tool). In a few words, the algorithm calculates the relative height and duration of each syllable in a given utterance by comparing the value of the analyzed syllable with the average of the six adjacent syllables (i.e. the three preceding and the three following ones); the pitch rise slope is then processed and the presence of a subsequent silent pause is considered. Regarding duration, a simple relativization was carried out. Ideally, syllable duration models based on syllable intrinsic properties and normalizing local speech rate for spontaneous speech should be used (see for a first experiment on read aloud sentences Obin et al. [21] [22]). Given that such a model still has to be reliably developed and tested, we only took the number of phonemes of the syllable into account. By doing so, we could avoid the bias of syllable weight, a syllable compounded of five phones being by essence longer than a mono-phonemic one. Pitch measures (average height and rise) are in semitones.

According to thresholds which were automatically estimated on the basis of the annotation of a 70-minute corpus by two experts (see Simon et al. [START_REF] Simon | La détection des proéminences syllabiques. Un aller-retour entre l'annotation manuelle et le traitement automatique[END_REF] and Avanzi et al. [START_REF] Avanzi | C-PROM. An Annotated Corpus for French Prominence Studies[END_REF] for an exhaustive presentation of this material); the ANALOR software program marks the prominent syllables and estimates the perceptual degree of each one. These calculations are described in the following section (see Figure 3 and Figure 4 §8 for illustrations).

Analysis and results

This section is focused on the presentation of the prosodic analysis. Considering that in French, prosodic boundaries coincide with prominent terminal phrases' syllables (Rossi [START_REF] Rossi | Le français, langue sans accent ?[END_REF], Vaissière [START_REF] Vaissière | Rythm, accentuation and final lenghtening in French[END_REF] and Lacheret-Dujour & Beaugendre [START_REF] Lacheret-Dujour | La prosodie du français[END_REF]), we used the ANALOR software to conduct a prosodic analysis in two steps. First, we searched to see whether dislocated NPs systematically ended on a prosodic boundary, as the Prosodic Theory predicts. Next, we compared the degree of prominence of each syllable of our two classes to see if we found a significant difference between them.

On the presence of a boundary

In the first step of the study, we used the ANALOR tool to check whether the dislocated NPs of our corpus were always followed by a prosodic break, and compared the results with those for the heavy subjects. We considered that a syllable was prominent or bearing a boundary tone if one of the following four parameters was activated: the relative pitch height reaches at least 2.5 st., relative duration is 1.5 longer, vocalic rise is equal or greater than 2.5 st. and/or a silent pause, whatever its duration, follows the syllable. As one can see in the diagram in Figure 1, contrary to the predictions developed above in 1. a boundary tone is not systematically detected at the end of the left-dislocated NP: A binomial logistic regression was used in order to predict the probability of prominence detection (yes/no) according to subject's type and speaker. This makes apparent that the impact of the subject's type on the prominence detection is not significant (Chi2 = 2.46, p=0.11), and neither is the impact of speaker (Chi2 = 12.6, p=0.76).

On the boundary degree

A second analysis has been conducted to see if the difference between heavy and left-dislocated subjects had to be searched in the prominence degree of the boundary tone. Estimating the degree of salience of accentual prominence in French is not so easy: prosodic parameters involved in French accentual prominence are numerous, and there are many compensatory phenomena between them (that is to say, for example, that in some contexts, a pitch rise can be equivalent perceptively to a vocalic lengthening, see t'Hart et al. [START_REF] Hart | A Perceptual Study of Intonation. An Experimental-Phonetic Approach to Speech Melody[END_REF], inter alia). Nowadays, devoted tools that can deal with it are still missing. In this study, we propose a method to estimate the degree of prominence of a syllable. Our study is based upon the following hypothesis: the more the acoustic parameters are mobilized (number of parameters and/or prominence thresholds), the more the prominence will be perceived. The thresholds mentioned above ( §4.1.) have been used to assign a score to each syllable. 1 was given to a syllable which had an approaching threshold (25% less than the chosen threshold), 2 if the value reached but not exceeding 25% of the original threshold, 3 if the threshold exceeded more than 25% of the reference measure. We attributed 0 if the threshold did not reach at least 25% less than the training value; 1 if the syllable was followed by a silent pause. This allowed for an ad hoc non linear rescaling of prominence between 0 and 10 (in most 3 for relative height, 3 for duration, 3 for pitch rise and 1 for silent pause). The calculation had been applied to all utterances of the corpus, which gave the following distribution: A linear mixed effects model with crossed random effects was run, with the "degree of prominence" as the dependent variable and subject's type as a predictor. Speakers were entered as a random term. The advantage of a mixed effects model here is that it takes into account the randomness induced by the different number of realizations produced by each speaker but also their variations. The model shows no effect of subject's type on the prominence degree (F (1,132) = 3.75, p = 0.055), however it does show a tendency towards higher degree of prominence for dislocated subjects. Separate measurements of normalized duration, normalized f0 and normalized f0 rise displayed in Figure 2 also follow the same tendency. The standard error bars give a hint about why this tendency doesn't reveal significant differences. A great variability within and between speakers is observed for the production of both subject-NPs. Furthermore 6 out of 18 speakers have higher values for dislocated subjects than for heavy subjects. This variability is a strong clue to invalidate [START_REF] Bally | Linguistique générale et Linguistique française[END_REF]'s intuitions and [START_REF] Rossi | L'intonation, le système du français -description et modélisation[END_REF]'s studies on spoken aloud utterances.

Discussion & Conclusion

The aim of this study was to answer the following question: is there a difference regarding the prosodic boundary between heavy and left-dislocated subjects. In order to bring new arguments, we analyzed a spontaneous corpus of 67 pairs of utterances pronounced in semi-directed conditions by Parisian French.

Contrary to the traditional predictions (see §2. above), the first analysis showed that a strong prosodic break is not obligatory at the end of left-dislocated subjects. The second analysis showed that the breaks which end the left-dislocated NPs are slightly more prominent than the breaks which end the Heavy NPs, but that the difference is not significant, for neither concerning the "global" score, nor any of the acoustic parameters investigated (relative height, relative duration and intra-vocalic pitch rise). It also revealed that two strategies were used by the Parisian speakers of our corpus. One consists of making a greater prominence on the left-dislocated NP (12/18 speakers); the other consists of making a greater prominence on the heavy-subject NP (6/18 speakers). While the first can be explained by taking syntactic information into account (extra-sentential elements are always ended by a boundary tone), the second can be explained by an economy principle (Lacheret-Dujour [START_REF] Lacheret-Dujour | The Intonational Marking of Topical Salience in Spontaneous Speech -Evidence from Spoken French[END_REF] and [START_REF] Lacheret-Dujour | De la notion de détachement topical à celle de constituant thématique extrapropositionnel[END_REF]), which is relatively close to Gussenhoven [START_REF] Gussenhoven | Intonation and interpretation: phonetics and phonology[END_REF]'s effort code. Applied to the utterances we studied here, it stipulates that a right edge of a discourse constituent need not be marked at the same time by syntax and by prosody if one of the two parameters already operates. That is to say that a subject NP need not be prosodically salient if the syntax already marks it as dislocated, and on the contrary, a subject NP already prosodically salient is not obligatory marked by the syntax.

