On higher Gauss maps

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Contents lists available atScienceDirect

Journal

of

Pure

and

Applied

Algebra

www.elsevier.com/locate/jpaa

On

higher

Gauss

maps

✩

Pietro De Poia_, _{Giovanna Ilardi}b,∗

a _Dipartimento_di_Matematica_e_Informatica,_Università_degli_Studi_di _Udine,_via_delle_Scienze,_206,

33100Udine,Italy

b_Dipartimento_di_Matematica_e_{Applicazioni,}_Università_degli_Studi_di_Napoli_“Federico_II”,

80126 Napoli,Italy

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received8December2014 Receivedinrevisedform10April 2015

Availableonline27May2015 CommunicatedbyA.V.Geramita

MSC:

Primary:53A20;secondary:14N15; 51N35;53B99

Weprovethatthegeneralﬁbreofthei-thGaussmaphasdimensionm ifandonly ifatthegeneralpointthe(i+ 1)-thfundamentalformconsistsofconeswithvertex aﬁxedPm−1,extendingaknowntheoremfortheusualGaussmap.Weprovethis viaarecursiveformulaforexpressinghigherfundamentalforms.Wealsoshowsome consequencesoftheseresults.

1. Introduction

Let V ⊂ PN be a projective embeddedalgebraic variety and P ∈ V a point of it;with P it associates the i-th osculating space T_P(i)(V ) ⊂ PN generated by the (i+ 1)-th inﬁnitesimal neighbourhood (see e.g.

[9,Example II.3.2.5]);letdi be itsdimension.Werecallthatthei-th Gaussmap γi:V G(di,N ) isthe

rationalmapwhichassociatestothegeneralpointP itsi-thosculatingspaceT_P(i)(V ).Fori= 1 thisreduces totheusualGauss map.

The study of the Gauss map has been rather intense, both in classical (see for example [5,2,13]) and modernalgebraicgeometry (seeforexample[8,1]).Instead,thestudyof higherorderGaussmap hasbeen muchmorescarce; asfaras weknow,we canonlycite theclassicalpaperofM. Castellani[3],reviewedin modernformin[7,11]andtheveryrecentpaper[6].CloselyrelatedresultshavebeenobtainedbyR. Piene onhigherdualvarieties,seeforexample[12].WenotemoreoverthatadiﬀerentnotionofhigherGaussmap hasbeengivenbyF. Zak:seeforexample[15].

✩ _The_authors_were_partially_supported_by_MIUR,_PRIN_2010–11_project_“Geometria_delle_varietà_algebriche_e_dei_loro_spazi_di

moduli”.

* Correspondingauthor.

E-mailaddresses:pietro.depoi@uniud.it(P. De Poi),giovanna.ilardi@unina.it(G. Ilardi). http://dx.doi.org/10.1016/j.jpaa.2015.05.009

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The main basicdifference between the first and higher Gauss maps is—as shownin[3,7,12]—thatthe fibresoftheGaussmap are(Zariskiopensubsetsof)linearspaces,whileforhigherGaussmapsthisisnot trueingeneral.

Inthispaper,thestudyofhigherGaussmapsistakenupfurther:inparticular,weanalysethe infinites-imal behavioursof these maps. Surprisingly enough,it turns out, see Theorem 3.6,that thegeneral fibre of thei-th Gaussmaphasdimensionm ifandonlyifatthegeneralpoint the(i+ 1)-thfundamentalform consistsofconeswithvertexafixedPm−1,asithappensfortheusualGauss map:see[8,(2.6)].Thisresult is obtainedbyvirtueofarecursiveformulaforexpressingthe(i+ 1)-th fundamentalformintermsofthe

i-th fundamental one:see Lemma 3.1. This last result isinteresting initsown right: forexample, wecan proveimmediatelyas acorollarythe(well-known)factthatJacobiansystemofthe(i+ 1)-thfundamental form is containedin thei-th fundamentalform. Finally,we show some resultsthatfollow from themain resultofthepaper:inparticular, asanexampleoftheconsequencesofthetheorem,westudy thevarieties whose image of ahigher Gauss map has as image a curve, describingcompletely theirvarieties of higher osculatingspaces.

