Duality with expanding maps and shrinking maps, and its applications to Gauss maps
aa r X i v : . [ m a t h . AG ] A p r DUALITY WITHEXPANDING MAPS AND SHRINKING MAPS, ANDITS APPLICATIONS TO GAUSS MAPS
KATSUHISA FURUKAWA
Abstract.
We study expanding maps and shrinking maps of subvarietiesof Grassmann varieties in arbitrary characteristic. The shrinking map wasstudied independently by Landsberg and Piontkowski in order to charac-terize Gauss images. To develop their method, we introduce the expandingmap, which is a dual notion of the shrinking map and is a generalization ofthe Gauss map. Then we give a characterization of separable Gauss mapsand their images, which yields results for the following topics: (1) Linearityof general fibers of separable Gauss maps; (2) Generalization of the charac-terization of Gauss images; (3) Duality on one-dimensional parameter spacesof linear subvarieties lying in developable varieties. Introduction
For a projective variety X ⊂ P N over an algebraically closed field of arbitrarycharacteristic, the Gauss map γ = γ X of X is defined to be the rational map X G (dim X, P N ) which sends each smooth point x to the embedded tan-gent space T x X at x in P N . The shrinking map of a subvariety of a Grassmannvariety was studied independently by Landsberg and Piontkowski in order tocharacterize Gauss images in characteristic zero, around 1996 according to [11,p. 93] (see [1, 2.4.7] and [11, Theorem 3.4.8] for details of their results). Todevelop their method, we introduce the expanding map of a subvariety of aGrassmann variety, which is a generalization of the Gauss map and is a dualnotion of the shrinking map (see § Linearity of general fibers of separable Gauss maps.Theorem 1.1 (= Corollary 3.7) . Let γ be a separable Gauss map of a projectivevariety X ⊂ P N . Then the closure of a general fiber of γ is a linear subvarietyof P N . Date : April 8, 2013.2000
Mathematics Subject Classification.
Primary 14N05; Secondary 14M15.
Key words and phrases.
Gauss map, separable, birational. In the first version of this paper (arXiv:1110.4841v1), we proved the linearity by usingduality on subspaces of tangent spaces with linear projection techniques. After that, we have
According to a theorem of Zak [27, I, 2.8. Corollary], the Gauss map isfinite if X is smooth (and is not a linear subvariety of P N ). Combining withTheorem 1.1, we have that, if the projective variety X ⊂ P N is smooth andthe Gauss map γ is separable, then γ is in fact birational (Corollary 3.8).Geometrically, the birationality of γ means that a general embedded tangentspace is tangent to X at a unique point.In characteristic zero, it was well known that the closure F ⊂ X of a generalfiber of the Gauss map γ is a linear subvariety of P N (Griffiths and Harris[7, (2.10)], Zak [27, I, 2.3. Theorem (c)]). In positive characteristic, γ canbe inseparable , and then F can be non-linear (see Remark 3.9); this leadsus to a natural question: Is F a linear subvariety if γ is separable ? (Kajiasked, for example, in [16, Question 2] [17, Problem 3.11].) The curve casewas classically known (see Remark 3.10). Kleiman and Piene [20, pp. 108–109]proved that, if X ⊂ P N is reflexive , then a general fiber of the Gauss map γ is scheme-theoretically (an open subscheme of) a linear subvariety of P N . Incharacteristic zero, their result gives a reasonable proof of the linearity of F ,since every X is reflexive. In arbitrary characteristic, in terms of reflexivity,the linearity of a general fiber F of a separable γ follows if codim P N ( X ) = 1 ordim X
2, since separability of γ implies reflexivity of X if codim P N ( X ) = 1(due to the Monge-Segre-Wallace criterion [8, (2.4)], [19, I-1(4)]), dim X = 1(Voloch [25], Kaji [14]), or dim X = 2 (Fukasawa and Kaji [6]). On the otherhand, for dim X >
3, Kaji [15] and Fukasawa [4] [5] showed that separabilityof γ does not imply reflexivity of X in general. For any X , by Theorem 1.1,we finally answer the question affirmatively.1.2. Generalization of the characterization of Gauss images.
We gener-alize the characterization of Gauss images given by Landsberg and Piontkowskito the arbitrary characteristic case, as follows:
Theorem 1.2 (= Corollary 3.15) . Let σ be the shrinking map from a closedsubvariety Y ⊂ G ( M, P N ) to G ( M − , P N ) with integers M, M − ( M > M − ),and let U G ( M − , P N ) ⊂ G ( M − , P N ) × P N be the universal family of G ( M − , P N ) .Then Y is the closure of a image of a separable Gauss map if and only if M − = M − dim Y holds and the projection σ ∗ U G ( M − , P N ) → P N is separable andgenerically finite onto its image. Here the generalized conormal morphism , induced from a expanding map,plays an essential role; indeed, we give a generalization of the Monge-Segre-Wallace criterion to the morphism (Proposition 3.13).1.3.
Duality on one-dimensional developable parameter spaces.
Laterin the paper, instead of the subvariety X ⊂ P N , we focus on X ⊂ G ( m, P N ), aparameter space of m -planes lying in X , and study developability of ( X , X ) (seeDefinition 4.1). It is classically known that, in characterize zero, a projective examined what is essential in the proof. In the second version, considering expanding mapsand shrinking maps, we give a more evident proof of the result. UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 3 variety having a one-parameter developable uniruling (by m -planes) is obtainedas a cone over an osculating scroll of a curve ([1, 2.2.8], [11, Theorem. 3.12.5];the arbitrary characteristic case was investigated by Fukasawa [2]). Applyingour main theorem, we find duality on one-dimensional developable parameterspaces via expanding maps and shrinking maps, in arbitrary characteristic, asfollows. Here γ i = γ i X is defined inductively by γ := γ, γ i := γ γ i − X ◦ γ i − ,with the closure γ i X of the image of X under γ i . In a similar way, σ i is defined. Theorem 1.3 (= Theorem 4.18) . Let
X ⊂ G ( m, P N ) and X ′ ⊂ G ( m ′ , P N ) beprojective curves. Then, for an integer ε > , the following are equivalent: (a) X ′ is developable, the map γ ε = γ ε X ′ is separable and generically finite,and γ ε X ′ = X . (b) X is developable, the map σ ε = σ ε X is separable and generically finite,and σ ε X = X ′ .In this case, m = m ′ + ε . As a corollary, if σ m is separable and X is not a cone, then C := σ m X is aprojective curve in P N such that γ m is separable and X = γ m C ; in particular, X is equal to the osculating scroll of order m of C (Corollary 4.19). On theother hand, if γ X is separable, then an equality T (( T X ) ∗ ) = X ∗ in ( P N ) ∨ holds(Corollary 4.20; cf. for osculating scrolls of curves, this equality was deducedfrom Piene’s work in characterization zero [23], and was shown by Hommaunder some conditions on the characteristic [9] (see [9, Remark 4.3])).This paper is organized as follows: In § γ : X G ( m + , P N ) of a subvariety X ⊂ G ( m, P N ). In addition, setting Y to be the closure of the image of X ,we investigate properties of composition of the expanding map γ of X and theshrinking map σ of Y . Then, in § § X as a parameter space of m -planes lying in X ⊂ P N , andstudy developability of X in terms of γ .2. Expanding maps of subvarieties of Grassmann varieties
In this section, we denote by γ : X G ( m + , P N ) the expanding map of asubvariety X ⊂ G ( m, P N ) with integer m, m + ( m m + ), which is defined asfollows: Definition 2.1.
Let Q G ( m, P N ) and S G ( m, P N ) be the universal quotient bundleand subbundle of rank m + 1 and N − m on G ( m, P N ) with the exact sequence0 → S G ( m, P N ) → H ( P N , O (1)) ⊗ O G ( m, P N ) → Q G ( m, P N ) →
0. We set Q X := Q G ( m, P N ) | X and call this the universal quotient bundle on X , and so on. Wedenote by X sm the smooth locus of X . A homomorphism ϕ is defined by thecomposition: ϕ : S X sm → H om ( H om ( S X sm , Q X sm ) , Q X sm ) → H om ( T X sm , Q X sm ) , K. FURUKAWA where the first homomorphism is induced from the dual of Q X ⊗ Q ∨X → O X , andthe second one is induced from T X sm ֒ → T G ( m, P N ) | X sm = H om ( S X sm , Q X sm ). Wecan take an integer m + = m + γ with m m + N such that a general point x ∈ X satisfies dim(ker ϕ ⊗ k ( x )) = N − m + . Let ( P N ) ∨ := G ( N − , P N ), the space of hyperplanes. Then ker ϕ | X ◦ is asubbundle of H ( P N , O (1)) ⊗ O X ◦ ≃ H (( P N ) ∨ , O (1)) ∨ ⊗ O X ◦ of rank N − m + for a certain open subset X ◦ ⊂ X . By the universality of the Grassmannvariety, under the identification G ( N − m + − , ( P N ) ∨ ) ≃ G ( m + , P N ), we havean induced morphism, γ = γ X / G ( m, P N ) : X ◦ → G ( m + , P N ) . We call γ the expanding map of X . Here ker ϕ | X ◦ ≃ γ | ∗X ◦ ( S G ( m + , P N ) ). Remark 2.2.