These findings deserve further consideration. (i) We will have to integrate speech rate and the syntactic position of the NP in the sentence as predictor variables, the sentences pronounced with a faster speech rate will tend to contain fewer prominences, non sentence initial NPs (embedded clauses, NPs following a connector) do not have the same metrical structure as initial ones. (ii) We will also have to see if, for a certain number of cases, the absence of a detected prosodic break is not a bias from the software. Indeed, the actual algorithm does not allow us to take other prosodic characteristics, such as the low tones, into account. As [START_REF] Martin | L'intonation du français[END_REF] and Le Gac & Yoo [START_REF] Gac | De l'invariante invariablité des topiques en français[END_REF] recently argued, such realizations exist: they are motivated by slope inversion. (iii) From the informational structure point of view, we will have to cross the results we get in this study with a categorization of topics, and see how syntactic dislocation and prosodic detachment interact. We will focus our attention on these three points in future work.

Figure 1 :

 1 Figure 1: Distribution between the number of Heavy Subjects (on the left) and the number of Left-Dislocated Subjects (on the right) according to the

Figure 2 :

 2 Figure 2: Degree of prominence; normalized duration; normalized f0 and normalized f0 rise (from left to right) for heavy vs. dislocated subjects.

Acknowledgements

Two institutional frameworks financed this research: The Swiss National Science Foundation (SNF Project 100012-113726/1, "La structure interne des périodes", Neuchâtel University), and the Agence Nationale de la Recherche (ANR-07-CORP-030-01, "Rhapsodie -Corpus prosodique de référence du français parlé"). Thanks also to B. Victorri (Lattice/ENS, Paris) for friendly help in the different stages of the ANALOR software conception.

Annexes

The figures below are ANALOR screen shots. In the abscissa, temporal values are given in milliseconds; in the ordinate, the values of F0 in a logarithmic scale can be seen. The transcription tiers are, from top to bottom: phone tier, syllable tier (both in SAMPA alphabet); automatic prominence detection tire (prominence characteristics can be posted by clicking on red crosses upside the phone tier) and the orthographic word tier. Figure 3 shows a non-prominent leftdislocated subject, Figure 4 shows a prominent heavy subject. Both utterances are performed by the same speaker (a 60 years-old female living in the 12 th district of Paris). The final NP syllables (les parents) are framed in red.


A Bitri

S Brûlé

E Javelaud

K Samyn 1 1-Brgm

ASSESSMENT OF SOILS COMPACTION USING MULTICHANEL ANALYSIS OF SURFACE WAVE

Keywords: Surface waves, Shear velocity, Rayleigh wave, Inversion, Compaction. EVALUATION Ondes de surface, Ondes de cisaillement, Ondes de Rayleigh, Inversion, compaction

ABSTARCT

The shear properties of soils are of fundamental interest in civil or environmental engineering, particularly for designing building foundations or predicting site effects in seismic hazard studies. Dynamic compaction improves the soils mechanical characteristics obtained by insitu investigations (cone penetration tests (CPTs). Vs profiles obtained by inversion of surface wave before and after compaction have allowed a good evaluation of the compaction performance and demonstrate the potential of this technique in geotechnical investigations

INTRODUCTION

Evaluation of soil geotechnical parameters is a preliminary task to be conducted either for designing building foundations, or for studying site effects in seismic hazard evaluation. Both of these applications show the importance of knowing soil elastic properties, which govern its behavior in presence of natural or human solicitations. Shear-wave velocity (Vs) is usually taken as a good indicator of the elastic behavior of soils, and is classically estimated from cross-hole measurements. In order to avoid such destructive -and expensive -testing, we study here the possibility to estimate the Vs distribution with depth by using Spectral Analysis of Surface Waves (SASW) method. The determination of Vs profiles from Rayleigh waves is attractive since their propagation velocity, i.e. the phase velocity Vph, highly depends on the stiffness-depth profile. Several inverse methods used seismology has already been proposed to resolve such a problem. [START_REF] Dorman | Numerical inversion of seismic surface waves dispersion data and crust-mantle structures in the New-York-Pennsylvania area[END_REF] used surface wave to determine elastic properties of the crustal-mantle structures. Engineering problems were tackled in [START_REF] Nazarian | In-situ shear wave velocities from spectral analysis of surface waves[END_REF] where a hammer is used as impulsive source for generating Rayleigh waves while ground motions are recorded from two seismic sensors. The dispersion is afterwards derived from the spectrum of recorded vertical motions. [START_REF] Addo | Shear-wave velocity measurements of soils using Rayleigh waves[END_REF] also used the same approach and validated the method on six different sites by comparing SASW inverted results and cone penetration tests data. [START_REF] Gabriels | In-situ measurements of shear wave velocity in sediments with higher mode Rayleigh waves[END_REF] improved the method by taking into account higher modes of Rayleigh waves and by increasing the distance between source and sensors. Recently, Xia et al. (1999a) proposed an inverse method based on Levenberg-Marquart technique. The Linearized Least-Squares (LLS) technique used here is adapted from [START_REF] Hermann | Computer programs in seismology[END_REF] and was already tested in [START_REF] Bitri | Shear wave velocity determination of soils from in-situ Rayleigh waves measurements[END_REF]. The purpose of this paper is to present and discuss the results of surface wave method for the assessment site compaction.

FIELD ACQUISITION AND RESULTS

The project is located at the site of an old glass factory, near Givors (France). The site was occupied for over a century of industrial activity. The plant was dismantled recently and new construction-type industrial buildings are provided. To build these structures, a technique for improving soil has been considered by the designers. This is the technique of dynamic compaction "high energy" was chosen as an alternative to deep piles to address the dual problem of soil with low mechanical properties and by environmental stress resulting from a chemical residual impact of soil and water In fact, deep piles could impact the water resource, blending the superficial polluted soil and water with the deep ones.