2. Notationandpreliminaries

Weusenotationasin[8,9],andmostofthepreliminariesaretakenfrom[4].LetV ⊂ PN beaprojective variety of dimension k over C that will be always irreducible. For any point P ∈ V we use the following notation:TP(V )⊂ PN istheembeddedtangentprojectivespacetoV inP andTP(V ) istheZariskitangent

space.

As in[8],we abusenotation byidentifyingtheembeddedtangentspaceinPN withtheaﬃnecone over it inCN +1_. _With_this _convention_T

P(V )∼=

TP(V )

C .Wedenote by G(t,N ) theGrassmannian oft-planes of

PN_.

TostudythebehaviourofV inP ,following[8](andreferences[2,6,7,10]therein),weconsiderthemanifold

F(V ) of framesinV .Anelementof F(V ) isaDarboux frame centredinP .This meansan(N + 1)-tuple

{A0; A1, . . . , Ak; . . . , AN}

whichisabasisofCN +1suchthat,ifπ :CN +1\ {0}→ PN isthecanonicalprojection,

π(A0) = P, and π(A0), π(A1), . . . , π(Ak) span TP(V ).

LetthisframemoveinF(V );thenwehavethefollowingstructureequations(intermsoftherestrictions to V oftheMaurer–Cartan1-formsωi, ωi,j onF(PN))fortheexteriorderivativesofthis movingframe

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ωμ= 0 ∀μ > k dA0= k i=0ωiAi dAi= N j=0ωi,jAj i = 1, . . . , N dωj= k h=0ωh∧ ωh,j j = 0, . . . , k dωi,j= N h=0ωi,h∧ ωh,j i = 1, . . . , N, j = 0, . . . , N. (2.1)

Remark 2.1. Geometrically, the frame {Ai} deﬁnesacoordinate simplexin PN. The 1-forms ωi,ωi,j give

the rotationmatrixwhen thecoordinate simplexisinﬁnitesimally displaced; inparticular, moduloA0,as dA0 ∈ TP∗(PN) (the cotangent space), the1-forms ω1,. . . ,ωk give abasisfor the cotangentspace TP∗(V ),

thecorresponding π(Ai)= vi∈ TP(V ) giveabasisforTP(V ) suchthatvi istangentto thelineA0Ai,and

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Wecandeﬁnenow thet-thfundamentalform andthet-thosculatingspaceat P ∈ V ,fort≥ 2,see [8, §1.(b)and(d)]and[4,§1]:

Deﬁnition 2.2. Let P ∈ V , let t ≥ 2 be an integer and let I = (i1,. . . ,ik) be such that |I| ≤ t. The

t-th osculating space to V at P is the subspace T˜_P(t)(V ) ⊂ PN _spanned _by _A

0 and by all the derivatives d|I|A0 dvi1 1 · · · dv ik k ,wherev1,. . . ,vk spanTP(V ).

IfthepointP ∈ V isgeneral,wewillput

dt:= dim( ˜TP(t)(V )).

Remark2.3.Obviously,the“expected”dimensionofthet-thosculatingspace atageneralpointP for our

k-dimensional variety V ∈ PN _is _the _minimum _among _{N and} _k t :=

_k+t

k

− 1;indeed, for the “general” variety,dt hasthis expecteddimension,but,asthenextexampleshows,thereareexceptions.

Example2.4. Letusconsider theso-called Togliatti surface S⊂ P5,whichisaparticularprojectionof the delPezzosexticsurfaceembeddedinP6_,_see_[14]_and_[10]_;_it_is_the_closure_of_the_image_of_the_rational_map

ψ:P2 P5

(x : y : z) → (x2y : xy2: x2z : xz2: y2z : yz2).

FortheTogliattisurfacewehaved2= 4:thiscanbeeasilyseenfromthefactthattheparametrisationofS

given above,restrictedto theopen affineset x= 0, satisfiesthefollowing second order partial differential equation(classicallycalledLaplaceequation):

u2∂ 2_x ∂u2 + uv ∂2_x ∂u∂v + v 2∂2x ∂v2 − 2u ∂x ∂u− 2v ∂x ∂v + 2x = 0,

whereu= y_x, v =z_x,aretheaﬃnecoordinates,etc.