Suppose that m = 0 and X ⊂ P N = G (0 , P N ). Then γ = γ X/ P N coincides with the Gauss map X G (dim( X ) , P N ); in other words, γ ( x ) = T x X for each smooth point x ∈ X . The reason is as follows: In this setting, itfollows that S P N = Ω P N (1) and Q P N = O P N (1), and that ϕ is the homomorphismΩ P N (1) | X → Ω X (1). Therefore ker ϕ | X sm = N ∨ X/ P N (1) | X sm , which implies theassertion.The shrinking map σ : Y G ( M − , P N ) of a subvariety Y ⊂ G ( M, P N )with integers M, M − ( M > M − ) is defined similarly, as follows: Definition 2.3.
Let Q Y and S Y be the universal quotient bundle and subbundleof rank M + 1 and N − M on Y . A homomorphism Φ is defined by thecomposition:Φ : Q ∨Y sm → H om ( H om ( Q ∨Y sm , S ∨Y sm ) , S ∨Y sm ) → H om ( T Y sm , S ∨Y sm ) , where the second homomorphism is induced from T Y sm ֒ → T G ( M, P N ) | Y sm = H om ( Q ∨Y sm , S ∨Y sm ). We can take an integer M − = M − σ with − M − M such that a general point y ∈ Y satisfiesdim(ker Φ ⊗ k ( y )) = M − + 1 . Since ker Φ | Y ◦ is a subbundle of H ( P N , O (1)) ∨ ⊗ O Y ◦ of rank M − + 1 for acertain open subset Y ◦ ⊂ Y , we have an induced morphism, called the shrinkingmap of Y , σ = σ Y / G ( M, P N ) : Y ◦ → G ( M − , P N ) . Here we have ker Φ | Y ◦ = σ | ∗Y ◦ ( Q ∨ G ( M − , P N ) ). Remark 2.4.
Let ¯
X ⊂ G ( N − m − , ( P N ) ∨ ) be the subvariety correspondingto X under the identification G ( m, P N ) ≃ G ( N − m − , ( P N ) ∨ ), and so on.Then γ X / G ( m, P N ) is identified with the shrinking map σ ¯ X / G ( N − m − , ( P N ) ∨ ) : ¯ X G ( N − m + − , ( P N ) ∨ )under G ( m + , P N ) ≃ G ( N − m + − , ( P N ) ∨ ). In a similar way, σ Y / G ( M, P N ) isidentified with the expanding map γ ¯ Y / G ( N − M − , ( P N ) ∨ ) . UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 5
Let U G ( m, P N ) ⊂ G ( m, P N ) × P N be the universal family of G ( m, P N ). Wedenote by U X := U G ( m, P N ) | X ⊂ X × P N the universal family of X , and by π X : U X → P N the projection, and so on. (Recall that, for each x ∈ X , the m -plane x ⊂ P N is equal to π X ( L x ) for the fiber L x of U X → X at x .) Remark 2.5.
A general point x ∈ X gives an inclusion x ⊂ γ ( x ) of linearvarieties in P N , and a general point y ∈ Y gives an inclusion σ ( y ) ⊂ y of linearvarieties in P N . Lemma 2.6.
Let X , Y , U X , U Y be as above. Then the following holds: (a) If m + N − and the image of γ is a point L ∈ G ( m + , P N ) , then π X ( U X ) ⊂ P N is contained in the m + -plane L . (b) If M − > and the image of σ is a point L ∈ G ( M − , P N ) , then π Y ( U Y ) is a cone in P N such that the M − -plane L is a vertex of the cone.Proof. (a) For general x ∈ X , we have x ⊂ γ ( x ) = L as in Remark 2.5. Itfollows that π X ( U X ) is contained in the m + -plane L .(b) For general y ∈ Y , we have σ ( y ) = L ⊂ y . We set Y ′ := π Y ( Y ) ⊂ P N .Then a general point y ′ ∈ Y ′ is contained in some M -plane y , so that also h y ′ , L i is contained in y , where h y ′ , L i is the linear subvariety of P N spannedby y ′ and L . Hence Y ′ is a cone with vertex L . (cid:3) We denote by P ∗ ( A ) := Proj( L Sym d A ∨ ) the projectivization of a locallyfree sheaf or a vector space A . Definition 2.7.
Let V G ( M, P N ) := P ∗ ( S G ( M, P N ) ) , which is contained in G ( M, P N ) × ( P N ) ∨ = P ∗ ( H ( P N , O (1)) ⊗ O G ( M, P N ) ) andis regarded as the universal family of G ( N − M − , ( P N ) ∨ ). We set V Y := V G ( M, P N ) | Y and set ¯ π = ¯ π Y : V Y → ( P N ) ∨ to be the projection.In the case where Y is the closure of the image of X under the expandingmap γ , the following commutative diagram is obtained: γ ∗ V Y / / ❴❴❴ γ ∗ ¯ π * * (cid:15) (cid:15) V Y ¯ π / / (cid:15) (cid:15) ( P N ) ∨ X γ / / ❴❴❴❴ Y ,where we call the projection γ ∗ ¯ π : γ ∗ V Y → ( P N ) ∨ the generalized conormal mor-phism , and where γ ∗ V Y ⊂ X × ( P N ) ∨ is the closure of the pull-back ( γ | X ◦ ) ∗ V Y .Note that we have ( γ | X ◦ ) ∗ V Y = P ∗ (ker ϕ | X ◦ ), because of γ | ∗X ◦ ( S Y ) ≃ ker ϕ | X ◦ .2.1. Standard open subset of the Grassmann variety.
Let us denote by( Z : Z : · · · : Z N ) K. FURUKAWA the homogeneous coordinates on P N . To fix our notation, we will prepare adescription of a standard open subset G ◦ m ⊂ G ( m, P N ) which is the set of m -planes not intersecting the ( N − m − Z = Z = · · · = Z m = 0). Letus denote by Z , Z , . . . , Z N ∈ H ( P N , O (1)) ∨ the dual basis of Z , Z , . . . , Z N ∈ H ( P N , O (1)), and so on.(A) The sheaves Q G ◦ m and S ∨ G ◦ m are free on G ◦ m , and are equal to Q ⊗ O G ◦ m and S ∨ ⊗ O G ◦ m , for the vector spaces Q := M i m K · η i and S ∨ := M m +1 j N K · ζ j , where K is the ground field, η i is the image of Z i under H ( P N , O (1)) ⊗ O → Q G ( m, P N ) , and ζ j is the image of Z j under H ( P N , O (1)) ∨ ⊗ O → S ∨ G ( m, P N ) .We have a standard isomorphism G ◦ m ≃ Hom( Q ∨ , S ∨ ) : x X i m,m +1 j N a ji · η i ⊗ ζ j = ( a ji ) i,j , (1)as follows. We take an element x ∈ G ◦ m . Under the surjection H ( P N , O (1)) ∨ → S ∨ G ( m, P N ) ⊗ k ( x ), for each 0 i m , we have Z i
7→ − P m +1 j N a ji · ζ j with some a ji = a ji ( x ) ∈ K . This induces a linear map Q ∨ → S ∨ : η i P a ji · ζ j , whichis regarded as a tensor P a ji · η i ⊗ ζ j under the identification Hom( Q ∨ , S ∨ ) ≃ Q ⊗ S ∨ . This gives the homomorphism (1).In this setting, the linear map Q ∨ G ◦ m ⊗ k ( x ) → H ( P N , O (1)) ∨ is given by η i Z i + P j a ji · Z j , and hence, for each x ∈ G ( m, P N ), the m -plane x ⊂ P N is spanned by the points of P N corresponding to the row vectors of the( m + 1) × ( N + 1) matrix, a m +10 a m +20 · · · a N a m +11 a m +21 · · · a N . . . ... ... ... a m +1 m a m +2 m · · · a Nm . (B) Let U G ( m, P N ) := P ∗ ( Q ∨ G ( m, P N ) ) in G ( m, P N ) × P N , which is the universalfamily of G ( m, P N ). Then we have an identification U G ◦ m ≃ G ◦ m × P m ≃ Hom( Q ∨ , S ∨ ) × P m . Regarding ( η : · · · : η m ) as the homogeneous coordinates on P m = P ∗ ( Q ∨ ),under the identification (1), we can parametrize the projection U G ◦ m → P N bysending (( a ji ) i,j , ( η : · · · : η m )) to the point( η : · · · : η m : X i η i a m +1 i : · · · : X i η i a Ni ) ∈ P N . UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 7
This is also expressed as X i m η i Z i + X i m,m +1 j N η i a ji · Z j ∈ P N . (2)(C) The m -plane x ∈ G ◦ m , which is expressed as ( a ji ) i,j under (1), is alsogiven by the set of points ( Z : Z : · · · : Z N ) ∈ P N such that (cid:2) Z Z · · · Z N (cid:3) a m +10 a m +20 · · · a N a m +11 a m +21 · · · a N ... ... ... a m +1 m a m +2 m · · · a Nm − − − = 0 . Let V G ( m, P N ) := P ∗ ( S G ( m, P N ) ) in G ( m, P N ) × ( P N ) ∨ , which is the universalfamily of G ( N − m − , ( P N ) ∨ ). Then we have an identification V G ◦ m ≃ G ◦ m × P N − m − ≃ Hom( Q ∨ , S ∨ ) × P N − m − . Regarding ( ζ m +1 : · · · : ζ N ) as homogeneous coordinates on P N − m − = P ∗ ( S ),we can parametrize V G ◦ m → ( P N ) ∨ by sending (( a ji ) i,j , ( ζ m +1 : · · · : ζ N )) to thehyperplane defined by the homogeneous polynomial X i m,m +1 j N ζ j a ji · Z i + X m +1 j N − ζ j · Z j . (3)2.2. Parametrization of expanding maps.