Fig.1: Location map of project (top left) and schematic geological cross-section (top right) Bottom, location and plan view of the test board. The five circles indicate the position of the static penetrometers

The Vs initial model before compaction was established in September 2009 along 4 linear lines. To increase the speed and efficiency of data recording and thereby keep acquisition costs down, a multichannel seismic cable has been designed and manufactured. It consists of 24 takeouts at fixed 2m intervals. Each takeout is attached to a single self-orientating, gimbals-mounted, vertical geophone. To help ensure proper coupling, each gimbal geophone is housed in a heavy casing (~1 kg). To damp the motion of the sensor around its rotational axis, the inside of the casing is filled with viscous oil. The seismic cable is towed behind a vehicle. A 24-channel Geometrics Stratavizor seismograph was used to record impacts of 10kg hammer seismic source. The source-to-nearest-receiver offset was 4m, while the source stations were separated by 10m along the seismic lines. The record length was selected as 1024 ms at 1 ms sample interval. The data required for inversion are the values of phase velocity as a function of frequency and surface wave mode. Phase velocity dispersion curve (phase velocity versus frequency) were determined using the slant-stack method in common shot gathers, followed by a 1-D Fourier transform over the intercept time to obtain the wave field in the f-v plane. The surface wave investigations were carried out in the end of October 2009, after heavy dynamic compaction process on the same lines in order to verify the effect of compaction. A seismic shot gather before and after compaction is presented in figure 2a, with the related dispersion curves showing the fundamental mode (Fig. 2b). The initial model for inversion was calculated with an empirical relation already used by (Xia et al., 1999). Ten iterations for inverting velocities and ten more for inverting layers thickness were necessary to converge to the final model A 2D contour plot of the shear-wave velocity field was produced by gathering all the velocity profiles into sequential order, according to half shot station. Figure 3 shows the difference of results after and before the compaction. As shown in figure, shear velocity increases of about 40-50m/s after compaction for the depth up to 7m. The same results were obtained by geotechnical investigations (heavy dynamic penetrometer) Fig. 3 Difference between shear wave velocity model (line a) after and before compaction.

CONCLUSIONS

Nonintrusive multichannel surface wave method presents a great interest in geotechnical investigation as it offers the potential to determine the shear wave profiles from the surface allowing the characterization of soils in their in situ state. This can be used as an alternative to the penetration tests for assessment of compaction. In the Givors site the shear wave velocity increase by 22% after the heavy dynamic compaction. The results show a net increase up to 7m and little or no significant at greater depth.

  Fig 2: a) Typical shot gather before and after compaction and b) the related dispersion diagram with clear fundamental Rayleigh mode.


Edoardo Ballico

Alessandra Bernardi

Maria Virginia Catalisano

Luca Chiantini

GRASSMANN SECANTS AND LINEAR SYSTEMS OF TENSORS

For any irreducible non-degenerate variety X ⊂ P r , we relate the dimension of the s-th secant varieties of the Segre embedding of P k × X to the dimension of the (k, s)-Grassmann secant variety GS X (k, s) of X. We also give a criterion for the s-identifiability of X.

Introduction

In 1915, in an elegant XIX-century style Italian language ([Ter15, p. 97]), A. Terracini pointed out that the defectiveness of the s-th secant varieties of a Segre product Seg(P k × V d ) between a projective space P k and a Veronese surface V d is related to the fact that the set of all P k 's lying in the span of s independent points of V d has not the dimension that one can expect from an obvious count of parameters. That pioneering work (also known as the second Terracini's Lemma) waited for a century before finding a first generalization. In 2001, C. Dionisi and C. Fontanari proved that Terracini's result can be formulated by replacing Veronese surface with any irreducible non-degenerate projective variety X ([DF01, Proposition 1.3]). In analogy with the notion of defectiveness, which is set forth for secant varieties, they utilized the concept of (k, s)-Grassmann defect that holds for the varieties X for which the set of all P k 's lying in the span of s independent points of X has dimension smaller than the expected one. The Zariski closure GS X (k, s) of such a set, in the corresponding Grassmannian, is called (k, s)-Grassmann secant variety of X. Indeed, what they proved is the following result. Proposition 0.1. [DF01, Proposition 1.3] Let X ⊂ P r be an irreducible nondegenerate projective variety of dimension n. Then X is (k, s)-defective with defect δ k,s (X) = δ if and only if Seg(P k ×X) is s-defective with defect δ s (Seg(P k ×X)) = δ.

We will show that [DF01, Proposition 1.3] holds because of a precise relation among the dimension of GS X (k, s) and the dimension of the s-th secant variety σ s (Seg(P k × X)) the Segre embedding of P k × X into P rk+r+k (see Theorem 5.1 below).

Theorem Let X ⊂ P r be an irreducible non-degenerate projective variety of dimension n. Set w = min{k, s -1} ≤ r. Then, dim σ s (Seg(P k × X)) = dim GS X (w, s) + (w + 1)(k + 1) -1. Such a theorem allows to compute the dimension of σ s (Seg(P k × X)), in the case of k ≥ s -1, for any irreducible non-degenerate projective variety X ⊂ P r (see Theorem 5.4). Moreover, such a result and its consequences endorse Conjecture 5.5 [START_REF] Abo | On the dimensions of secant varieties of segre-veronese varieties[END_REF].

The key issue for the proof of Theorem 5.1 is the existence of a rational map Φ between σ s (Seg(P k × X)) and GS X (w, s). Given the natural way in which Φ arises, it defies belief that its importance is limited to the proof of our Theorem 5.4. We devote the whole Section 2 to a detailed description of this map because we are persuaded that it will be a key tool for further investigations on secant varieties and applications. As an example, we show how it can be exploited to shed light on another problem whose importance is probably due to its numerous applications. Given s distinct points P 1 , . . . , P s ∈ X, fix a projective linear subspace Π ⊂ P 1 , . . . , P s . How many more P s-1 containing Π can be found among those that are s-secants to X? When the answer to this question is: "No one besides P 1 , . . . , P s ", then Π is said to be X-identifiable. Moreover if the general element of GS X (k, s) is contained in a unique P s-1 s-secant to X, then we say that the (k, s)-identifiability holds for X. The identifiability properties are studied, for the case k = 0, because of their many applications (see, e.g. [START_REF] Kruskal | Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics[END_REF]

, [DL06], [AMR09], [CC03], [Com02], [BC11], [CC02], [Mel09], [CO11], [KB09], [ERSS05], [ CC11
]). Our main contribution towards a solution of this problem follows from a direct application of the map Φ above, and is summarized in the following two theorems (see Theorems 3.1 and 3.3 respectively).