Example 2.5. It is immediate to see that for the Veronesesurface V2 ⊂ P5 we have dim ˜TP(2)(V2) = 5 for

everypointP ∈ V2: thereisnotahyperplanesectionofV2 withatriplepoint.

Deﬁnition 2.6.Lett ≥ 2 andV0 ⊆ V bethequasiprojective varietyofpoints where T˜P(t)(V ) hasmaximal

dimension.Thevariety

Tant(V ) :=

P∈V0 ˜

T_P(t)(V )

iscalled thevariety ofosculating t-spacestoV .

Remark2.7.Obviouslywehave

dt≤ dt−1+ k− 1 + t t ≤ · · · ≤ t i=1 k + i− 1 i = kt.

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Following [8]wegive

Deﬁnition 2.8. The t-th fundamental form of V in P is the linear system |It_{| in} _the _projective _space

P(TP(V ))∼=Pk−1 ofhypersurfacesof degreet deﬁnedsymbolicallybytheequations:

dtA0= 0.

Moreintrinsically,wewriteIt_as _the_map

It: Sym(t)T (V )→ Nt(V ) where Nt_{(V ) is} _the _bundle _deﬁned _locally _as _Nt

P(V ) :=

CN +1

˜

T_P(t−1)(V ) and the map I

t _is _deﬁned _locally _on eachv∈ TP(V ) as vt→ d t_A 0 dvt mod ˜T (t−1) P (V ).

Example2.9.Withanexplicitcalculation(orfromExample 2.11)wecanshowthatthethirdfundamental form at thegeneral point P of theTogliatti surfaceS ⊂ P5 _is _given _by_a _single_cubic _in_P(T

P(V ))∼=P1,

while for a “general” surface (i.e. if d2 = 5) of P5 the third fundamental form is the empty set; this is

obviously truefortheVeronesesurfaceV2⊂ P5.

Wewill denotethedimensionofthet-thfundamentalformbyΔt:

Δt:= dim(|It|),

i.e. |It|∼=PΔt_.

Werecallthat(see[4,Corollary1.15]) Proposition 2.10.Wehave that

dt= dt−1+ Δt+ 1,

and viceversa:if dt= dt−1+ Δ+ 1,then thet-th fundamental formhas dimensionΔ.

Example 2.11.FortheTogliattisurfaceS⊂ P5 _we_have_that_d

3= 5 andd2= 4,and thereforeΔ3= 0.

Analogously, for a “general” surfaceof P5, d3 = d2 = 5, and Δ3 = −1, i.e. |III | = ∅; this is true for

examplefortheVeronesesurfaceV2⊂ P5.

From nowon,wewill supposethatourDarboux frame

{A0; A1, . . . , Ak; Ak+1, . . . , Ad2; Ad2+1, . . . , Ads; . . . , Adt; . . . , AN} (2.2)

is suchthatA0,A1,. . . ,Ads spanT˜ (s)

P (V ) foralls= 1,. . . ,t,withd1:= k.

Deﬁnition2.12.Lett≥ 1.Thet-th (projective) Gaussmap istherationalmap

γt: V G(dt,PN)

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Werecallthat(seeforexample[4,Theorem1.18])

Theorem2.13.The ﬁrstdiﬀerential of γt atP isthe(t+ 1)-thfundamental formatP .

Theideaof theproof isthefollowing:wehave,bythedeﬁnitionofγt,that

dγ_Pt: TPV T_T˜(t)

P VG(dt,P

N_),

andwe recallthatT_T_˜(t) P VG(dt

,PN₎∼_{= Hom( ˜}_T(t)

P V,N t+1

P (V ));moreoverifwechooseaDarbouxframe asin

(2.2),wehavethatdA0∈ ˜TPV ⊂ ˜TP(t)V and

˜ T_P(t)V CA0 = T_P(t)V andthereforedγt P ∈ Hom(TPV ⊗ TP(t)V,NPt+1(V )).

