Varieties n–covered by curved of degree δ

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Varieties n–covered by curved of degree δ

Francesco Russo

Universit`a degli Studi di Catania

Francesco Russo Varieties n–covered by curved of degree δ

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Definitions and notation

Joint work with Luc Pirio.

X denotes an irreducible projective (or proper) complex variety.

dim(X ) = r + 1 ⇐⇒ X = X^{r +1}.

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Definitions and notation

dim(X ) = r + 1 ⇐⇒ X = X^{r +1}.

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Definitions and notation

dim(X ) = r + 1 ⇐⇒ X = X^{r +1}.

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Definitions and notation

Remarks

p₁, . . . , p_n∈ X

n ≥ 2 general points =⇒

∃ C = C_p₁_,...,p_n irreducible curve : p1, . . . , pn∈ C

if we put restrictions on C natural obstructions appear. n = 2&

C rational curve ⇐⇒ X rationally connected variety ( dubbed RC).

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Definitions and notation

Remarks

p₁, . . . , p_n∈ X

∃ C = C_p₁_,...,p_n irreducible curve :

p1, . . . , pn∈ C

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Definitions and notation

Remarks

p₁, . . . , p_n∈ X

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Definitions and notation

Remarks

p₁, . . . , p_n∈ X

if we put restrictions on C natural obstructions appear.

n = 2&

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Definitions and notation

Remarks

p₁, . . . , p_n∈ X

if we put restrictions on C natural obstructions appear.

n = 2&

C rational curve ⇐⇒ X rationally connected variety ( dubbedRC).

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Definitions and notation

Theorem (Koll´ar–Miyaoka–Mori)

1 X RC⇐⇒

∃C = C_p₁_,...,p_n ⊆ X rational curve passing through n ≥ 2

general points p₁, . . . , p_n∈ X

2

X^{r +1} RC SMOOTH r ≥ 2

=⇒

∃C ⊆ X SMOOTH rational curve passing through n ≥ 2

3

C any SMOOTH curve

=⇒ ∃f : C → X EMBEDDING : p₁, . . . , p_n∈ f (C )

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Definitions and notation

1 X RC⇐⇒

2

=⇒

3

C any SMOOTH curve

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Definitions and notation

1 X RC⇐⇒

2

=⇒

3

C any SMOOTH curve

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Definitions and notation

Definition

Fixed n ≥ 2, δ ≥ n − 1 and an embedding X ⊂ P^N

X ⊂ P^N

n–covered by curves of degree δ

⇐⇒

∃ C = C_p₁_,...,p_n ⊆ X

through n ≥ 2 general points with deg(C ) = δ.

In this case we shall use the notation : X = X (n, δ) ⊂ P^N.

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Definitions and notation

Definition

Fixed n ≥ 2, δ ≥ n − 1 and an embedding X ⊂ P^N X ⊂ P^N

⇐⇒

∃ C = C_p₁_,...,p_n ⊆ X

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Definitions and notation

Definition

⇐⇒

∃ C = C_p₁_,...,p_n ⊆ X

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Definitions and notation

Definition

⇐⇒

∃ C = C_p₁_,...,p_n ⊆ X

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Examples/reinterpretation of known results

We shall also assume X ⊆ P^N non-degenerate.

Examples

X = X^{r +1}(2, 1) ⊂ P^N ⇐⇒ N = r + 1 and X = P^{r +1}; X = X^{r +1}(3, 2) ⊂ P^N ⇐⇒ (a) N = r + 2

(b) X^{r +1} ⊂ P^{r +2} quadric ;

X = X^{r +1}(n, n − 1) ⊂ P^N ⇐⇒

N = r + n − 1 X^{r +1}⊂ P^{r +n−1} deg(X ) = n − 1 (minimal degree)

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Examples/reinterpretation of known results

Examples

X = X^{r +1}(2, 1) ⊂ P^N ⇐⇒ N = r + 1 and X = P^{r +1};

X = X^{r +1}(3, 2) ⊂ P^N ⇐⇒ (a) N = r + 2

(b) X^{r +1} ⊂ P^{r +2} quadric ;

