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+22} )⁻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 ?].

Francesco Russo Varieties n–covered by curved of degree δ

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+22} )⁻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 ?].

Francesco Russo Varieties n–covered by curved of degree δ

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+22} )⁻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 ?].

Francesco Russo Varieties n–covered by curved of degree δ

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+22} )⁻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 ?].

Francesco Russo Varieties n–covered by curved of degree δ

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+22} )⁻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 ?].

Francesco Russo Varieties n–covered by curved of degree δ

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+22} )⁻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 ?].

Francesco Russo Varieties n–covered by curved of degree δ

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+δ2} )⁻¹

Francesco Russo Varieties n–covered by curved of degree δ

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+δ2} )⁻¹

Francesco Russo Varieties n–covered by curved of degree δ

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+δ2} )⁻¹

Francesco Russo Varieties n–covered by curved of degree δ

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+δ2} )⁻¹

Francesco Russo Varieties n–covered by curved of degree δ

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+δ2} )⁻¹

Francesco Russo Varieties n–covered by curved of degree δ

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

Francesco Russo Varieties n–covered by curved of degree δ

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

Francesco Russo Varieties n–covered by curved of degree δ

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

Francesco Russo Varieties n–covered by curved of degree δ

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

Francesco Russo Varieties n–covered by curved of degree δ

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

Francesco Russo Varieties n–covered by curved of degree δ

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 .

Francesco Russo Varieties n–covered by curved of degree δ

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 .

Francesco Russo Varieties n–covered by curved of degree δ

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 ; πT : X 99KXT ⊆ P^{N−r −2}

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

Francesco Russo Varieties n–covered by curved of degree δ

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 ,p1,p2} with deg(C ) = 3 ;

π_T(C_{x ,p1,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.

Francesco Russo Varieties n–covered by curved of degree δ

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 ,p1,p2} with deg(C ) = 3 ;

π_T(C_{x ,p1,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.

Francesco Russo Varieties n–covered by curved of degree δ

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 ,p1,p2} with deg(C ) = 3 ;

π_T(C_{x ,p1,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.

Francesco Russo Varieties n–covered by curved of degree δ

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 ,p1,p2} with deg(C ) = 3 ;

π_T(C_{x ,p1,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.

Francesco Russo Varieties n–covered by curved of degree δ

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 ,p1,p2} with deg(C ) = 3 ;

π_T(C_{x ,p1,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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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).

Francesco Russo Varieties n–covered by curved of degree δ

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).

Francesco Russo Varieties n–covered by curved of degree δ

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).

Francesco Russo Varieties n–covered by curved of degree δ

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).

Francesco Russo Varieties n–covered by curved of degree δ

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).

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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.

Francesco Russo Varieties n–covered by curved of degree δ

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 )

Francesco Russo Varieties n–covered by curved of degree δ

X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

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

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 )

Francesco Russo Varieties n–covered by curved of degree δ

X = X

^{r +1}

(3, 3) ⊂ P

^{2(r +1)+1}

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

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 )

Francesco Russo Varieties n–covered by curved of degree δ

Nel documento Varieties n–covered by curved of degree δ (pagine 25-65)