Now,weremark that,inourDarbouxframe,wecaninterpretγt_as

γt(P ) = A0∧ · · · ∧ Adt, andthereforeby(2.1), dγt_P ≡ 1≤i≤dt dt+1≤j≤N (−1)dt−i+1_ω i,jA0∧ · · · ∧ ˆAi∧ · · · ∧ Adt∧ Aj, mod ˜T (t) P V ;

now,abasisforTPV ⊗ TP(t)V canbeexpressed by(Aα⊗ Aμ)α=1,...,k μ=1,...,dt , and dγ_Pt(Aα⊗ Aμ) = dt+1≤j≤N ωμ,j(Aα)Aj ∈ N_Pt+1(V );

ontheotherhand,forthe(t+ 1)-thfundamentalform wehave

dAμ

dvα ≡

dt+1≤j≤N

ωμ,j(Aα)Aj mod ˜TP(t)(V ).

3. Fibresof higherGauss maps

Westartbywritinghigherfundamentalformsexplicitly:chooseaDarbouxframe

{A0; A1, . . . , Ak; Ak+1, . . . , Ad2; Ad2+1, . . . , Ads; . . . , Adt; . . . , AN}

suchthatA0,A1,. . . ,Ads spanT˜ (s)

P (V ) for alls= 1,. . . ,t,with d1:= k, T˜P(1)(V )= ˜TP(V ) (and d0 := 0).

Weuseindices1≤ i(1),i(1)₁ ,. . . ,i(1)_h ≤ k = d1,ds−1+ 1≤ i(s)≤ ds fors= 2,. . . t anddt+ 1≤ i(t+1)≤ N.

Withthesenotations,thesecond fundamentalform isgivenbythequadrics

V_i(2)() = i(1)1 ,i (1) 2 q_i(1) 1 ,i (1) 2 ;i()ωi (1) 1 ωi (1) 2 , (3.1) with> k,whereq i(1)1 ,i (1) 2 ;i() (= q i(1)2 ,i (1) 1 ;i() ) aredeﬁnedby

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ω_i(1) 1 ,i()= i(1)2 q_i(1) 1 ,i (1) 2 ;i()ωi (1) 2 , (3.2)

whichareobtainedviatheCartanlemma from 0 = dω_i()= i(1)1 ω i(1)1 ∧ ωi (1) 1 ,i(), sinceωi()= 0 onT˜_P(V ).

Thehigherfundamentalformscanbeexpressed asinthefollowing Lemma3.1.The (s+ 1)-th fundamental formisgivenby thepolynomials

V_i(s+1)() = k i(1)1 ,...,i (1) s+1=1 q_i(1) 1 ,...,i (1) s+1;i()ωi (1) 1 · · · ωi (1) s+1, (3.3)

with ≥ s+ 1,whichinductively satisfy therelations

ds i(s)_=d s−1+1 q_i(1) 1 ,...,i (1) s ;i(s)ωi(s),i()= k i(1)s+1=1 q_i(1) 1 ,...,i (1) s+1;i()ωi (1) s+1, (3.4) where q_i(1) 1 ,...,i (1)

s ;i(s) are the coeﬃcients of the s-th fundamental form (and with thenatural symmetries of

theindices:q_...,i(1) j ,...,i (1) k ...;i() = q_...,i(1) k ,...,i (1) j ,...;i()

),andthebasisof theinduction isthesecond fundamental form (i.e.s= 1)forwhichrelations(3.3)and (3.4)aretobe readas, respectively,(3.1)and(3.2).

Proof. BythedeﬁnitionofT˜_P(s)(V ),where s∈ {1,. . . ,t− 1},we havethat

dA_i(s−1) ≡ 0 mod ˜T_P(s)(V ) (3.5)

and from(2.1)wecanwrite

dAi(s−1) =

N

j=ds−1+1

ωi(s−1),jAj

from whichwededuce,by(3.5)

ω_i(s−1)_,i(s+1) = ω_i(s−1)_,i(s+2) =· · · = ω_i(s−1)_,i(t) = ω_i(s−1)_,i(t+1) = 0. (3.6) These equationsimplythat

0 = dω_i(s−1)_,i()=

i(s)

ω_i(s−1)_,i(s)∧ ω_i(s)_,i() ≥ s + 1;

now, bytheinductivehypothesis,wehave i(s−1) q_i(1) 1 ,...,i (1) s−1;i(s−1)ωi(s−1),i() = i(1)s q_i(1) 1 ,...,i (1) s ;i()ωi(1)s (3.7)