X = X^{r +1}(n, n − 1) ⊂ P^N ⇐⇒

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Examples/reinterpretation of known results

Examples

(b) X^{r +1} ⊂ P^{r +2} quadric ;

X = X^{r +1}(n, n − 1) ⊂ P^N ⇐⇒

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Examples/reinterpretation of known results

Examples

(b) X^{r +1} ⊂ P^{r +2} quadric ;

X = X^{r +1}(n, n − 1) ⊂ P^N ⇐⇒

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Pirio–Trepr´ eau bound

Theorem (Pirio–Trepr´eau, 2009) Let X = X^{r +1}(n, δ) ⊂ P^N. Then

N ≤ π(r , n, δ) − 1 = π(r , n, δ + r (n − 1) + 2) − 1,

where

π(r , n, d ) =X

σ≥0

σ + r − 1 σ

maxn

0, d − (σ + r ) (n − 1) − 1o

is the Castelnuovo–Harris function bounding the genus g (V ) of an irreducible variety

V^r ⊂ P^{r +n−1} of degree deg(V ) = d .

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Pirio–Trepr´ eau bound

N ≤ π(r , n, δ) − 1 = π(r , n, δ + r (n − 1) + 2) − 1, where

π(r , n, d ) =X

σ≥0

σ + r − 1 σ

maxn

0, d − (σ + r ) (n − 1) − 1o

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Pirio–Trepr´ eau bound

π(r , n, d ) =X

σ≥0

σ + r − 1 σ

maxn

0, d − (σ + r ) (n − 1) − 1o

is the Castelnuovo–Harris function

bounding the genus g (V ) of an irreducible variety

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Pirio–Trepr´ eau bound

π(r , n, d ) =X

σ≥0

σ + r − 1 σ

maxn

0, d − (σ + r ) (n − 1) − 1o

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Previously known results and some classifications

Let us begin with δ = 2 (or n = 2). Recall that δ ≥ n − 1.

X = X (3, 2) ⊂ P^N =⇒ X^{r +1}⊂ P^{r +2} quadric. (C. Segre &) Scorza (1909) : π(r , 2, 2) ≤ ^{r +1+2}₂ Scorza (1909) : X arbitrary

π(r , 2, 2) = ^{r +1+2}₂ ⇐⇒ X = ν₂(P^{r +1}) ⊂ P(^{r +1+2}² )⁻¹. Next case X = X (2, 2) ⊂ P^N.

Scorza (1909) : classification X = X (2, 2) ⊂ P^N in some cases reconsidered in [Chiantini, Ciliberto, — ;201 ?].

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Previously known results and some classifications

X = X (3, 2) ⊂ P^N =⇒ X^{r +1} ⊂ P^{r +2} quadric.

(C. Segre &) Scorza (1909) : π(r , 2, 2) ≤ ^{r +1+2}₂ Scorza (1909) : X arbitrary

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Previously known results and some classifications

(C. Segre &) Scorza (1909) : π(r , 2, 2) ≤ ^{r +1+2}₂

Scorza (1909) : X arbitrary

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Previously known results and some classifications

π(r , 2, 2) = ^{r +1+2}₂ ⇐⇒ X = ν₂(P^{r +1}) ⊂ P(^{r +1+2}² )⁻¹.

Next case X = X (2, 2) ⊂ P^N.

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Previously known results and some classifications

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Previously known results and some classifications

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Previously known results and some classifications

X = X (2, 2) ⊂ P^N SMOOTH (conic-connected manifold)

=⇒

CLASSIFIED : Fano & b₂(X ) ≤ 2 etc, etc

(Ionescu, —– ; 2008) Bompiani ( ?) (1921), Pirio & — (2010) :

X^{r +1} ⊂ P^N arbitrary (a) π(r , 2, δ) = ^{r +1+δ}_{r +1}

(b) N = ^{r +1+δ}_{r +1} − 1 ⇐⇒ X = νδ(P^{r +1}) ⊂ P(^{r +1+δ}² )⁻¹

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Previously known results and some classifications