Let
X ⊂ G ( m, P N ) be a sub-variety with m >
0. We will give a local parametrization of the expandingmap γ : X G ( m + , P N ) around a general point x o ∈ X in the following twosteps. Step 1.
Changing the homogeneous coordinates ( Z : · · · : Z N ) on P N , wecan assume that x o ∈ G ( m, P N ) and γ ( x o ) ∈ G ( m + , P N ) are linear subvarietiesof dimensions m and m + such that x o = ( Z m +1 = · · · = Z N = 0) , (4) γ ( x o ) = ( Z m + +1 = · · · = Z N = 0) , (5)in P N . As in § G ◦ m ⊂ G ( m, P N ), and takea system of regular parameters z , . . . , z dim( X ) of the regular local ring O X ,x o .Then, under the identification (1), X ∩ G ◦ m is parametrized around x o by X i m,m +1 j N f ji · η i ⊗ ζ j = ( f ji ) i,j K. FURUKAWA with regular functions f ji ’s. From (4), we have f ji ( x o ) = 0. For a general point x ∈ X near x o , we identify Q with Q X ⊗ k ( x ), and S with S X ⊗ k ( x ). Then thelinear map T x X ֒ → T x G ( m, P N ) = Hom( Q ∨ , S ∨ ) is represented by ∂∂z e X i m,m +1 j N f ji,z e ( x ) · η i ⊗ ζ j (1 e dim( X )) , (6)where f ji,z e := ∂f ji /∂z e . Therefore Hom(Hom( S, Q ) , Q ) → Hom( T x X , Q ) isrepresented by ζ j ⊗ η i ⊗ η i ′ X e dim( X ) f ji,z e ( x ) · dz e ⊗ η i ′ (0 i, i ′ m, m + 1 j N ) . Since S → Hom(Hom(
S, Q ) , Q ) is given by ζ j ζ j ⊗ P i ( η i ⊗ η i ), it followsthat ϕ x : S → Hom( T x X , Q ) is represented by ϕ x ( ζ j ) = X i m, e dim( X ) f ji,z e ( x ) · dz e ⊗ η i ( m + 1 j N ) . (7)Recall that m + is the integer such that N − m + = dim(ker ϕ x ), which impliesthat dim( ϕ x ( S )) = m + − m . By (5), we have ϕ x o ( ζ m + +1 ) = · · · = ϕ x o ( ζ N ) = 0.It follows that ϕ x ( ζ m +1 ) , . . . , ϕ x ( ζ m + ) give a basis of the vector space ϕ x ( S ),and that ϕ x ( ζ µ ) = X m +1 ν m + g µν ( x ) · ϕ x ( ζ ν ) ( m + + 1 µ N )(8)with regular functions g µν ’s. As a result, we have f µi,z e = X m +1 ν m + g µν f νi,z e (0 i m, e dim( X ))(9)for m + + 1 µ N . Since ϕ x o ( ζ µ ) = 0 for m + + 1 µ N , in this setting,it follows that g µν ( x o ) = 0 and that f µi,z e ( x o ) = 0 (0 i m, e dim( X ))(10)for m + + 1 µ N . Lemma 2.8.
Let
X ⊂ G ( m, P N ) be of dimension > with m < N . For theinteger m + given with γ : X G ( m + , P N ) , we have m + > m .Proof. Assume m + = m . Then, as above, we have dim( ϕ x ( S )) = 0, whichmeans that ϕ x ( ζ j ) = 0 for any m + 1 j N . Then we have f ji,z e = 0 forany i, j, e . This contradicts that f ji ’s are regular functions parametrizing theembedding X ֒ → G ( m, P N ) around the point x o . (cid:3) Step 2.
We set
Y ⊂ G ( m + , P N ) to be the closure of X under γ . As in § V G ( m, P N ) := P ∗ ( S G ( m, P N ) ) and consider the generalized conormalmorphism γ ∗ ¯ π | X ◦ : P ∗ (ker ϕ | X ◦ ) ⊂ V G ( m, P N ) → ( P N ) ∨ UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 9 given in Definition 2.7. Let ℓ x be the fiber of P ∗ (ker ϕ | X ◦ ) → X ◦ at x , and let v ∈ ℓ x be a point. Here v is expressed as (( f ji ( x )) i,j , ( s m +1 : · · · : s N )). Since P m +1 j N s j · ϕ x ( ζ j ) = ϕ x ( P m +1 j N s j ζ j ) = 0, the equality (8) implies X m +1 ν m + s ν · ϕ x ( ζ ν ) = − X m + +1 µ N s µ · ϕ x ( ζ µ )= X m +1 ν m + ,m + +1 µ N − s µ g µν ( x ) · ϕ x ( ζ ν ) . Thus s ν = P m + +1 µ N − s µ g µν for m + 1 ν m + . Then it follows from(3) that each point v ∈ ℓ x is sent to the hyperplane γ ∗ ¯ π ( v ) ∈ ( P N ) ∨ which isdefined by the homogeneous polynomial(11) X i mm +1 ν m + X m + +1 µ N − s µ g µν f νi ( x ) · Z i + X i mm + +1 µ N s µ f µi ( x ) · Z i + X m +1 ν m + m + +1 µ N s µ g µν ( x ) · Z ν + X m + +1 µ N − s µ · Z µ . Note that, for the m + -plane γ ( x ) ⊂ P N , the image γ ∗ ¯ π ( ℓ x ) is equal to γ ( x ) ∗ ⊂ ( P N ) ∨ , the set of hyperplanes containing γ ( x ).Now, the parametrization of the expanding map γ : X Y is obtained,as follows: Let Q G ( m + , P N ) and S G ( m + , P N ) be the universal quotient bundle andsubbundle of rank m + + 1 and N − m + on G ( m + , P N ). In a similar way to § G ◦ m + as the set of m + -planes notintersecting the ( N − m + − Z = · · · = Z m + = 0). Then Q G ◦ m + and S ∨ G ◦ m + are equal to Q + ⊗ O G ◦ m + and S ∨ + ⊗ O G ◦ m + , for vector spaces Q + = M λ m + K · q λ and S ∨ + = M m + +1 µ N K · s µ , where q λ and s µ correspond to Z λ and Z µ .In this setting, by (3) and (11), γ ( x ) ∈ G ◦ m + = Hom( Q ∨ + , S ∨ + ) is expressed as X i mm + +1 µ N ( f µi + X m +1 ν m + − g µν f νi )( x ) · q i ⊗ s µ + X m +1 ν m + m + +1 µ N g µν ( x ) · q ν ⊗ s µ (12)for a point x ∈ X near x o . Example 2.9. (i) We set
X ⊂ G (1 , P ) to be the surface which is the closureof the image of a morphism Spec K [ z , z ] → G ◦ defined by( z , z ) ( f ji ) i , j = (cid:20) f f f f f f (cid:21) = (cid:20) z z z z z ( z ) (cid:21) , (13) where ( z e ) a means that the a -th power of the parameter z e , and so on. Assumethat the characteristic is not 2.(ii) Let γ X : X G ( m + , P ) be the expanding map of X . Then m + = 3and γ X is expressed on G ◦ by t (cid:2) − z z − ( z ) z z (cid:3) . (14)This is calculated as follows: First, we take the following 4 × A : A = " ( f ji,z )( f ji,z ) with ( f ji,z ) = (cid:20) z z (cid:21) , ( f ji,z ) = (cid:20) z (cid:21) . By (7), the rank of ϕ x coincides with that of A , which is equal to 2. Sincedim( ϕ x ( S )) = m + − m , and since m = 1 in the setting, we have m + = 3.Moreover, the following equality holds in the matrix A :2 z + 2 z = z z z . It follows from (9) that g = 2 z , g = 2 z . Now, (12) yields the expression(14) of γ X . More precisely, the calculation of (11) is given by − g f f − − g f f − + f f − = − z z − ( z ) z z − . (iii) Let Y ⊂ G (3 , P ) be the surface which is the closure of the image of X under γ X . Then the shrinking map σ of Y is a map from Y to G (1 , P ) and isindeed expressed on G ◦ by (13). Hence the closure of the image of Y under σ is equal to X , and σ ◦ γ X coincides with the identity map on an open subsetof X .We note that one can calculate the expression of σ in a similar way to (ii),since σ is identified with γ ¯ Y / G (0 , ( P ) ∨ ) as in Remark 2.4. Here ¯ Y ⊂ G (0 , ( P ) ∨ ) =( P ) ∨ is the subvariety corresponding to Y ⊂ G (3 , P ), and is parametrized by(1 : − z : − z : ( z ) : 2 z z ) = ( − z : 2 z : − ( z ) : − z z ) , where the right hand side is given by transposing and reversing (14).(iv) For later explanations, we consider a 3-fold X ⊂ P which is the imageof the projection π X : U X → P , where U X ⊂ X × P is the universal family of X . As in § X is the closure of the image of amorphism Spec K [ z , z , η ] → P defined by(1 : η : z : z + η z : 2 z z + η ( z ) ) . UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 11 (v) This 3-fold X ⊂ P is called a twisted plane (see [1, 2.2.9]), and is definedby a homogeneous polynomial of degree 3,( Z ) Z + Z ( Z ) − Z Z Z , where ( Z : Z : · · · : Z ) is the homogeneous coordinates on P .(vi) Let γ X : X G (3 , P ) be the Gauss map of X . Then, in a similar wayto (ii), one can obtain the expression of γ X , which indeed coincides with (14).In particular, the closure of the image of X under γ X is equal to Y . Example 2.10.