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whereq_i(1) 1 ,...,i

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s−1;i(s−1) arethecoeﬃcientsofthe(s− 1)-thfundamentalformandqi(1)1 ,...,i (1)

s ;i()arethoseof

thes-th fundamentalform (with theconventionthatq_i(1) 1 ,...,i

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s ;i() = 1 if s≤ 1,and ifs= 2, (3.7)is just (3.2)i.e.theCartan lemmausedtoobtainthesecondfundamentalform(3.1))andthen,applyingrelations

(3.7)to(a linearcombination of)(3.6),wededuce i(s−1)_,i(s) q_i(1) 1 ,...,i (1) s−1;i(s−1)ωi(s−1),i(s)∧ ωi(s),i()= i(1)s ,i(s) q_i(1) 1 ,...,i (1) s ;i(s)ωi(1)s ∧ ωi(s),i()

then,bytheCartan lemma i(s) q_i(1) 1 ,...,i (1) s ;i(s)ωi(s),i()= i(1)s+1 q_i(1) 1 ,...,i (1) s+1;i()ωi (1) s+1;

andthe(s+ 1)-th fundamentalform is

V_i(s+1)() = i(1)1 ,...,i (1) s+1 q_i(1) 1 ,...,i (1) s+1;i()ωi (1) 1 · · · ωi (1) s+1. 2

Remark3.2. Alternatively,we can write—for example writingback relations (3.4)in (3.3) or by adirect computation

V_i(s+1)() =

i(1)_,...,i(s)

ωi(1)ω_i(1)_,i(2)ω_i(1)_,i(3)· · · ω_i(s−1),i(s)ω_i(s)_,i(),

with≥ s+ 1 andthesumvaries,as above,for1≤ i(1)_{≤ k = d}

1,. . . ,ds−1+ 1≤ i(s)≤ ds.

Werecall

Deﬁnition3.3. LetΣ bethelinearsystemofdimension d ofhypersurfacesof degreen> 1 inPN _{(N > 1),}

generatedbythed+ 1 hypersurfacesf0= 0,. . . ,fd= 0.TheJacobiansystem J (Σ) ofΣ isthelinearsystem

deﬁnedbythepartial derivativesoftheforms f0,. . . ,fd:

J (Σ) := (∂fi/∂xj) i = 0, . . . , d; j = 0, . . . , r.

Remark 3.4. Obviously, theJacobian system J (Σ) does not depend on the choice of f0,. . . ,fd, butonly

on Σ.

Corollary 3.5. The (s+ 1)-th fundamental form is a linear system of polynomials of degree s+ 1 whose

Jacobiansystemis containedinthes-thfundamental form.

Proof. Wehave,byLemma 3.1

∂V_i(s+1)() ∂ω_i(1) j = i(1)1 ,...,i (1) s ,i(s) q_i(1) 1 ,...,i (1) s ;i()ωi(1)1 · · · ωi (1) s ∂ω_i(s)_,i() ∂ω_i(1) j . 2 (3.8)

Wearenowabletoprovethefollowing fact(see[8,(2.6)])

Theorem 3.6. The s-th Gaussmap γs _(s_{≥ 1)} _has _ﬁbres _of _dimension _{m if} _and _only _if _at _a _general _point

P ∈ V theformsofdegrees+ 1 ofthe(s+ 1)-thfundamental form|Is+1_{| are}_cones_over_a_Pm−1_{⊂ PT} P(V ).

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Proof. We chooseaDarbouxframeas above,i.e.

{A0; A1, . . . , Ak; Ak+1, . . . , Ad2; Ad2+1, . . . , Ads; . . . , Adt; . . . , Adt+1; . . . , AN}

suchthatA0,A1,. . . ,Ads spanT˜ (s)

P (V ) andindicesds−1+ 1≤ i(s)≤ dsforalls= 1,. . . ,t+ 1,withd1:= k,

etc.