Theorem Let k ≤ s -1 < r. The (k, s)-identifiability holds for X if and only if the s-identifiability holds for Seg(P k × X).

Theorem Let s be an integer such that r > sn + s -1 and X is not s-defective. Then for all integers k > 0, k ≤ s -1 such that

sn + (k + 1)(s -1 -k) < (k + 1)(r -k)
the (k, s)-identifiability holds for X.

In the particular case in which the variety X itself is a standard Segre variety Seg(P(V 1 ) × • • • × P(V t )), for certain vector spaces V i , then the identifiability properties allow to deduce many peculiar examples on linear systems of tensors

E ⊂ P(V 1 ⊗ • • • ⊗ V t ) (see Section 4
). We like to stress here the Example 4.6 where we show that the general linear system of dimension 3 of matrices of type 4 × 4 and rank s = 6 is not identifiable and it is computed by exactly two sets of decomposable tensors (if the rank of the general linear system of dimension 3 of matrices of type 4 × 4 is smaller than 6 then it is identifiable, while the general linear system of dimension 3 of matrices of type 4 × 4 has rank 7).

In Section 1 we introduce all the basic definitions that will be needed throughout the paper and we fix some notation. Section 2 is entirely devoted to the description of the key tool of this paper, namely the ration map Φ : σ s (Seg(P k × X)) GS X (w, s). In Section 3 we make use of Φ to study the identifiability properties of Seg(P k × X) that will be applied in Section 4 for the particular case of linear systems of tensor. In Section 5 we prove the already quoted Theorems 5.1 and 5.4 and we show some interesting consequences of them in the particular case of X being a Segre-Veronese variety.

Preliminaries, Notation and Basic Definitions

Throughout this paper we will always work over an algebraically closed field K of characteristic 0. All the definitions that we give in this section olds for any irreducible non-degenerate projective variety Y contained in P m .

Let us recall the classical definition of secant varieties and the more modern concept of Grassmann secant varieties. When σ s (Y ) does not have the expected dimension, Y is said to be s-defective, and the positive integer

δ s (Y ) := exp dim σ s (Y ) -dim σ s (Y ) is called the s-defect of Y .
The fact that σ s (Y ) can have dimension smaller than the expected one, is clearly explained by the well known Terracini's Lemma (the first one). We remark here a consequence that arises when interpreting Terracini's Lemma in terms of fat points (see [CGG11, Section 2]).

Remark 1.2. Let P 1 , . . . , P s ∈ Y ⊂ P m be generic distinct points and consider the 0-dimensional scheme of s 2-fat points Z ⊂ Y defined by the ideal sheaf Notice that, for k = 0, the Grassmann secant variety GS Y (k, s) coincides with the secant variety σ s (Y ).

I Z = I 2 P1 ∩ • • • ∩ I 2 Ps ⊂ O Y . i) If H(Z, 1) = m + 1, then dim σ t (Y ) = m for all t ≥ s. ii) If H(Z, 1) = s(dim Y + 1), then dim σ t (Y ) = t(dim Y + 1) -1 for all t ≤ s. Definition 1.3. Let 0 ≤ k ≤ s -1 < m be
The expected dimension of GS Y (k, s) is the following (see eg. [START_REF] Ciliberto | On Grassmann secant extremal varieties[END_REF]):

(2

) exp dim GS Y (k, s) = min{s(dim Y ) + (k + 1)(s -1 -k); (k + 1)(m -k)}.
In analogy with the theory of classical secant varieties, we define the (k, s)-defect of Y as the integer:

δ k,s (Y ) := exp dim GS Y (k, s) -dim GS Y (k, s).
We end this section by introducing the concept of identifiability which will be the core of Sections 3 and 4. Definition 1.4. Fix a linear subspace Π ⊂ P m (possibly a point) and let P 1 , . . . , P s ∈ Y be distinct points. We say that Π is computed by P 1 , . . . , P s ∈ Y if Π belongs to the linear span of the points P i 's.

In this case, we say that P 1 , . . . , P s provide a decomposition of Π. The minimum integer s for which there exist s distinct points P 1 , . . . , P s ∈ Y such that Π is computed by P 1 , . . . , P s , is called the Y -rank of Π. We indicate it with r Y (Π).

Definition 1.5. Let Y and Π be as in Definition 1.4 and let s be the Y -rank of Π. We say that Π is Y -identifiable if there is a unique set of distinct points {P 1 , . . . , P s } ⊂ Y whose span contains Π.

Definition 1.6. Let Y ⊂ P m as above. We say that the (k, s)-identifiability holds for Y if the general element of GS Y (k, s) has Y -rank equal to s and it is Y -identifiable.

When k = 0, we will often omit k and we will simply say that the s-identifiability holds for Y .

The map Φ

From now on, with X we will always denote an irreducible non-degenerate projective variety of dimension n contained in P r . For any integer k ≥ 0, set N = rk+r+k and let ϕ :

P k × X → P N be the Segre embedding of P k × X. The image of ϕ is the Segre variety Seg(P k × X) ⊂ P N .
The aim of this section is to study a projective rational map Φ = Φ(X, k, s) from the s-th secant variety σ s (P k × X) of the Segre variety Seg(P k × X) into the Grassmann secant variety GS X (k, s).

We will give a definition of the morphism, in terms of local coordinates. Then, we will show how it allows to link the main secant properties of Seg(P k × X) with the Grassmann-secant properties of X.

Notation 2.1. For any choice of t points

A i = (a i,0 , . . . , a i,r ) ∈ K r+1 , 1 ≤ i ≤ t,
we denote by (A 1 , . . . , A t ) the t(r + 1)-uple (a 1,0 , . . . , a 1,r , . . . , a t,0 , . . . , a t,r ).

Let (λ 0 , . . . , λ k ) and (x 0 , . . . , x r ) be sets of homogeneous coordinates for the points Λ ∈ P k and P ∈ X, respectively.

Consider the point ϕ(Λ, P ) ∈ Seg(P k × X), so that, in coordinates:

ϕ(Λ, P ) = (λ 0 x 0 , . . . , λ 0 x r , λ 1 x 0 , . . . , λ 1 x r , . . . , λ k x 0 , . . . , λ k x r ).

Accordingly with the previous notation, we have:

ϕ(Λ, P ) = (λ 0 P, . . . , λ k P ).

Let A be a general point in σ s (Seg(P k × X)). Then there exist s distinct points Λ 1 , . . . , Λ s ∈ P k and s distinct points P 1 , . . . , P s ∈ X such that A ∈ ϕ(Λ 1 , P 1 ), . . . , ϕ(Λ s , P s ) .