We set
X ⊂ G (1 , P ) to be the surface which is the closureof the image of a morphism Spec K [ z , z ] → G ◦ defined by( z , z ) (cid:20) f f f f f f f f (cid:21) = (cid:20) z z a · ( z ) a − z h z ( z ) a (cid:21) , where a is an integer greater than 1, and where h ∈ K [ z ]. Assume that a ( a − γ X is a mapfrom X to G (3 , P ) and is expressed on G ◦ by t (cid:20) − a ( a − · ( z ) a − z − ( a − · ( z ) a a ( a − · ( z ) a − z a · ( z ) a − h − z h z h z (cid:21) . Composition of expanding maps and shrinking maps.
We set
X ⊂ G ( m, P N ) to be a quasi-projective smooth variety, and set Y ⊂ G ( m + , P N ) tobe the closure of the expanding map γ : X G ( m + , P N ). For this Y , wewill investigate the homomorphism Φ given in Definition 2.3, by consideringthe pull-back of Φ via γ . Now we have the following commutative diagram: γ ∗ Q ∨Y / / Ψ γ ∗ Φ , , H om ( H om ( γ ∗ Q ∨Y , γ ∗ S ∨Y ) , γ ∗ S ∨Y ) / / −◦ dγ + + ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ H om ( γ ∗ T Y , γ ∗ S ∨Y ) (cid:15) (cid:15) H om ( T X , γ ∗ S ∨Y ) , (15)where Ψ : γ ∗ Q ∨Y γ ∗ Φ −−→ H om ( γ ∗ T Y , γ ∗ S ∨Y ) → H om ( T X , γ ∗ S ∨Y ) is the compositehomomorphism, and dγ : T X → γ ∗ T G ( m + , P N ) ≃ H om ( γ ∗ Q ∨Y , γ ∗ S ∨Y )is the homomorphism of tangent bundles induced by γ .We recall that a rational map f : A B of varieties is said to be separable ifthe field extension K ( A ) /K ( f ( A )) is separably generated. Here, the followingthree conditions are equivalent: (i) f is separable; (ii) the linear map d x f : T x A → T f ( x ) f ( A ) of Zariski tangent spaces is surjective for general x ∈ A ; (iii)a general fiber of f is reduced. In characteristic zero, every rational map mustbe separable. A rational map is said to be inseparable if it is not separable. Remark 2.11. If γ is separable, then we haveker γ ∗ Φ | X ◦ = ker Ψ | X ◦ for a certain open subset X ◦ ⊂ X . This is because the vertical arrow in (15),Hom( T γ ( x ) Y , S ∨Y ⊗ γ ( x )) → H om ( T x X , S ∨Y ⊗ γ ( x )), is injective at a general point x ∈ X .Let x o ∈ X be a general point. In the setting of § x ∈ X near x o , it follows from (12) that d x γ : T x X → T γ ( x ) G ( m + , P N ) is represented by ∂∂z e X i mm + +1 µ N X m +1 ν m + − g µν,z e f νi ( x ) · q i ⊗ s µ + X m +1 ν m + m + +1 µ N g µν,z e ( x ) · q ν ⊗ s µ , (16)where we apply the following equality obtained by (9):( f µi + X m +1 ν m + − g µν f νi ) z e = X m +1 ν m + − g µν,z e f νi . Then, from the diagram (15), the linear map Ψ x : Q ∨ + → Hom( T x X , S ∨ + ) isrepresented byΨ x ( q i ) = X e dim( X ) m + +1 µ N X m +1 ν m + − g µν,z e f νi ( x ) · dz e ⊗ s µ (0 i m ) , Ψ x ( q ν ) = X e dim( X ) m + +1 µ N g µν,z e ( x ) · dz e ⊗ s µ ( m + 1 ν m + ) . (17) Lemma 2.12.
The ranks of the above linear maps are obtained as follows. (a) rk d x γ is equal to the rank of the dim( X ) × ( N − m + ) · ( m + − m ) matrix g m + +1 m +1 ,z ( x ) · · · g µν,z ( x ) · · · g Nm + ,z ( x ) ... ... ... g m + +1 m +1 ,z dim( X ) ( x ) · · · g µν,z dim( X ) ( x ) · · · g Nm + ,z dim( X ) ( x ) . (b) rk Ψ x is equal to the rank of the ( m + − m ) × ( N − m + ) · dim( X ) matrix g m + +1 m +1 ,z ( x ) · · · g µm +1 ,z e ( x ) · · · g Nm +1 ,z dim( X ) ( x ) ... ... ... g m + +1 m + ,z ( x ) · · · g µm + ,z e ( x ) · · · g Nm + ,z dim( X ) ( x ) . UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 13
Proof.
Considering the matrix description of (16), we find that, each columnvector not belonging to the matrix of (a) is expressed as P m +1 ν m + − g µν,z f νi ( x )... P m +1 ν m + − g µν,z dim( X ) f νi ( x ) . This vector is linearly dependent on column vectors of the matrix of (a); hencethe assertion of (a) follows. In the same way, considering the matrix descriptionof (17), we have the assertion of (b). (cid:3)
Proposition 2.13. Q ∨X ◦ ⊂ ker Ψ | X ◦ for a certain open subset X ◦ ⊂ X Proof.
It is sufficient to show that Q ∨X ⊗ k ( x o ) ⊂ ker Ψ x o for a general point x o ∈ X . In the setting of § f ji ( x o ) = 0, it follows from (17) thatΨ x o ( q ) = · · · = Ψ x o ( q m ) = 0 . This implies that Q ∨X ⊗ k ( x o ) ⊂ ker Ψ x o in H ( P N , O (1)). (cid:3) Let σ = σ Y / G ( m + , P N ) : Y G ( m , P N ) be the shrinking map of Y ⊂ G ( m + , P N ), where we set m := ( m + ) − . Corollary 2.14.
Assume that γ is separable and assume that Ψ x is of rank m + − m for general x ∈ X . Then m = m and Q | ∨X ◦ = ker γ ∗ Φ | X ◦ for a certainopen subset X ◦ ; hence σ ◦ γ | X ◦ is an identity map of X ◦ ⊂ X .Proof. From Proposition 2.13, we have Q | ∨X ◦ ⊂ ker Ψ | X ◦ . Since ( m + + 1) − rk Ψ x = ( m + + 1) − ( m + − m ) = m + 1, we have Q | ∨X ◦ = ker Ψ | X ◦ . It followsfrom Remark 2.11 that Q | ∨X ◦ = ker γ ∗ Φ | X ◦ . We recall that, by universality, the morphism σ ◦ γ : X ◦ → G ( m , P N ) isinduced from ker γ ∗ Φ | X ◦ ⊂ H ( P N , O (1)) ∨ ⊗ O X ◦ . Therefore σ ◦ γ coincideswith the original embedding X ֒ → G ( m, P N ). (cid:3) For the universal family U X ⊂ X × P N , we define a rational map˜ γ : U X Y × P N , by sending ( x, x ′ ) ∈ U X with x ∈ X and x ′ ∈ P N to ( γ ( x ) , x ′ ) ∈ Y × P N . Let σ ∗ U G ( m , P N ) ⊂ Y × P N be the closure of the pull-back of U G ( m , P N ) under σ . Corollary 2.15.
Assume that γ is separable. Then the image of U X under ˜ γ is contained in σ ∗ U G ( m , P N ) , and hence we have the following inclusion of linearvarieties of P N : x ⊂ σ ◦ γ ( x ) ⊂ γ ( x ) . Proof.
As in Remark 2.11, ker γ ∗ Φ | X ◦ = ker Ψ | X ◦ . Then Proposition 2.13 im-plies that U X ◦ is contained in γ ∗ σ ∗ U G ( m , P N ) = P ∗ ker( γ ∗ Φ | X ◦ ). Hence we have˜ γ ( U X ◦ ) ⊂ σ ∗ U G ( m , P N ) . (cid:3) It is known that, in characteristic two, the Gauss map of every curve isinseparable. This also happens to the expanding map, as follows.
Lemma 2.16.
Assume that
X ⊂ G ( m, P N ) is a curve, and assume that γ isgenerically finite. If the characteristic is two, then γ is inseparable.Proof. Let x o ∈ X be a general point. In the setting of § X is a curve,it is locally parametrized around x o by one parameter z . It follows from (9)and g µν ( x o ) = 0 that f µi,z,z ( x o ) = X m +1 ν m + g µν,z f νi,z ( x o ) . In characteristic two, we have f µi,z,z ( x o ) = 0. Since the above formula vanishesfor each µ and i , it follows from (7) that P m +1 ν m + g µν,z ( x o ) · ϕ x o ( ζ ν ) = 0. Since ϕ x o ( ζ ν )’s are linearly independent, g µν,z ( x o ) = 0. By Lemma 2.12, rk d x o γ = 0,that is to say, γ is inseparable. (cid:3) Duality with expanding maps and shrinking maps
Let σ be the shrinking map from a subvariety Y ⊂ G ( M, P N ) to G ( m , P N )with integers M, m ( M > m ), as in Definition 2.3 (we set m := M − ). Let X ⊂ G ( m , P N ) be the closure of the image of the map σ , and let π be theprojection from the universal family U X ⊂ X × P N of X to P N . We set σ ∗ U X ⊂ Y × P N to be the closure of the pull-back of U X under σ , and set σ ∗ π to be theprojection from σ ∗ U X to P N . Note that these constructions of σ and σ ∗ π depend only on Y . For a subvariety X ⊂ P N , we setΓ( X ) := { ( T x X, x ) ∈ G ( M, P N ) × P N | x ∈ X sm } , the incidence variety of embedded tangent spaces and their contact points,where X sm is the smooth locus of X . Theorem 3.1 (Main theorem) . Let
N, M be integers with < M < N , let X ⊂ P N be an M -dimension closed subvariety, and let Y ⊂ G ( M, P N ) bea closed subvariety. We set σ as above, and so on. Then the following areequivalent: (a) The Gauss map γ = γ X : X G ( M, P N ) is separable, and the closureof its image is equal to Y . (b) Γ( X ) = σ ∗ U X in G ( M, P N ) × P N . (c) σ ∗ π : σ ∗ U X → P N is separable and generically finite, and its image isequal to X (in particular, the image is of dimension M .) (c’) σ ∗ π : σ ∗ U X → P N is separable and its image is equal to X , and σ isseparable. UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 15
Corollary 3.2.