Incoordinates,thes-thGauss mapγs is

γs(P ) = A0∧ · · · ∧ Ads,

and thereforeby(2.1), itsdiﬀerential, whichisthe(s+ 1)-thfundamentalform,isgivenby

dγ_Ps ≡ 1≤i≤ds ds+1≤j≤N (−1)ds−i+1_ω i,jA0∧ · · · ∧ ˆAi∧ · · · ∧ Ads∧ Aj, mod ˜T (s) P V ;

therefore, sinceγs_has_ﬁbres _of_dimension_{m if} _and_only _if_dγs

P hasrankk− m (recallthatP isageneral

point) γshasﬁbresofdimensionm ifandonlyifthespace

U∗:=ωi,j 1≤i≤ds

ds+1≤j≤N

⊂ T∗ P(V )

has dimension k− m. Dually, this space deﬁnes a subspace U ⊂ TP(V ) of dimension m, deﬁned by the

equations

ω_i(h)_,i() = 0, h = 1, . . . , s; ≥ s + 1.

Now, we prove that V_i(s+1)() are cones with vertex P(U): let us suppose to choose a frame such that

ωm+1,. . . ,ωk form abasisforU∗,thatis

ωi,j 1≤i≤ds

ds+1≤j≤N

=ωm+1, . . . , ωk. (3.9)

Moreover, wechooseindices 1≤ j₁(1),. . . ,j_h(1)≤ m,andm+ 1≤ k(1)₁ ,. . . ,k(1)_h ≤ k.

From (3.4),thepointsofU satisfyalso 0 = i(h) q_i(1) 1 ,...,i (1) h ;i(h)ωi (h)_,i()= i(1)h+1 q_i(1) 1 ,...,i (1) h+1;i()ωi (1) h+1, (3.10)

whichimplies,foreverychoiceoftheindicesasabove,that

q_i(1) 1 ,...,j

(1) h+1;i()

= 0. From thisweinfer,from (3.3), that

V_i(s+1)() = k(1)1 ,...,k (1) s+1 q_k(1) 1 ,...,k (1) s+1;i()ωk (1) 1 · · · ωk (1) s+1, (3.11) thatis,V(s+1)

i() isaconewithvertex P

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Viceversa,let ussuppose thatthe polynomials V_i(s+1)() aresingular alongaPm−1 =P(U);by choosing ourframe,wecansupposethatU isdeﬁnedby

ωj = 0, j = m + 1, . . . , k,

orthat(3.9)holds.Then,we have

0 = dω_i(h)_,i()= ds i=1 ω_i(h)_,i∧ ω_i,i()+ N j=ds+1 ω_i(h)_,j∧ ω_j,i() (3.12)

and ω_i,i(),ω_i(h)_,j ∈ U∗ and thereforethe Frobeniusintegrability conditionsfor ω_m+1,. . . ,ω_k are satisﬁed; then,from dA_i(s) = N j=ds+1 ω_i(s)_,jAj

we conclude that the s-th osculating space remains constant along the leaves of the foliation deﬁned by

ωm+1,. . . ,ωk. 2

Remark3.7.Aswerecalledintheintroduction,onecandeduce,fromTheorem 3.6,that,fors= 1 (i.e.the usual Gaussmap), theﬁbres ofγ1 _are_(Zariski_open_subsets _of)_linear_spaces: _see_[8, _(2.10)]_. _Moreover,_in [8,§3]they giveageneraldescriptionofthese varieties;we will pursuefurtherthis kindofdescription for highers inSubsection 3.1.

Example 3.8. Theﬁrst nontrivial exampleof applicationof the precedingtheorem with s≥ 2 is given by surfaceswhosesecondGaussmapγ2hasas imageacurve;thiscasehasbeenstudiedin[3]andin[7]:they proved thatthe curveswhich are ﬁbres of γ2 _are _indeed_contained _in_{3-dimensional} _projective _spaces; _by Theorem 3.6,wededucethatthethirdfundamentalform atthegeneralpointisgivenbyaperfectcube.

Letusseethefollowing consequenceoftheprecedingresult.

Corollary 3.9. If thes-th Gaussmapγs _(s_{≥ 1)}_for _a_{k-dimensional}_variety _{V has} _ﬁbres_of _dimension _m,

then ds+1≤ ds+ k− m + s s + 1 , (3.13)

andtheequalityisreachedifandonlyifthe(s+ 1)-thfundamental form|Is+1_{| is}_given_by_the_linear_system

of allthecones overaPm−1⊂ PTP(V ).