X = X (2, 2) ⊂ P^N SMOOTH (conic-connected manifold) =⇒

(Ionescu, —– ; 2008)

Bompiani ( ?) (1921), Pirio & — (2010) : X^{r +1} ⊂ P^N arbitrary

(a) π(r , 2, δ) = ^{r +1+δ}_{r +1}

(b) N = ^{r +1+δ}_{r +1} − 1 ⇐⇒ X = νδ(P^{r +1}) ⊂ P(^{r +1+δ}² )⁻¹

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Previously known results and some classifications

X^{r +1} ⊂ P^N arbitrary

(a) π(r , 2, δ) = ^{r +1+δ}_{r +1}

(b) N = ^{r +1+δ}_{r +1} − 1 ⇐⇒ X = νδ(P^{r +1}) ⊂ P(^{r +1+δ}² )⁻¹

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Previously known results and some classifications

(b) N = ^{r +1+δ}_{r +1} − 1 ⇐⇒ X = νδ(P^{r +1}) ⊂ P(^{r +1+δ}² )⁻¹

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Previously known results and some classifications

(b) N = ^{r +1+δ}_{r +1} − 1 ⇐⇒ X = νδ(P^{r +1}) ⊂ P(^{r +1+δ}² )⁻¹

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Previously known results and some classifications

Let us consider δ = 3. Recall δ ≥ n − 1.

X = X (4, 3) ⊂ P^N =⇒ X^{r +1}⊂ P^{r +3} & deg(X ) = 3 (CLASSIFIED).

π(r , 3, 3) = 2r + 4, as we shall see in a moment.

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1} =⇒

CLASSIFIED : (Pirio, — ; 2010) object of this talk

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Previously known results and some classifications

X = X (4, 3) ⊂ P^N =⇒ X^{r +1} ⊂ P^{r +3} & deg(X ) = 3 (CLASSIFIED).

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1} =⇒

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Previously known results and some classifications

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1} =⇒

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Previously known results and some classifications

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}

=⇒

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Previously known results and some classifications

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1} =⇒

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π(r , 3, 3) = 2r + 4

Notation

x ∈ X = X^{r +1}(3, 3) ⊂ P^N general point ;

T = T_xX = P^{r +1} projective tangent space to X at x ; π_T : X 99KXT ⊆ P^{N−r −2}

projection of X from T , not defined along T ∩ X .

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π(r , 3, 3) = 2r + 4

Notation

T = T_xX = P^{r +1} projective tangent space to X at x ;

π_T : X 99KXT ⊆ P^{N−r −2} projection of X from T , not defined along T ∩ X .

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π(r , 3, 3) = 2r + 4

Notation

T = T_xX = P^{r +1} projective tangent space to X at x ; πT : X 99KXT ⊆ P^{N−r −2}

projection of X from T , not defined along T ∩ X .

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π(r , 3, 3) = 2r + 4

p16= p₂ ∈ X general =⇒ π_T(p1) 6= πT(p2).

(otherwise N = r + 2 < 2r + 3).

∃C = C_{x ,p}₁_,p₂ with deg(C ) = 3 ;

π_T(C_{x ,p}₁_,p₂) = hπ_T(p₁), π_T(p₂)i ⊆ X_T, that is

X_T = X (r + 1, 2, 1) = P^{r +1−σ} = P^{N−r −2}.

In conclusion N = 2r + 3 − σ, σ ≥ 0 and π(r , 3, 3) = 2r + 4.

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π(r , 3, 3) = 2r + 4

(otherwise N = r + 2 < 2r + 3).

∃C = C_{x ,p}₁_,p₂ with deg(C ) = 3 ;

X_T = X (r + 1, 2, 1) = P^{r +1−σ} = P^{N−r −2}.

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π(r , 3, 3) = 2r + 4

(otherwise N = r + 2 < 2r + 3).

∃C = C_{x ,p}₁_,p₂ with deg(C ) = 3 ;

X_T = X (r + 1, 2, 1) = P^{r +1−σ} = P^{N−r −2}.