Assume that one of the conditions (a-c’) holds. Then m = M − dim( Y ) , and the diagram U X π / / (cid:15) (cid:15) X γ X (cid:15) (cid:15) ✤✤✤ X γ X / / ❴❴❴ Y (18) is commutative, where γ X is the expanding map of X and is indeed a birationalmap whose inverse is the shrinking map σ , and where σ ( y ) ∈ X correspondsto the closure of the fiber γ − X ( y ) ⊂ P N for a general point y ∈ Y . In this setting, we will call ( X , X ) the maximal developable parameter space(see Definition 4.8). Example 3.3.
Let X ⊂ P be the 3-fold given in (iv) of Example 2.9. Here,the above X is obtained as the surface X ⊂ G (1 , P ) in (i), which is equalto the image of Y under σ as in (iii). By (vi), we can directly verify that thediagram (18) is commutative.In § ⇒ (b) of Theorem 3.1, which leadsto the linearity of general fibers of separable Gauss maps (Corollary 3.7). In § ⇒ (a), and complete the proof of Theorem 3.1. Herethe implication (c) or (c’) ⇒ (a) gives a generalization of the characterizationof Gauss images (Corollary 3.15). We note that both implications (a) ⇒ (b)and (c’) ⇒ (a) will be discussed in the same framework, given in § Remark 3.4. (b) ⇒ (a) of Theorem 3.1 holds, as follows. Let ˜ γ : X Γ( X ) be the rational map defined by x ( γ ( x ) , x ). Since σ ∗ U X → Y isseparable and since ˜ γ is birational, the composite map γ is separable. Inaddition, (b) ⇒ (c) holds, since σ ∗ π is identified with the birational projectionΓ( X ) → X under the assumption. On the other hand, (c) ⇒ (c’) holds, since,if σ ∗ π : σ ∗ U X → P N is separable and generically finite onto its image, then σ ∗ U X U X is separable, and then so is σ .3.1. Separable Gauss maps and shrinking maps.
In this subsection, weconsider the Gauss map X G (dim( X ) , P N ) of a quasi-projective smoothsubvariety X ⊂ P N , which coincides with the expanding map of X , as inRemark 2.2. We denote by γ the map, and by Y ⊂ G (dim( X ) , P N ) the closureof the image of γ . In the setting of § m = 0, we have a naturalhomomorphism ξ : γ ∗ Q ∨Y → T X in the following commutative diagram with exact rows and columns:0 (cid:15) (cid:15) (cid:15) (cid:15) O X ( − (cid:15) (cid:15) O X ( − (cid:15) (cid:15) / / γ ∗ Q ∨Y / / ξ (cid:15) (cid:15) H ( P N , O (1)) ∨ ⊗ O X / / (cid:15) (cid:15) γ ∗ S ∨Y / / / / T X ( − / / (cid:15) (cid:15) T P N ( − | X / / (cid:15) (cid:15) N X/ P N ( − / / P N .We note that the diagram yields ker( ξ ) = O X ( − § Proposition 3.5.
Assume that m = 0 . Then we have an equality, ker Ψ | X ◦ = ker dγ ( − ◦ ξ | X ◦ for a certain open subset X ◦ ⊂ X .Proof. It is sufficient to proveker Ψ ⊗ k ( x o ) = ker( dγ ( − ◦ ξ ) ⊗ k ( x o )for a general point x o ∈ X . In the setting of § P N ,we can assume x o = (1 : 0 : · · · : 0) and γ ( x o ) = ( Z dim( X )+1 = · · · = Z N = 0).Then, by taking z , . . . , z dim( X ) as f , . . . , f dim( X )0 , the original embedding X ֒ → P N can be locally parametrized by (1 : z : · · · : z dim( X ) : f dim( X )+10 : · · · : f N ).We find ϕ x ( ζ ) = dz ⊗ η , ϕ x ( ζ ) = dz ⊗ η , · · · , ϕ x ( ζ dim( X ) ) = dz dim( X ) ⊗ η in (7). Hence we have g µν = f µ ,z ν in (9). As in (16), the linear map d x o γ : T x o X → T γ ( x o ) G (dim( X ) , P N ) is represented by ∂∂z e X ν dim( X )dim( X )+1 µ N f µ ,z ν ,z e ( x o ) · s µ ⊗ q ν (1 e dim( X )) . It follows from (17) that Ψ x o : Q ∨ + → Hom( T x X, S ∨ + ) is represented byΨ x o ( q ) = 0 , Ψ x o ( q ν ) = X e dim( X )dim( X )+1 µ N f µ ,z ν ,z e ( x o ) · dz e ⊗ s µ (1 ν dim( X )) . UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 17
Since ξ x o : Q ∨ + → T x o X is obtained by ξ x o ( q ) = 0 and ξ x o ( q ν ) = ∂/∂z ν with1 ν dim( X ), the linear maps d x o γ ◦ ξ x o and Ψ x o can be identified; inparticular, their kernels coincide. (cid:3) Theorem 3.6.
The implication (a) ⇒ (b) of Theorem 3.1 holds.Proof. Let X ⊂ P N be a projective variety, let Y ⊂ G (dim( X ) , P N ) be theclosure of the image of X under γ , and let Y G ( m , P N ) be the shrinkingmap with m := dim( X ) − . We apply the previous argument to the quasi-projective variety X sm .Assume that γ is separable. Then ker γ ∗ Φ | X ◦ = ker Ψ | X ◦ as in Remark 2.11.Therefore Proposition 3.5 impliesker γ ∗ Φ | X ◦ = ker( dγ ( − ◦ ξ ) | X ◦ , where the right hand side is of rank dim( X ) − dim( Y ) + 1 because of theseparability of γ . This implies m = dim( X ) − dim( Y ).On the other hand, it follows from Corollary 2.15 that Γ( X ) ⊂ σ ∗ U X . Infact, Γ( X ) and σ ∗ U X coincide, since both have the same dimension, dim( X ). (cid:3) Now, we give the proof of Theorem 1.1; more precisely, we have:
Corollary 3.7.
Assume that the Gauss map γ : X Y is separable, and let y ∈ Y be a general point. Then the m -plane σ ( y ) ⊂ P N is contained in X .Moreover, the fiber γ − ( y ) ⊂ X sm coincides with σ ( y ) ∩ X sm , whose closure isequal to σ ( y ) . This implies that the Gauss map is separable if and only if its general fiberis scheme-theoretically (an open subscheme of) a linear subvariety of P N . Thelatter condition immediately implies the former one, since a fiber is reduced ifit is scheme-theoretically a linear subvariety. Proof of Corollary 3.7.
Let y ∈ Y be a general point, and denote by F y thefiber of σ ∗ U X → Y at y . We recall that the image σ ∗ π ( F y ) is equal to the m -plane σ ( y ) ⊂ P N . If γ is separable, then Theorem 3.6 implies Γ( X ) = σ ∗ U X ,and then the following commutative diagram holds:Γ( X ) (cid:15) (cid:15) σ ∗ U X (cid:15) (cid:15) σ ∗ π z z ✉✉✉✉✉✉✉✉✉✉ X γ / / ❴❴❴❴❴ Y .In particular, the m -plane σ ( y ) = σ ∗ π ( F y ) is contained in X . On the otherhand, it follows that ( σ ∗ π ) − ( γ − ( y )) = F y ∩ ( σ ∗ π ) − ( X sm ). Since σ ∗ π issurjective, we have γ − ( y ) = σ ( y ) ∩ X sm , where the right hand side is an opendense subset of σ ( y ) ⊂ X . (cid:3) Combining [27, I, 2.8. Corollary] and Corollary 3.7, we have:
Corollary 3.8.
If the projective variety X ⊂ P N is smooth (and is non-linear),then the separable Gauss map γ is in fact birational onto its image. Remark 3.9.
In positive characteristic, the Gauss map γ can be inseparable (Wallace [26, § F of γ can be non-linear . Several authors gave examples where F of aninseparable γ is not a linear subvariety (Kaji [12, Example 4.1] [13], Rathmann[24, Example 2.13], Noma [21], Fukasawa [2] [3]). Remark 3.10.
If dim X = 1 and the Gauss map γ is separable, then γ isbirational. This fact was classically known for plane curves in terms of dualcurves (for example, see [18, p. 310], [10, § Generalized conormal morphisms.