Proof. By Proposition 2.10 we have to estimate the dimension Δs+1 of the linear system given by the

(s+ 1)-th fundamental form, butthis, by Theorem 3.6, is formed bydegree (s+ 1) hypersurfaces which are cones over a Pm−1 in a Pk−1(= P(TP(V ))). Since the linear systems of these cones has (projective)

dimensionk−m+s_s+1 −1 (isequaltothecompletelinearsystemofdegree(s+ 1) hypersurfacesinaPk−m−2), wehavetheassertion. 2

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3.1. Varietieswith acurveastheimage of thes-thGaussmap

Letus startwiththefollowing

Lemma 3.11. If ds+1 = ds+ 1, then ds+2 ≤ ds+ 2. Moreover, if ds+2 = ds+ 2, then |Is+1| and|Is+2| are

generatedbypowersofthesamelinearform,andifds+ 1= ds+2,thenourvarietyV iscontainedinPds+2.

Proof. We apply Proposition 2.10 to deducethat the(s+ 1)-th fundamentalform |Is+1| is generated by onlyoneform.Callthisform f ;byCorollary 3.5,ifg isaform in|Is+2_|, _then_all_its_partial_derivatives_are

in|Is+1_|;_therefore,_by_Euler’s_formula

g = 1 s + 2 k i=1 ωi ∂g ∂ωi = 1 s + 2 k i=1 ωiaif

ai ∈ C,i= 1,. . . ,k,andwededucethateitherg iszeroorg = f ,where is alinearform.Ifg = f ,bya

changeofcoordinateswecansupposethat= ω1;then,wehave ∂g ∂ω1 = f + ω1 ∂f ∂ω1 ∈ |I s+1_{| = (f)} ∂g ∂ωj = ω1 ∂f ∂ωj ∈ |I s+1_{| = (f)} _{j = 2, . . . , k}

whichimplythateitherf doesnotdependonω1,buttheng isthezeroform,orf isreducibleoftheform f = ω1h.Wecannowproceedbyinductiontoprovethatf = ω1s+1:iff = ω1φ,then

∂g ∂ω1 = ( + 1)ωφ + ω+1₁ ∂φ ∂ω1 = nω₁φ ∂g ∂ωj = ω₁+1 ∂φ ∂ωj = njω1φ j = 2, . . . , k

n,nj ∈ C,j = 1,. . . ,k, and,as above,eitherφ doesnotdependonω1,buttheng isthezeroform,or φ is

reducible oftheformφ= ω1φ.

Ifg = 0, thenwecanconcludefrom Theorem 3.6(or, better,from Example 3.10). 2

Corollary3.12.Ifds+2= ds+1+ 1= ds+ 2,thenthes-thand(s+ 1)-thGaussmapshaveﬁbresofdimension

k− 1.

Proof. ByLemma 3.11wehavethatthe|Is+1_{| and}_|Is+2_{| are}_cones_over_a_Pk−2_,_therefore,_by_{Theorem 3.6}_,

thethesisfollows. 2

Thecaseweareinterestedinnowistheonewithm= k− 1 (i.e. thenextafterExample 3.10),therefore, by(3.13),(ds≤)ds+1≤ ds+ 1;ifds+1< ds+ 1,thends+1= ds,butinthiscasewewouldhavenoelements

in |Is+1_|, _and _therefore_we _would _be _in _{Example 3.10}_. _We _deduce _that_d

s+1 = ds+ 1, and we are inthe

caseof Lemma 3.11, andbyTheorem 3.6theimage ofthes-thGauss map hasdimension 1;the complete description ofsuch caseswill followfrom [8, (2.4)].To usethis result,we needsomenotations andresult: we startbyrecallingthenotationintroducedin[8,§2(a)];letB beanr-dimensionalvarietyand let