Choose a set of homogeneous coordinates (a 0 , . . . , a N ) for A. By a suitable choice of the homogeneous coordinates (λ i,0 , . . . , λ i,k ) of the points Λ i , we can write:

A = (a 0 , . . . , a N ) = ϕ(Λ 1 , P 1 ) + • • • + ϕ(Λ s , P s ) = = (λ 1,0 P 1 , . . . , λ 1,k P 1 ) + • • • + (λ s,0 P s . . . , λ s,k P s ).
In the previous notation, this amounts to write

A = (λ 1,0 P 1 + • • • + λ s,0 P s , . . . , λ 1,k P 1 + • • • + λ s,k P s ).
For any general point A as above, we set:

(3) Φ(A) = λ 1,0 P 1 + • • • + λ s,0 P s , . . . , λ 1,k P 1 + • • • + λ s,k P s .
Observe that, since A is general, then the right side of the equality represents a linear space of dimension w = min{s -1, k}.

We want to show that, in this way, we get indeed a rational map

Φ : σ s (Seg(P k × X)) GS X (w, s).
The map Φ is well defined, if

(αa 0 , . . . , αa N ), α ∈ K -{0}
is another set of homogeneous coordinates of A, then

α(a 0 , . . . , a N ) = α(λ 1,0 P 1 + • • • + λ s,0 P s , . . . , λ 1,k P 1 + • • • + λ s,k P s ),
and in this case, obviously, λ 1,i P 1 + • • • + λ s,i P s and α(λ 1,i P 1 + • • • + λ s,i P s ), for 0 ≤ i ≤ k, represent the same projective points. Moreover, if there exist points M i = (µ i,0 , . . . , µ i,k ) ∈ P k and Q i = (y i,0 , . . . , y i,r ) ∈ X such that

A = (a 0 , . . . , a N ) = ϕ(M 1 , Q 1 ) + • • • + ϕ(M s , Q s ),
we get the following equality of (r + 1)(k + 1)-tuples:

(λ 1,0 P 1 + • • • + λ s,0 P s , . . . , λ 1,k P 1 + • • • + λ s,k P s ) = = (µ 1,0 Q 1 + • • • + µ s,0 Q s , . . . , µ 1,k Q 1 + • • • + µ s,k Q s ). Hence (4) λ 1,i P 1 + • • • + λ s,i P s = µ 1,i Q 1 + • • • + µ s,i Q s ; i = 0, . . . , k.
It follows that Φ is consistent.

Next, we give a characterization of points belonging to the inverse image Φ -1 (Π) of a space Π ∈ GS X (k, s).

Lemma 2.2. Let w = min{k, s -1} ≤ r and take a general point Π ∈ GS X (w, s). Assume Π ⊂ P 1 , . . . , P s for P 1 , . . . , P s ∈ X distinct points. Let B be a general element in Φ -1 (Π). Hence there exist points N 1 , . . . , N s ∈ P k such that

B = ϕ(N 1 , P 1 ) + • • • + ϕ(N s , P s ).
Proof. Let us stress, before beginning the proof, that the generality hypothesis on Π and B are crucial, for the argument.

By definition, we know that there are points Q 1 , . . . , Q s ∈ X and M 1 , . . . , M s ∈ P k , with M i = (µ i,0 , . . . , µ i,k ) for i = 1, . . . , s, such that

B = ϕ(M 1 , Q 1 ) + • • • + ϕ(M s , Q s ) = = (µ 1,0 Q 1 , . . . , µ 1,k Q 1 ) + • • • + (µ s,0 Q s . . . , µ s,k Q s ). Since Φ((µ 1,0 Q 1 , . . . , µ 1,k Q 1 ) + • • • + (µ s,0 Q s . . . , µ s,k Q s )) = = µ 1,0 Q 1 + • • • + µ s,0 Q s , . . . , µ 1,k Q 1 + • • • + µ s,k Q s and Π = λ 1,0 P 1 + • • • + λ s,0 P s , . . . , λ 1,k P 1 + • • • + λ s,k P s it follows that each point µ 1,i Q 1 + • • • + µ s,i Q s , (i = 0, . . . , k)
, lies in the span of the points λ 1,j P 1 + • • • + λ s,j P s , (j = 0, . . . , k).

By the definition of w and by the generality of Π, we may assume that the points λ 1,j P 1 + • • • + λ s,j P s , (j = 0, . . . , w) are independent. It follows that, both for w = s -1 or for w = k, there are coefficients α i,j ∈ K, such that

µ 1,i Q 1 + • • • + µ s,i Q s = w j=0 α i,j (λ 1,j P 1 + • • • + λ s,j P s ) = =   w j=0 α i,j λ 1,j   P 1 + • • • +   w j=0 α i,j λ s,j   P s for i = 0, . . . , k.
So, by setting ν h,i = w j=0 α i,j λ h,j , we have

µ 1,i Q 1 + • • • + µ s,i Q s = ν 1,i P 1 + • • • + ν s,i P s , i = 0, . . . , k.
Hence we get:

B = (µ 1,0 Q 1 + • • • + µ s,0 Q s , . . . , µ 1,k Q 1 + • • • + µ s,k Q s ) = = (ν 1,0 P 1 + • • • + ν s,0 P s , . . . , ν 1,k P 1 + • • • + ν s,k P s ) = = (ν 1,0 P 1 , . . . , ν 1,k P 1 ) + • • • + (ν s,0 P s , . . . , ν s,k P s ) = = ϕ(N 1 , P 1 ) + • • • + ϕ(N s , P s ),
where N i = (ν i,0 , . . . , ν i,k ) ∈ P k for all i.

Some consequences on the identifiability of general points

The previous construction of the map Φ in Section 2, as well as Lemma 2.2, determine the following analogue of the main Theorem in [START_REF] Dionisi | Grassman defectivity à la Terracini[END_REF], for identifiability.

Theorem 3.1. Let k ≤ s -1 < r. The (k, s)-identifiability holds for X if and only if the s-identifiability holds for Seg(P k × X).