We denote by γ the expandingmap from a closed subvariety X ⊂ G ( m, P N ) to G ( m + , P N ), and by Y ⊂ G ( m + , P N ) the closure of the image of X under γ . We consider the generalizedconormal morphism γ ∗ ¯ π : γ ∗ V Y → ( P N ) ∨ given in Definition 2.7. Let Y be theimage of γ ∗ V Y under γ ∗ ¯ π , which is a subvariety of G ( N − , P N ) = ( P N ) ∨ .We denote by σ Y = σ Y/ G ( N − , P N ) the shrinking map from Y to G ( N − − dim Y, P N ). As in Remarks 2.2 and 2.4, the map σ Y is identified with the Gaussmap Y G (dim Y, ( P N ) ∨ ) which sends y ∈ Y to T y Y ⊂ ( P N ) ∨ . Denoting by A ∗ ⊂ ( P N ) ∨ the set of hyperplanes containing a linear subvariety A ⊂ P N , wehave σ Y ( y ) ∗ = T y Y . Theorem 3.11.
Let X be as above. Assume that γ ∗ ¯ π is separable and itsimage Y is of dimension N − m − , and assume that γ is separable. Then theshrinking map σ Y of Y is separable and the closure of its image is equal to X . Remark 3.12.
By considering the dual of the above statement, it follows thatTheorem 3.11 is equivalent to “(c’) ⇒ (a)” of Theorem 3.1. This is because, inthe setting of Theorem 3.1, we have σ Y / G ( M, P N ) = γ ¯ Y / G ( N − M − , ( P N ) ∨ ) , γ X/ P N = σ ¯ X/ G ( N − , ( P N ) ∨ ) , and so on (see Remark 2.4).The following result is essential for the proof of Theorem 3.11, and is indeeda generalization of the Monge-Segre-Wallace criterion. Here we recall thatΨ x : Q ∨ + → Hom( T x X , S ∨ + ) is the linear map given in (15), (17) in § Proposition 3.13.
Let
X ⊂ G ( m, P N ) , let v ∈ γ ∗ V Y be a general point, andlet x ∈ X be the image of v under γ ∗ V Y → X . Then the following holds: (a) rk d v γ ∗ ¯ π − ( N − m + − rk d x γ and rk d v γ ∗ ¯ π − ( N − m + − rk Ψ x . (b) If rk d v γ ∗ ¯ π = N − m + − , then rk d x γ = 0 . (c) If γ ∗ ¯ π is separable, then T γ ∗ ¯ π ( v ) Y ⊂ x ∗ in ( P N ) ∨ . To prove Proposition 3.13, we will describe the linear map d v o γ ∗ ¯ π for a gen-eral point v o ∈ γ ∗ V Y , as follows. Under the setting of § γ ∗ ¯ π is expressed as (11), where we have V G ◦ m + ≃ G ◦ m + × P N − m + − as in § UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 19 v o = ( x o , s o ) ∈ γ ∗ V Y with x o ∈ X and s o ∈ P N − m + − . Changing homogeneouscoordinates on P N , we can assume that x o = ( Z m +1 = · · · = Z N = 0) ⊂ v ′ o := γ ∗ ¯ π ( v o ) = ( Z N = 0) in P N . We regard ( s m + +1 : · · · : s N ) and ( Z : · · · : Z N ) as homogeneous coordinateson P N − m + − = P ∗ ( S + ) and ( P N ) ∨ . Then, since v ′ o = ( Z N = 0), we have s o = ( s m + +1 = · · · = s N − = 0) ∈ P N − m + − . For affine coordinates ¯ s µ := s µ /s N on { s N = 0 } ⊂ P N − m + − , we regard z , . . . , z dim( X ) , ¯ s m + +1 , . . . , ¯ s N − as a system of regular parameters of O γ ∗ V Y ,v o .In addition, we set ¯ s N := 1, and take affine coordinates ¯ Z α := Z α /Z N on { Z N = 0 } ⊂ ( P N ) ∨ .Now, for a general point v = ( x, s ) ∈ γ ∗ V Y near v o , which is expressed as(( f ji ) i,j , (¯ s m + +1 , · · · , ¯ s N − )), it follows from (11) that the linear map d v γ ∗ ¯ π : T v γ ∗ V Y → T v ′ ( P N ) ∨ is represented by ∂∂z e X i m X m +1 ν m + m + +1 µ N − ¯ s µ g µν,z e f νi · ∂∂ ¯ Z i + X m +1 ν m + X m + +1 µ N ¯ s µ g µν,z e · ∂∂ ¯ Z ν ,∂∂ ¯ s ¯ µ X i m ( f ¯ µi + X m +1 ν m + − g ¯ µν f νi ) · ∂∂ ¯ Z i + X m +1 ν m + g ¯ µν · ∂∂ ¯ Z ν − ∂∂ ¯ Z ¯ µ , (19)for 1 e dim( X ) and m + + 1 ¯ µ N −
1. Here the N − m + − d v γ ∗ ¯ π ( ∂/∂ ¯ s m + +1 ) , . . . , d v γ ∗ ¯ π ( ∂/∂ ¯ s N − ) are linearly independent, since each ofthem has ∂/∂ ¯ Z ¯ µ as its tail term. Moreover, setting a dim( X ) × ( m + − m )matrix G d v γ ∗ ¯ π := P µ ¯ s µ g µm +1 ,z ( x ) · · · P µ ¯ s µ g µm + ,z ( x )... ... P µ ¯ s µ g µm +1 ,z dim( X ) ( x ) · · · P µ ¯ s µ g µm + ,z dim( X ) ( x ) , we have rk d v γ ∗ ¯ π = N − m + − G d v γ ∗ ¯ π . Proof of Proposition 3.13. (a) From Lemma 2.12(a), it follows that rk G d vo γ ∗ ¯ π rk d x o γ . In addition, from Lemma 2.12(b), considering the transpose of the ma-trix, we have rk G d vo γ ∗ ¯ π rk Ψ x o .(b) Assume that rk d x o γ >
0. Then it follows from Lemma 2.12(a) that g µν,z e ( x o ) = 0 for some µ, ν, e . Hence G d v γ ∗ ¯ π = 0 for some v with v x o under γ ∗ V Y → X . This implies that rk d v γ ∗ ¯ π > N − m + − f ji ( x o ) = 0, the description (19) implies that im( d v o γ ∗ ¯ π ) is con-tained in the vector subspace of T v ′ o ( P N ) ∨ spanned by ∂/∂ ¯ Z m +1 , · · · , ∂/∂ ¯ Z N − .If γ ∗ ¯ π is separable, then T v ′ o Y ⊂ ( Z = · · · = Z m = 0) in ( P N ) ∨ , where theright hand side is equal to x ∗ o . (cid:3) Recall that σ : Y G ( m , P N ) is the shrinking map with m := ( m + ) − . Corollary 3.14.
Assume that γ ∗ ¯ π is separable and its image is of dimension N − m − , and assume that γ is separable. Then m = m and σ ◦ γ | X ◦ is anidentity map of a certain open subset X ◦ ⊂ X .Proof. Since rk d v γ ∗ ¯ π = N − m −
1, it follows from Proposition 3.13(a) that m + − m rk Ψ x (the equality indeed holds, due to Lemma 2.12(b)). ThenCorollary 2.14 implies the result. (cid:3) Proof of Theorem 3.11.
Corollary 3.14 implies that m = m and that σ ◦ γ | X ◦ isan identity map of X ◦ . On the other hand, since Y is of dimension N − m −
1, inthe statement of Proposition 3.13(c), we have T γ ∗ ¯ π ( v ) Y = x ∗ in ( P N ) ∨ . Since σ Y is identified with the Gauss map Y G (dim Y, ( P N ) ∨ ), it follows that σ Y ( γ ∗ ¯ π ( v )) = x in P N . Now the diagram γ ∗ V Y γ ∗ ¯ π ( ( / / ❴❴❴ (cid:15) (cid:15) V Y ¯ π / / (cid:15) (cid:15) Y σ Y (cid:15) (cid:15) ✤✤✤ X γ / / ❴❴❴❴ id Y σ / / ❴❴❴ X is commutative. In particular σ Y is separable, since γ ∗ V Y → X is. (cid:3) Proof of Theorem 3.1. (b) ⇒ (c) ⇒ (c’) follows from Remark 3.4. (a) ⇒ (b) follows from Theorem 3.6. (c’) ⇒ (a) follows from Theorem 3.11 andRemark 3.12. (cid:3) Proof of Corollary 3.2. γ X ◦ σ is identity map due to the dual statement ofCorollary 3.14. As in Corollary 3.7, for general y , σ ( y ) corresponds to theclosure of the fiber γ − X ( y ). Thus the diagram (18) is commutative. (cid:3) From the equivalence (c) ⇔ (a), we have: Corollary 3.15.
Let σ be the shrinking map from a closed subvariety Y ⊂ G ( M, P N ) to G ( M − , P N ) . Then Y is the closure of a image of a separableGauss map if and only if M − = M − dim Y holds and σ ∗ U G ( M − , P N ) → P N isseparable and generically finite onto its image. Remark 3.16.
In the case where m = 0, Proposition 3.13(c) gives the state-ment of the Monge-Segre-Wallace criterion [8, (2.4)], [19, I-1(4)]. Remark 3.17.
For
X ⊂ G ( m, P N ), in the diagram of Definition 2.7, γ ∗ ¯ π canbe inseparable even if γ is separable. The reason is as follows: If m = 0, then γ ∗ ¯ π coincides with the conormal map C ( X ) → ( P N ) ∨ in the original sense.Then, as we mentioned in §
1, Kaji [15] and Fukasawa [4] [5] gave examplesof non-reflexive varieties (i.e., γ ∗ ¯ π ’s are inseparable) whose Gauss maps arebirational. This implies the assertion.Considering the dual of the above statement, in the setting for Theorem 3.1,we find that σ ∗ π can be inseparable even if σ is separable; in other words, UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 21 separability of σ is not sufficient to give an equivalent condition for separabilityof the Gauss map of X .4. Developable parameter spaces
In this section, we set π = π X to be the projection π : U X := U G ( m, P N ) | X → P N for a closed subvariety X ⊂ G ( m, P N ), and set X := π ( U X ) in P N . Definition 4.1.