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π(r , 3, 3) = 2r + 4

(otherwise N = r + 2 < 2r + 3).

∃C = C_{x ,p}₁_,p₂ with deg(C ) = 3 ;

π_T(C_{x ,p}₁_,p₂) = hπ_T(p₁), π_T(p₂)i ⊆ X_T,

that is

X_T = X (r + 1, 2, 1) = P^{r +1−σ} = P^{N−r −2}.

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π(r , 3, 3) = 2r + 4

(otherwise N = r + 2 < 2r + 3).

∃C = C_{x ,p}₁_,p₂ with deg(C ) = 3 ;

X_T = X (r + 1, 2, 1) = P^{r +1−σ} = P^{N−r −2}.

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

C = Cp1,p2,p3⊂ hC i is a twisted cubic ;

π_{T |C} : C 99KL isomorphism, L ⊂ XT = P^{r +1} line ; π_T : X 99KP^{r +1} is birational ;

π⁻¹_T = φ_Λ: P^{r +1}99KX ⊂ P^{2r +3}, Λ ⊂ |O_P^{r +1}(3)| complete linear system.

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

π_{T |C} : C 99KL isomorphism, L ⊂ XT = P^{r +1} line ;

π_T : X 99KP^{r +1} is birational ;

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

α : eX = BlxX → X blow-up of X at x ;

E = P^r exceptional divisor ;

˜

π_T : eX → X 99KP^{r +1} induced rational map ;

˜

π_T(E ) = Πx = P^r ⊂ P^{r +1} HYPERPLANE

(π_T⁻¹(Π_x) = x & π_T⁻¹(L) is smooth at x for L general).

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

α : eX = BlxX → X blow-up of X at x ; E = P^r exceptional divisor ;

˜

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

˜

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

˜

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

Remarks

˜

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψ_x = ˜π_{T |E} : E 99KΠx is a Cremona transformation given by Ω ⊂ O_P^r(2) because

α^∗(O(1)) ⊗ O(−2E ) ⊗ O_E = O_P^r(2).

Ω = |II_{x ,X}|Second fundamental form of X at x; Bs(|II_{x ,X}|) = B_x ⊂ E =asymptotic directions to X at x.

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

α^∗(O(1)) ⊗ O(−2E ) ⊗ O_E = O_P^r(2).

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

α^∗(O(1)) ⊗ O(−2E ) ⊗ O_E = O_P^r(2).

Ω = |II_{x ,X}|Second fundamental form of X at x;

Bs(|II_{x ,X}|) = B_x ⊂ E =asymptotic directions to X at x.

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

α^∗(O(1)) ⊗ O(−2E ) ⊗ O_E = O_P^r(2).

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

dim(|II_{x ,X}|) = r < codim(X ) − 1 = expected dimension.

ψ_x⁻¹= π⁻¹_{T |Π}

x : Πx99KE given by ˜Ω ⊂ |O_P^r(2)|. (recall that π⁻¹_T (Πx) = x . !)

B˜_x = Bs(ψ⁻¹_x )

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψ⁻¹_x = π⁻¹_{T |Π}

x : Πx99KE given by ˜Ω ⊂ |O_P^r(2)|.

(recall that π⁻¹_T (Πx) = x . !)

B˜_x = Bs(ψ⁻¹_x )

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X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψ⁻¹_x = π⁻¹_{T |Π}

x : Πx99KE given by ˜Ω ⊂ |O_P^r(2)|.