Proof. Let Π be a general element of GS X (k, s). If there exist two different sets of distinct points {P 1 , . . . , P s }, {Q 1 , . . . , Q s } ⊂ X such that Π ⊂ P 1 , . . . , P s and Π ⊂ Q 1 , . . . , Q s , then, by Lemma 2.2, for a general point B in Φ -1 (Π), we have points M i 's and N i 's in P k , i = 1, . . . , s, with:

B = ϕ(M 1 , Q 1 ) + • • • + ϕ(M s , Q s ) = ϕ(N 1 , P 1 ) + • • • + ϕ(N s , P s ).
Since {P 1 , . . . , P s } = {Q 1 , . . . , Q s }, we get that B lies in the span of two distinct sets of points of Seg(P k × X). Now let A be a general element of σ s (Seg(P k × X)). If

A = ϕ(Λ 1 , P 1 ) + • • • + ϕ(Λ s , P s ) = ϕ(M 1 , Q 1 ) + • • • + ϕ(M s , Q s )
are two different decompositions of A, then, by the definition of Φ, Π = Φ(A) lies in the span of the two sets of points {P 1 , . . . , P s } and {Q 1 , . . . , Q s }. It suffices to prove that these two sets of points are distinct. Since A is general, we may assume that the two sets of points are both independent. Since:

λ 1,i P 1 + • • • + λ s,i P s = µ 1,i Q 1 + • • • + µ s,i Q s , i = 0, . . . , k,
then {P 1 , . . . , P s } = {Q 1 , . . . , Q s } implies λ j,i = µ j,i for all j, i. This contradicts the fact that the two decompositions of A are different.

Corollary 3.2. If the codimension of X is bigger than s, then Seg(P s-1 × X) is s-identifiabile.

Proof. Enough to observe that the general s-secant (s -1)-space cuts X only in s points, thus it is obvious that a general (s -1)-space contained in a s-secant (s -1)-space, is contained in just one of them! Then, since under our numerical assumptions we have r-n > s (hence s-1 < r), we may use the previous theorem to get the conclusion.

Using a result of [START_REF] Bocci | On the identifiability of binary segre products[END_REF], which, in turn, is based on the main result of [START_REF] Virginia Catalisano | Secant varieties of P 1 ו • •×P 1 (n-times) are not defective for n ≥ 5[END_REF], we are able to prove a criterion for the Grassmann identifiability.

Theorem 3.3. Let s be an integer such that r > sn+s-1 and X is not s-defective. Then for all integers k > 0, k ≤ s -1 such that

sn + (k + 1)(s -1 -k) < (k + 1)(r -k)
the (k, s)-identifiability holds for X.

Proof. Theorem 3.1 says that (k, s)-identifiability holds for X when s-identifiability holds for Seg(P k × X). Under our numerical assumptions, the s-secant variety of Seg(P k × X) cannot cover the linear span of Seg(P k × X). Thus we may apply the main theorem of [START_REF] Bocci | On the identifiability of binary segre products[END_REF], and conclude that Seg(P k × X) is s-identifiable.

Indeed, the proof of Theorem 3.1 can be enhanced, to give the following, more precise result: Proposition 3.4. Let w, Π, B be as in Lemma 2.2. The following two sets: E(Π) = {(P 1 , . . . , P s ) ∈ X s | Π ⊂ P 1 , . . . , P s } E(B) = {(P 1 , . . . , P s ) ∈ X s | ∃ N 1 , . . . , N s ∈ P k with B ∈ (N 1 , P 1 ), . . . , (N s , P s ) } have the same cardinality.

Proof. Almost immediate, following the proof of Theorem 3.1. The unique warning is that the set {P 1 , . . . , P s } that we use in the argument, must be independent. Since Π, B are general, this is true for all the elements of E(B) or E(Π), when these sets are finite, and for infinitely many elements, when they are infinite.

To be even more precise, the sets E(Π) and E(B) can be endowed with a quasiprojective structure and Lemma 2.2 shows indeed that there exists a birational map E(Π) → E(B).

We will not explore this point of view any further, because we do not need it in the sequel.

Linear systems of tensors

In this section, we collect some consequences of the previous theory, trying to explain properly its range of application.

We consider a linear space V over K of tensors of type n 1 + 1, . . . , n t + 1. A linear system of tensors is just a linear subspace of V . In the projective setting, tensors of type n 1 +1, . . . , n t +1, up to scalar multiplication, determine a projective space P M , where M = (Πn i ) -1. A linear system of tensors is a linear subspace E of P M .

We take the dimension of E to be the projective dimension of the linear subspace associated to E (i.e. the affine dimension, minus 1).

Inside the space of tensors, there is the subvariety X of decomposable tensors, which corresponds to the Segre embedding X = Seg(P n1 × • • • × P nt ) ⊂ P M . Definition 4.1. We say that the linear system E of tensors is computed by s decomposable tensors P 1 , . . . , P s ∈ X if for all P ∈ E there are scalars a 1 , . . . , a s such that:

P = a 1 P 1 + • • • + a s P s .
Geometrically, this means that the linear space associated to E lies in the span of the points P 1 , . . . , P s . We say that E has rank s if s is the minimum such that there are s tensors in X which compute E.

We say that a linear system E of rank s is identifiable if there exists a unique set of s decomposable tensors, that compute E.

We say that tensors of type n 1 + 1, . . . , n t + 1 are (k, s)-identifiable if the general linear system E of such tensors, of dimension k and rank s, is identifiable.

It is immediate to see that the previous terminology is consistent with the general terminology of the paper, once one consider the linear subspace associated to a linear system.

The map Φ constructed in the previous sections maps a tensor P of type k + 1, n 1 + 1, . . . , n t + 1 to a linear system of dimension k of tensors of type n 1 + 1, . . . , n t + 1. Roughly speaking, the map takes the tensor T to the linear space generated by its k + 1 slices along the first direction.

Thus, all the results in the previous section apply to the identifiability of linear systems of tensors. In particular, for instance, we see that: (i) The general linear systems of dimension k of tensors of type n 1 +1, . . . , n t +1 has rank s if and only if s is the minimum such that the secant variety σ s (P k × P n1 × • • • × P nt ) covers the projective space P N , N = (M + 1)(k + 1) -1. (ii) There are exactly q sets of decomposable tensors that compute a general linear system of tensors of type n 1 + 1, . . . , n t + 1 if and only if there are exactly q decomposable tensors that compute a general tensors of type k + 1, n 1 + 1, . . . , n t + 1. (iii) Tensors of type of type n 1 + 1, . . . , n t + 1 are (k, s)-identifiable if and only if tensors of type k + 1, n 1 + 1, . . . , n t + 1 are s-identifiable.

Let us see how the previous remarks allows to translate some known facts about tensors to facts about linear systems of tensors.

Example 4.3. For m > 4, the general linear pencil of tensors of type 2 × • • • × 2, (m-times) has rank 2 m /(m + 1) .

The general linear pencil as above, of rank s ≤ 2 m-1 /m, is identifiable.