We say that ( X , X ) is developable if X = π ( U X ) and, forgeneral x ∈ X , the embedded tangent space T x ′ X is the same for any smoothpoints x ′ ∈ X lying in the m -plane x ⊂ P N , i.e., the Gauss map γ X of X is constant on x ∩ X sm (cf. [1, 2.2.4]). We also say that X is developable if( X , π ( U X )) is developable. The variety X is said to be uniruled (resp. ruled) by m -planes if π is generically finite (resp. generically bijective).Note that, in the case where γ X is separable, there exists a developableparameter space ( X , X ) of m -planes with m > γ X is less than dim X ; this follows from existence of the maximal developable parameter space (see Definition 4.8). Example 4.2.
We take
X ⊂ G (1 , P ) and X ⊂ P to be the surface and 3-foldgiven in Example 2.9 (see also Example 3.3). Then ( X , X ) is developable dueto (vi); indeed, it is maximal.4.1. Expanding maps and developable parameter spaces.Proposition 4.3.
Let γ = γ X : X G ( m + , P N ) be the expanding map of X ⊂ G ( m, P N ) . We recall that d u π : T u U X → T u ′ P N is the linear map ofZariski tangent spaces at u ∈ U X , u ′ = γ ( u ) ∈ P N . Then the following holds: (a) rk d u π m + dim( X ) and rk d u π m + for general u ∈ U X . (b) If rk d u π = m for general u ∈ U X , then X is a point. (c) Assume that π is separable, and let x ∈ X be a general point. Then the m + -plane γ ( x ) ⊂ P N is spanned by dim( X ) -planes γ X ( u ′ ) with smoothpoints u ′ ∈ X lying in the m -plane x . To show Proposition 4.3, we will first describe the linear map d u π , as follows.As in § U G ◦ m ≃ G ◦ m × P m . Let u o = ( x o , η o ) ∈ U X be a generalpoint with x o ∈ X ∩ G ◦ m and η o ∈ P m . Changing homogeneous coordinates on P N , we can assume that x o = ( Z m +1 = · · · = Z N = 0) ⊂ P N and u ′ o := π ( u o ) =(1 : 0 : · · · : 0) ∈ P N . Then we have η o = (¯ η = · · · = ¯ η m = 0) ∈ P m . We can also assume γ ( x o ) = ( Z m + +1 = · · · = Z N = 0). From the expres-sion (2), the projection π : U X → P N sends an element u = ( x, η ) ∈ U X near u o , which is described as (( f ji ) i,j , ( η : · · · : η m )), to the point X i m η i · Z i + X i m,m +1 j N η i f ji · Z j ∈ P N . Let us take affine coordinates ¯ η i := η i /η on { η = 0 } ⊂ P m , and ¯ Z α := Z α /Z on { Z = 0 } ⊂ P N . Then we regard z , . . . , z dim( X ) , ¯ η , . . . , ¯ η m as a system ofregular parameters of O U X ,u o . We set ¯ η := 1.Now, for general u = ( x, η ) ∈ U X near u o , the linear map d u π : T u U X → T u ′ P N is represented by ∂∂ ¯ η ¯ ı ∂∂ ¯ Z ¯ ı + X m +1 j N f j ¯ ı · ∂∂ ¯ Z j (1 ¯ ı m ) ,∂∂z e X i m,m +1 j N ¯ η i f ji,z e · ∂∂ ¯ Z j (1 e dim( X )) . (20)Here the m elements d u π ( ∂/∂ ¯ η ) , . . . , d u π ( ∂/∂ ¯ η m ) are linearly independent.For a point u ∈ U X near u o such that u x under U X → X , setting adim( X ) × ( N − m ) matrix F d u π := P i ¯ η i f m +1 i,z ( x ) . . . P i ¯ η i f Ni,z ( x )... ... P i ¯ η i f m +1 i,z dim( X ) ( x ) . . . P i ¯ η i f Ni,z dim( X ) ( x ) , we have rk d u π = rk F d u π + m. Proof of Proposition 4.3. (a) From (6) in § F d uo π dim( X ).From (7), we have rk F d uo π rk ϕ x o = m + − m . Thus the assertion follows.(b) If dim X >
0, then f ji,z e ( x o ) = 0 for some i, j, e . It follows that F d u π = 0for some u with u x o . This implies that rk d u π > m .(c) Changing coordinates, we have that γ ( x o ) = ( Z m + +1 = · · · = Z N = 0) in P N . Then the equality (10) implies that, for each u ∈ U X near u o with u x o ,we have F d u π ( x o ) = P i ¯ η i f m +1 i,z ( x o ) . . . P i ¯ η i f m + i,z ( x o )... ... P i ¯ η i f m +1 i,z dim( X ) ( x o ) . . . P i ¯ η i f m + i,z dim( X ) ( x o ) . In addition, we recall that f ji ( x o ) = 0.Now, we find an inclusion of linear varieties γ X ( π ( u )) ⊂ γ ( x o ) in P N , asfollows: Considering the description (20), we have that im( d u π ) is containedin the vector subspace of T π ( u ) P N spanned by ∂/∂ ¯ Z , · · · , ∂/∂ ¯ Z m + . Since π isseparable, γ X ( π ( u )) is contained in γ ( x o ) = ( Z m + +1 = · · · = Z N = 0).Suppose that there exists an ( m + − L ⊂ P N contained in the m + -plane γ ( x o ), such that γ X ( u ′ ) ⊂ L holds for each smooth point u ′ ∈ X lyingin the m -plane x o . Then we find a contradiction, as follows: Changing coordi-nates, we can assume L = ( Z m + = 0) ∩ γ ( x o ) . Since π is separable and since γ X ( π ( u )) ⊂ L for each u ∈ U X with u x o ,considering the above matrix F d u π and the description (20) of d u π , we have UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 23 f m + i,z e ( x o ) = 0 for each i, e . Then ϕ x ( ξ m + ) = 0 due to (7). This contradicts thata basis of the vector space ϕ x ( S ) consists of ϕ x ( ξ m +1 ) , · · · , ϕ x ( ξ m + ). (cid:3) Corollary 4.4.
In the setting of Proposition 3.13, if the maps γ and ¯ π areseparable, then we have T γ ∗ ¯ π ( v ) Y ⊂ ( σ ◦ γ ( x )) ∗ ⊂ x ∗ in ( P N ) ∨ .Proof. From Corollary 2.15, we have x ⊂ σ ◦ γ ( x ) in P N . By applying thedual statement of Proposition 4.3(c) to ¯ π and γ ( x ), the inclusion T γ ∗ ¯ π ( v ) Y ⊂ ( σ ◦ γ ( x )) ∗ holds. (cid:3) We have the following criterion for developability (cf. [1, 2.2.4]), where recallthat m + is an integer given with the expanding map γ X : X G ( m + , P N ). Corollary 4.5.
Assume that π is separable. Then dim( X ) = m + if and only if ( X , X ) is developable. In this case, the following commutative diagram holds: U X π / / (cid:15) (cid:15) X γ X (cid:15) (cid:15) ✤✤✤ X γ X / / ❴❴❴ G ( m + , P N ) . (21) Proof.
In Proposition 4.3(c), γ X ( x ) = γ X ( u ′ ) holds if and only if the linearsubvariety γ X ( u ′ ) ⊂ P N is of dimension m + . Thus the assertion follows. (cid:3) In the case where π is generically finite, dim( X ) = dim( X ) + m ; hence wealso have: Corollary 4.6.
Assume that π is separable and generically finite. Then wehave dim( X ) = m + − m if and only if X is developable. Example 4.7.
In the setting of Example 4.2, we can also verify that ( X , X ) isdevelopable by using Corollary 4.6 (without calculation in (vi) of Example 2.9);this is because, we have m + − m = 2 in (ii), which implies that the equality“dim( X ) = m + − m ” holds. In a similar way, one can show that the space X ⊂ G (1 , P ) in Example 2.10 is developable. Definition 4.8.
Let X ⊂ P N be a projective variety whose Gauss map γ X isseparable. Then we set X ⊂ G ( m , P N )to be the closure of the space which parametrizes (closures of) general fibersof γ X , and call ( X , X ) the maximal developable parameter space.(a) From Corollary 3.2, X can be obtained as the closure of the image of X under the composite map σ Y ◦ γ X . In particular, the projection π : U X → X is birational and the expanding map γ X is birational.(b) For any developable ( X , X ) with X ⊂ G ( m, P N ), there exists a dominantrational map X X through which γ X : X Y factors. (This is because,for each x ∈ X , we have an inclusion x ∩ X sm ⊂ γ − X ( γ X ( x )) in P N . Indeed,since γ X ◦ σ Y = id , the map X X is given by σ Y ◦ γ X .) Remark 4.9.