(recall that π⁻¹_T (Πx) = x . !) B˜_x = Bs(ψ⁻¹_x )

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

(B_x = Bs(ψ_x) ⊂ E = P^r B˜x = Bs(ψ_x⁻¹) ⊂ Πx = P^r

Theorem (Pirio, — (2010))

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

A. ψ_x ∈ Lin ⊂ Bir₂₂(P^r) ⇐⇒X SMOOTH rational normal scroll B. X not a scroll :

1. Π⁻¹_T : P^{r +1}99K X^{bir .} is given by |3H − 2 ˜B_x|

2. B_x = Hilb^t+1(X , x ) ∼_projB˜_x (usually only Hilb^t+1(X , x ) ⊆ B_x holds !)

3. X SMOOTH =⇒ Bx et ˜Bx are SMOOTH

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

A. ψ_x ∈ Lin ⊂ Bir₂₂(P^r) ⇐⇒X SMOOTH rational normal scroll

B. X not a scroll :

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

1. Π⁻¹_T : P^{r +1}99K X^{bir .} is given by |3H − 2 ˜Bx| 2. B_x = Hilb^t+1(X , x ) ∼_projB˜_x

(usually only Hilb^t+1(X , x ) ⊆ B_x holds !)

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Structure Theorem for X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

ψx = ˜π_{T |E} =∈ Bir22(P^r)

X = X^{r +1}(3, 3) ⊂ P^{2(r +1)+1}. Then :

1. Π⁻¹_T : P^{r +1}99K X^{bir .} is given by |3H − 2 ˜Bx|

3. X SMOOTH =⇒ B_x et ˜B_x are SMOOTH

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From Bir

_(2,2)

(P

^r

) to X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

φ : P^r99KP^r ∈ Bir₂₂(P^r) \ Lin(P^r)

(B = Bs(φ) ⊂ P^r B = Bs(φ˜ ⁻¹) ⊂ P^r x = (x1 : . . . : xr +1) coordinates on P^r

φ = (φ₁(x) : . . . : φ_{r +1}(x)) : P^r99KP^r

φ⁻¹◦ φ = ϕ(x)(x₁: . . . : x_{r +1}) with ϕ(x) ∈ C[x1, . . . , x_r]₃; (φ⁻¹◦ φ ∼_bir I_P^r)

the cubic hypersurface V (ϕ(x)) ⊂ P^r hasdouble points along B = V (φ₁(x), . . . , φ_{r +1}(x)) = Bs(φ), that is

∂ϕ(x)

∂xi

∈ hψ₁(x), . . . , ψr +1(x)i ∀ i = 1, . . . , r + 1.

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From Bir

_(2,2)

(P

^r

) to X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

(B = Bs(φ) ⊂ P^r B = Bs(φ˜ ⁻¹) ⊂ P^r

x = (x1 : . . . : xr +1) coordinates on P^r φ = (φ₁(x) : . . . : φ_{r +1}(x)) : P^r99KP^r

∂ϕ(x)

∂xi

∈ hψ₁(x), . . . , ψr +1(x)i ∀ i = 1, . . . , r + 1.

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From Bir

_(2,2)

(P

^r

) to X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

φ = (φ₁(x) : . . . : φ_{r +1}(x)) : P^r99KP^r

∂ϕ(x)

∂xi

∈ hψ₁(x), . . . , ψr +1(x)i ∀ i = 1, . . . , r + 1.

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From Bir

_(2,2)

(P

^r

) to X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

φ = (φ₁(x) : . . . : φ_{r +1}(x)) : P^r99KP^r

∂ϕ(x)

∂xi

∈ hψ₁(x), . . . , ψr +1(x)i ∀ i = 1, . . . , r + 1.

(76)

From Bir

_(2,2)

(P

^r

) to X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

φ = (φ₁(x) : . . . : φ_{r +1}(x)) : P^r99KP^r

φ⁻¹◦ φ = ϕ(x)(x₁: . . . : x_{r +1}) with ϕ(x) ∈ C[x1, . . . , x_r]₃;

(φ⁻¹◦ φ ∼_bir I_P^r)

∂ϕ(x)

∂xi

∈ hψ₁(x), . . . , ψr +1(x)i ∀ i = 1, . . . , r + 1.

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From Bir

_(2,2)

(P

^r

) to X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

φ = (φ₁(x) : . . . : φ_{r +1}(x)) : P^r99KP^r

∂ϕ(x)

∂xi

∈ hψ₁(x), . . . , ψr +1(x)i ∀ i = 1, . . . , r + 1.