The first fact follows from the main result in [START_REF] Virginia Catalisano | Secant varieties of P 1 ו • •×P 1 (n-times) are not defective for n ≥ 5[END_REF]. The second, from the main result of [START_REF] Bocci | On the identifiability of binary segre products[END_REF].

Example 4.4. The general linear pencil of tensors of type 2 × 2 × 2 × 2 has rank 6.

The general linear pencil of tensors of type 2 × 2 × 2 × 2, of rank s < 5, is identifiable.

The general linear pencil of tensors of type 2 × 2 × 2 × 2, of rank 5, is NOT identifiable: it is computed by exactly two sets of decomposable tensors.

Just use the main results in [START_REF] Virginia Catalisano | Secant varieties of P 1 ו • •×P 1 (n-times) are not defective for n ≥ 5[END_REF], and Proposition 4.1 of [START_REF] Bocci | On the identifiability of binary segre products[END_REF].

There are also results for linear systems of matrices, which, as far as we know, cannot be found in the classical literature.

Example 4.5. The general linear system of rank s and dimension c-1, of matrices of type a × b, with a ≤ b ≤ c, is identifiable, as soon as s ≤ ab/16.

It follows from the main result in [START_REF] Chiantini | On generic identifiability of 3-tensors of small rank[END_REF].

Example 4.6. The general linear system of dimension 3 of matrices of type 4 × 4 has rank 7. The general linear system of dimension 3 of matrices of type 4×4 and rank s < 6 is identifiable.

The general linear system of dimension 3 of matrices of type 4×4 and rank s = 6 is NOT identifiable: it is computed by exactly two sets of decomposable tensors.

Use the main results in [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF], and Theorem 1.3 of [START_REF] Chiantini | On generic identifiability of 3-tensors of small rank[END_REF].

Tons of similar results, about the identifiability of linear systems of tensors, can be found by rephrasing, from the point of view of Remark 4.2 the examples that the reader can find in [START_REF] Kruskal | Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics[END_REF], [START_REF] De | A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization[END_REF], [START_REF] Lickteig | Typical tensorial rank[END_REF], [START_REF] Virginia Catalisano | Secant varieties of P 1 ו • •×P 1 (n-times) are not defective for n ≥ 5[END_REF], [START_REF] Bocci | On the identifiability of binary segre products[END_REF], [START_REF] Abo | Induction for secant varieties of Segre varieties[END_REF], [START_REF] Chiantini | On generic identifiability of 3-tensors of small rank[END_REF], [BCO] etc.

We will not get further on this subject.

Some consequences on the dimension of secant varieties of Segre varieties

The construction introduced with the map Φ in Section 2, as well as the obvious remark at the beginning of the proof of Corollary 3.2, allow us to reproduce the proof of the main result in [START_REF] Dionisi | Grassman defectivity à la Terracini[END_REF], and also provide some new results on the defectivity of Segre varieties, which seem missing in the literature.

Let us start with a result on the relation between the dimension of σ s (Seg(P k × X)) and GS X (w, s).

Theorem 5.1. Assume, as always, w = min{k, s -1} ≤ r. Then we have:

dim σ s (Seg(P k × X)) = dim GS X (w, s) + (w + 1)(k + 1) -1.
Proof. Let Π be a general element of GS X (w, s), that is, Π is a w-space contained in P 1 , . . . , P s , where the P i are independent points of X.

If we prove that dim Φ -1 (Π) = (w + 1)(k + 1) -1, we are done. Even if w < k, we can fix scalars λ i,j ∈ K, with i = 1, . . . , s and j = 0, . . . , k, such that

Π = λ 1,0 P 1 + • • • + λ s,0 P s , . . . , λ 1,k P 1 + • • • + λ s,k P s .
Consider the points Λ i = (λ i,0 , . . . , λ i,k ) ∈ P k , and let (5)

A = ϕ(Λ 1 , P 1 ) + • • • + ϕ(Λ s , P s ) ∈ σ s (Seg(P k × X)).
Obviously A ∈ Φ -1 (Π) and, for a general choice of the scalars, A will be a general point of Φ -1 (Π). Since s < r + 1, without loss of generality, we may assume that the P i are coordinate points, say P 1 = (1, 0, . . . , 0), P 2 = (0, 1, . . . , 0), . . . , P s = (0, . . . , 0, 1, . . . , 0). With this choice of coordinates, it is easy to see that Φ(A) = (λ 1,0 , λ 2,0 , . . . , λ s,0 , 0, . . . , 0), . . . , (λ 1,k , λ 2,k , . . . , λ s,k , 0, . . . , 0) , Now, fix another general point B ∈ Φ -1 (Π). By Lemma 2.2, we know that there are points M i = (µ i,0 , . . . , µ i,k ) ∈ P k with B = ϕ(M 1 , P 1 ) + • • • + ϕ(M s , P s ) and so: Φ(B) = (µ 1,0 , µ 2,0 , . . . , µ s,0 , 0, . . . , 0), . . . , (µ 1,k , µ 2,k , . . . , µ s,k , 0, . . . , 0) . Since Φ(A) = Φ(B), in the case w = k ≤ s -1, it follows that each point (µ 1,i , µ 2,i , . . . , µ s,i , 0, . . . , 0), (i = 0, . . . , k), lies in the span of the k + 1 points (λ 1,j , λ 2,j , . . . , λ s,j , 0, . . . , 0), (j = 0, . . . , k).

In case w = s -1 < k, each point (µ 1,i , µ 2,i , . . . , µ s,i , 0, . . . , 0) , (i = 0, . . . , k), lies in the span of w+1 independent points among the k+1 points (λ 1,j , λ 2,j , . . . , λ s,j , 0, . . . , 0), (j = 0, . . . , k), and we may assume that these w + 1 independent points are (λ 1,j , λ 2,j , . . . , λ s,j , 0, . . . , 0), with j = 0, . . . , w.

In other words, there exist (w + 1)(k + 1) elements α i,j ∈ K s.t.

(µ 1,i , µ 2,i , . . . , µ s,i , 0, . . . , 0) = w j=0 α i,j (λ 1,j , λ 2,j , . . . , λ s,j , 0, . . . , 0), where i = 0, . . . , k.