Let
X ⊂ G ( m, P N ) be a subvariety such that ( X , X ) is devel-opable.(a) Assume that π is separable and generically finite, and assume that γ X is generically finite. Then π is indeed birational (i.e, X is separably ruledby m -planes). The reason is as follows: From the diagram (21), for general x ∈ X , since m is equal to the dimension of the fiber of γ − X ( γ X ( x )), the m -plane x ⊂ P N is set-theoretically equal to an irreducible component of the closure ofthe fiber γ − X ( γ X ( x )). This implies that π is generically injective, and hence isbirational.(b) Assume that π and γ X are separable and generically finite. Then X is equal to the parameter space X given in Definition 4.8. The reason is asfollows: If γ X is separable, then so is γ X . It follows from Corollary 3.7 that theclosure of the fiber γ − X ( γ X ( x )) is irreducible, and hence is equal to the m -plane x . Thus X = X .For example, in the following situation, the maximal developable parameterspace for the dual variety of X ⊂ P N can be obtained: Proposition 4.10.
Let γ X : X Y ⊂ G (dim( X ) , P N ) be the Gauss map,and let Y := ¯ π ( V Y ) in ( P N ) ∨ , the dual variety of X . If X is reflexive and ¯ π is generically finite, then ( Y , Y ) is the maximal developable parameter spacewith the birational projection ¯ π : V Y → Y , and then the following diagram iscommutative: γ ∗ V Y / / ❴❴❴ (cid:15) (cid:15) V Y ¯ π / / (cid:15) (cid:15) Y σ Y (cid:15) (cid:15) ✤✤✤ X γ X / / ❴❴❴❴ Y σ Y / / ❴❴❴ X , where note that the shrinking map σ Y : Y X is identified with the Gaussmap γ Y/ ( P N ) ∨ : Y G ( N − m − , ( P N ) ∨ ) .Proof. Since X is reflexive, γ ∗ ¯ π is separable due to the Monge-Segre-Wallacecriterion, and so is ¯ π . Let M := dim X . Since dim Y = M − m = ( N − m − − ( N − M − Y , Y ) is developable andthat the diagram is commutative. Since γ X is separable, σ Y is birational (seeCorollary 3.2). Hence, considering the dual statement of Remark 4.9, we havethe assertion. (cid:3) Remark 4.11.
Suppose that Y is of dimension one. Then ¯ π is always separableand generically finite (see Lemma 4.12 below). In this case, X is reflexive ifand only if γ X is separable.4.2. One-dimensional developable parameter space.
In this subsection,we assume that
X ⊂ G ( m, P N ) is a projective curve. As above, we denote by π = π X : U X → P N the projection, and by X := π ( U X ) in P N . Here separabilityof π always holds; this is deduced from [14], and can be also shown, as follows: Lemma 4.12.
Let X be as above. Then π is separable and generically finite. UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 25
Proof.
Note that U X is of dimension m + 1. Since X is a curve, it follows fromProposition 4.3(b) that rk d u π > m + 1. Thus rk d u π = m + 1, which impliesthat π is separable and generically finite. (cid:3) Let us consider the expanding map γ : X G ( m + γ , P N ) and shrinking map σ : X G ( m − σ , P N ) of X , with integers m + γ and m − σ ( m − σ < m < m + γ ). Lemma 4.13. m + γ + m − σ = 2 m .Proof. In the setting of § F = f m +10 ,z . . . f N ,z ... ... f m +1 m,z . . . f Nm,z , where note that, since X is of dimension one, the system of parameters of O X ,x consists of one element z . Recalling the formula (7), we have rk F =dim( ϕ x ( S )) = m + γ − m . In the same way, we have rk F = m − m − σ . Thus theassertion follows. (cid:3) Corollary 4.14.
The following are equivalent: (a) X is developable. (b) m + γ = m + 1 . (c) m − σ = m − .Proof. The equivalence (a) ⇔ (b) follows from Corollary 4.6. The equivalent(b) ⇔ (c) follows from Lemma 4.13. (cid:3) Recall that γ ∗ ¯ π is the generalized conormal morphism given in Definition 2.7. Lemma 4.15.
Assume that γ is generically finite. Then, γ is separable if andonly if so is γ ∗ ¯ π : γ ∗ V G ( m + γ , P N ) → ( P N ) ∨ .Proof. For
Y ⊂ G ( m + γ , P N ), the closure of the image of γ , we set Y ⊂ ( P N ) ∨ to be the image of V Y under ¯ π . Since Y is of dimension one, ¯ π is separable andgenerically finite, due to Lemma 4.12. Hence the assertion follows. (cid:3) Considering the dual of the above statement, we also have:
Corollary 4.16.
Assume that σ is generically finite. Then σ is separable ifand only if so is σ ∗ π : σ ∗ U G ( m − σ , P N ) → P N . Remark 4.17. (a) If ( X , X ) is developable and γ is generically finite, then π is birational, due to Lemma 4.12 and Remark 4.9. Moreover, if γ is separable,then we have X = X .(b) If X ⊂ P N is non-degenerate and is not a cone, then it follows fromLemma 2.6 that γ and σ are generically finite.Recall that γ i and σ i are composite maps given in § T X = T X := S x ∈ X sm T x X ⊂ P N , the tangent variety , and by T X := X , T i X := T ( T i − X ). Theorem 4.18.
Let
X ⊂ G ( m, P N ) and X ′ ⊂ G ( m ′ , P N ) be projective curveswith projections π X : U X → X and π X ′ : U X ′ → X ′ . Then, for an integer ε > , the following are equivalent: (a) ( X ′ , X ′ ) is developable, γ ε = γ ε X ′ is separable, γ ε X ′ = X , and X ′ isnon-degenerate and is not a cone. (b) ( X , X ) is developable, σ ε = σ ε X is separable, σ ε X = X ′ , and X is non-degenerate and is not a cone.In this case, m = m ′ + ε and X = T ε X ′ .Proof. (b) ⇒ (a): It is sufficient to show the case ε = 1. Since X is developable,it follows from Corollary 4.14 that m ′ = m − σ is equal to m −
1. From Lemma 2.6, σ X is generically finite. From Corollary 4.16, σ ∗X π is separable. ApplyingCorollary 3.2, we have that σ X ◦ γ X ′ gives an identity map of an open subsetof X ′ , and that X ′ is developable. In addition, X is equal to the image of theGauss map γ X ′ ; hence the image of U X → P N is equal to T X ′ , which meansthat X = T X ′ .The converse (a) ⇒ (b) follows in the same way. (cid:3) In the statement of (a) of Theorem 4.18, if m ′ = 0 and C := X ′ ⊂ P N , thenwe regard C itself as a developable parameter space (of 0-planes).We denote by Tan ( i ) C the osculating scroll (= osculating developable ) oforder i of a curve C ⊂ P N (see [1, p. 76], [9, Definition 1.4], [22, § (1) C = T C holds. It is known that Tan ( i ) C coincideswith T i C if the characteristic is zero or satisfies some conditions (Homma [9, § Corollary 4.19.
Assume one of the conditions (a) and (b) of Theorem 4.18,and assume that m ′ = 0 , i.e., C := X ′ is a curve in P N . Then the followingholds: (c) C = σ m X and X = γ m C ; in particular, X = T m C . (d) T i C = Tan ( i ) C for < i m + 1 . (e) If γ X is separable (equivalently, so is γ X ) and m + 1 < N , then X is theclosure of the space parametrizing general fibers of γ X . In the case where X ⊂ P N is a cone with maximal vertex L , consideringthe linear projection from L and using Corollary 4.19, we have that X is acone over an osculating scroll of order m − dim( L ) − P N − dim( L ) − if ( X , X ) is developable and σ m − dim( L ) − is separable. Proof of Corollary 4.19. (c) The statement follows from Theorem 4.18; in par-ticular, γ m is separable, X = γ m C , and X = T m C .(d) For 0 i < m , it follows from the diagram (21) of Corollary 4.5 that γ T i C : T i C γ i +1 C is separable; then T i C is reflexive as in Remark 4.11. In-ductively, T i +2 C = Tan ( i +2) C follows from [9, Corollary 2.3 and Theorem 3.3].(e) If γ X is separable, then X = X as in Remark 4.17. (cid:3) UALITY WITH EXPANDING MAPS AND SHRINKING MAPS 27
Let us consider X ∗ ⊂ ( P N ) ∨ , the dual variety of X ⊂ P N . Then we have thefollowing relation with dual varieties and tangent varieties. Corollary 4.20.
Let m, ε be integers with ε > and m + ε + 1 < N , let X ⊂ P N be a non-degenerate projective variety of dimension m + 1 , and let X ⊂ G ( m, P N ) be a projective curve such that ( X , X ) is developable. If γ ε +1 = γ ε +1 X is separable, then we have T ε (( T ε X ) ∗ ) = X ∗ in ( P N ) ∨ .Proof. We can assume that X is not a cone. By definition, X is the image of U X → P N . From Theorem 4.18, T ε X is given by the image of U γ ε X → P N . Let Y := γ ε +1 X , and let Y be the image of V Y → ( P N ) ∨ . Since σ ε = σ ε Y is separable,considering the dual of the above statement, we have that T ε Y is given by theimage of V σ ε Y → ( P N ) ∨ . On the other hand, for each 0 i ε , from thediagram (21), since γ i +1 X is equal to the image of the Gauss map γ T i X , thedual variety ( T i X ) ∗ is given by the image of V γ i +1 X → ( P N ) ∨ (see Definition 2.7,Proposition 4.10). In particular, Y = ( T ε X ) ∗ . From Theorem 4.18, it followsthat σ ε Y = γ X . Hence T ε ( Y ) and X ∗ coincide, since these are given by theimage of V γ X → ( P N ) ∨ . (cid:3) References [1] G. Fischer and J. Piontkowski,
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