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beamorphism;lety∈ B beageneralpoint,andletSy ⊂ V (wherePN=P(V ))bethe(d+ 1)-dimensional

vectorspacewhichcorresponds tof (y).Then,thediﬀerentialoff iny can bethoughtof

dyf : TyB → Hom(Sy, Ny),

where Ny := _SV_y. Moreexplicitly, ify1,. . . ,yr are local coordinatesofB near y,and ife0(y),. . . ed(y) is a

basisforthe (d+ 1)-dimensionalvectorspacesnearSy, wehave

dyf ∂ ∂yi (ej(y))≡ ∂(ej(y)) ∂yi mod Sy

fori= 1,. . . ,r andj = 0,. . . , d.Then, fixedw∈ TyB,onecandefinethe“infinitelynear”space

Sy

dw := Im dyf (w)⊂ Ny,

andifwedenoteby[v] theclassof v∈ V inNy, weconsiderthefollowingsubspacesof V :

Sy+ Sy dw := v∈ V | v = s + t, s ∈ Sy, [t]∈ Sy dw Sy∩ Sy dw := s∈ Sy| ([0] =)[s] ∈ Sy dw = ker(dyf (w)). Obviously,we have rk(dyf ) = ρ ⇐⇒ dim(Sy+ Sy dw) = d + 1 + ρ ⇐⇒ dim(Sy∩ Sy dw) = d + 1− ρ.

WealsodenotetheprojectivesubspacesofPN _associated_to_the_vector_spaces_just_deﬁned_by

Pd y+ Pd y dw :=P(Sy+ Sy dw) Pd y∩ Pd y dw :=P(Sy∩ Sy dw).

Thecaseofr = 1 iscompletely solvedin[8];asinthecitedarticle,withabuseofnotationwedenote by

y thelocalcoordinateofB nearourgeneralpoint y:

Proposition 3.13.(See[8, (2.4)].) Withnotationsas above,if r = 1 anddim(Pd y∩

Pd y

dw)= d− ρ,thenthere

existρ curves inPN_,_C

1,. . . ,Cρ parametrised byy, a1,. . . ,aρ ∈ N, andaﬁxed (d−ρ_i=1ai)-dimensional

subspace H ⊂ PN _such _that _Pd

y is the span of H together with the (ai − 1)-osculating (ai − 1)-planes

˜ T(ai−1) P (Ci)⊂ PN,that is Pd y= ˜T (a1−1) P (C1), . . . , ˜T (aρ−1) P (Cρ), H.

Letusnowusethis resultinourcase:we takeB astheimage ofthes-thGaussmap:

γs: V → Im(γs) =: B→ G(di s, N ),

dim B = 1 and Sy denotes just the ds-dimensionalvector subspace T˜P(s)(V ) ⊂ CN +1, where y = γs(P ),

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havethatB deﬁnesalsoaruledvariety,whichisindeedthevarietyofosculatings-spacestoV :Tans(V ):=

∪y∈BPds

y ofdimensionds+ 1.ByCorollary 3.12,ifT˜P(s)(V )= ˜T

(s+1)

P (V ),thesamehappenswithT˜

(s+1)

P (V ):

we have,on B, theS_y’s,whichare theds+1-dimensionalvector subspacesT˜P(s+1)(V )⊂ CN +1,they deﬁne

the(ruled)varietyofosculating(s+ 1)-spacestoV Tans+1(V ):=∪y∈BPdys+1 ofdimensionds+ 2 suchthat

V ⊂ Tans(V )⊂ Tans+1(V ).

Since, bydeﬁnitionofthe(s+ 1) osculating space,wehavethat,fory∈ B,

Pds y + Pds y dy ⊂ P ds+1 y , whichimplies dim(Pds y ∩ Pds y dy ) = ds− 1. Then, byProposition 3.13

Proposition 3.14. If the s-th Gauss map has dimension one, then there exists a curve C ⊂ PN _having

(a− 1)-osculating(a− 1) planes Pa−1_y := ˜Ty(a−1)(C) andaﬁxed PN−a−1 suchthat

Pds

y =Pa−1y +PN−a−1.

In otherwords,Tans(V ),istheconeoverPN−a−1 _of_Tana−1_{(C) (and}_the_same_holds_true_for_higher_Gauss

maps).

Clearly, onecandeducemoregeneralresultson thevarietiesof osculatings-spacesinthe sameveinas at theendof[8,§3].

Acknowledgement

Wewouldlike tothanktheanonymousrefereeforvaluablesuggestionsand remarks. References

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