Equivalently, the following linear system

    M 0 0 . . . 0 0 0 M 0 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 . . . 0 . . . 0 M                     α 0,0 . . . α 0,k α 1,0 . . . α 1,k . . . α k,0 . . . α k,k                 =                 µ 1,0 . . . µ s,0 µ 1,1 . . . µ s,1 . . . µ 1,k . . . µ s,k                 where M =     λ 1,0 . . . λ
= ϕ(M 1 , P 1 ) + • • • + ϕ(M s , P s ) = (µ 1,0 P 1 , . . . , µ 1,k P 1 ) + • • • + (µ s,0 P s . . . , µ s,k P s ) = (µ 1,0 , µ 2,0 . . . , µ s,0 , 0, . . . , 0, µ 1,1 , µ 2,1 . . . , µ s,1 , 0, . . . , 0, . . . . . . , µ 1,k , µ 2,k , . . . , µ s,k , 0, . . . , 0), it immediately follows that the dimension of Φ -1 (Π) is (w + 1)(k + 1) -1.
As consequence of Theorem 5.1, we get Terracini's Theorem of [START_REF] Dionisi | Grassman defectivity à la Terracini[END_REF].

Corollary 5.2. Let k ≤ s -1 < r. Then X is (k, s)-defective with defect δ k,s (X) = δ if and only if Seg(P k × X) is s-defective with defect δ s (Seg(P k × X)) = δ.

Proof. By a direct computation we get

exp dim(σ s (Seg(P k × X))) = exp dim(GS X (k, s)) + k 2 + 2k. By Theorem 5.1, since w = k, we have dim(σ s (Seg(P k × X))) = dim(GS X (k, s)) + k 2 + 2k.
Hence, recalling Definition 1.1, we get that the s-defect of σ s (Seg(P k × X)) and the s-defect of GS X (k, s) are the same.

Next, we get some results about the defectivity or non-defectivity of the s-th higher secant variety of Seg(P k × X).

Lemma 5.3. For s -1 < k < r, we have dim σ s (Seg(P k × X)) = min{s(k + n + 1) -1; s(k + r -s + 2) -1} Proof. By Theorem 5.1 we get dim σ s (Seg(P k × X)) = dim GS X (s -1, s) + s(k + 1) -1.

It is well known, (see, for instance, [CC08, Section 2]), that the dimension of GS X (s -1, s) is the smallest between sn and the dimension of the Grassmannian G(s -1, r), i.e.

dim GS X (s -1, s) = min{sn; s(r -s + 1)}.

and so in this case σ s (Seg(P k × X)) is not defective.

In case (ii)(b) we have s(k + n + 1) -1 -dim σ s (Seg(P k × X))

= s(k + n + 1) -1 -(s(k + r -s + 2) -1) = s(n -r + s -1) > 0 and N -dim σ s (Seg(P k × X)) = (k + 1)(r + 1) -1 -(s(k + r -s + 2) -1) = = (r -s + 1)(k -s + 1) > 0.

Hence dim σ s (Seg(P k ×X)) < exp dim σ s (Seg(P k ×X)). It follows that σ s (Seg(P k × X)) is defective.

(iii) For s -1 = k, we have s(k + r -s + 2) -1 = N , hence by Lemma 5.3 and (6) we get the conclusion.

(iv) Obvious from Theorem 5.1.

If k = r -n, by applying the theorem above we get the following interesting result.

Corollary 5.5. If k = r -n, then σ s (Seg(P k × X)) is never defective.

Proof. First assume s -1 = k. In this case, by Theorem 5.4 (iii), the sth higher secant variety of Seg(P k × X) is not defective and we have: dim σ s (Seg(P k × X)) = min{s(n + s) -1, s(r + 1) -1}. Now, since r -n = s -1, we have s(n + s) -1 = s(r + 1) -1, moreover s(n + s) -1 = (k + 1)(r + 1) -1 = N, s(r + 1) -1 = s(k + n + 1) -1, and so, for s = k + 1, we have that dim σ s (Seg(P k × X)) = N = s(k + n + 1) -1 = s(dim Seg(P k × X) + 1) -1. Now assume that s = k + 1. In this case, by Remark 1.2, we get:

• for s > k + 1, dim σ s (Seg(P k × X)) = N ;

• for s < k + 1, dim σ s (Seg(P k × X)) = s(k + n + 1) -1, and the conclusion immediately follows.

Example 5.6. Let Y be the Segre-Veronese embedding of P ( n+1 2 ) × P n via divisors of bi-degree (1, 2). Then σ s Y is never defective. In fact, let X ⊂ P ( n+2

2 )-1 be the 2-uple Veronese embedding of P n . Since Y = Seg(P ( n+1 2 ) × X).

Definition 1. 1 .

 1 The s-th higher secant variety σ s (Y ) of Y , is the Zariski closure of the union of all projective linear spaces spanned by s distinct points of Y : σ s (Y ) := P1,...,Ps∈Y P 1 , . . . , P s ⊂ P m . The expected dimension of σ s (Y ) is (1) exp dim σ s (Y ) := min{s(dim Y + 1) -1; m}.

  integers and let G(k, m) be the Grassmannian of linear k-spaces contained in P m . The (k, s)-Grassmann secant variety of Y , denoted with GS Y (k, s), is the closure in G(k, m) of the set {Λ ∈ G(k, m)|Λ lies in the linear span of s independent points of Y }.

  Remark 4.2.

- 1

 1 -n, then from Corollary 5.5 we get the conclusion.Example 5.7. Let Y be the Segre-Veronese embedding ofP k × P n1 × • • • × P nt via divisors of multi-degree (1, d 1 , . . . , d t ). If k = Π t i=1 ni+di dit i=1 n i -1, Corollary 5.5 implies that σ s (Y ) is never defective.

Acknowledgments

All authors were supported by MIUR funds. The second author was partially supported by Project Galaad of INRIA Sophia Antipolis Méditerranée (France) and Marie Curie Intra-European Fellowships for Career Development (FP7-PEOPLE-2009-IEF): "DECONSTRUCT".

Hence

dim σ s (Seg(P k × X)) = min{sn; s(r -s + 1)} + s(k + 1) -1} and the conclusion follows.

Theorem 5.4. Let X ⊂ P r be an irreducible non-degenerate projective variety of dimension n.

Proof. (i) It is enough to prove this case for s -1 = r.

Let P 1 , . . . , P s be independent points in X. We may assume that they are the coordinate points P 1 = (1, 0, . . . , 0), P 2 = (0, 1, . . . , 0), . . . , P s = (0, . . . , 0, 1). Now let A be a general point in P N , (recall that N = (k + 1)(r + 1) -1) A = (λ 1,0 , λ 2,0 , . . . , λ s,0 , . . . , λ 1,k , λ 2,k , . . . , λ s,k ).

Then

A = ϕ(Λ 1 , P 1 ) + • • • + ϕ(Λ s , P s ) and we are done.

(ii) By Lemma 5.3 we immediately get the dimensions of σ s (Seg(P k × X)) both in case (a) and in case (b).

Since in case (ii)(a) we have