3 Onbirationaltransformationsof P ofdegree3and4

Introduction ACM examples Other examples inBir4,4(P3)

On birational transformations of P

of degree 3 and 4

Frederic Han, (j. w. Julie Déserti)

Mediterranean Complex Projective Geometry, Carry le Rouet

Mai 27, 2016

Introduction ACM examples Other examples inBir4,4(P3)

Denition Bir_d(P3) =

φ:P399KP3, birational linear system of degree d with base locus of codim > 1

Denition Bir_d1,d2(P₃) =

φ∈Bir_d1(P₃),φ⁻¹∈Bir_d2(P₃)

Introduction ACM examples Other examples inBir4,4(P3)

Denition Bir_d(P3) =

φ:P399KP3, birational linear system of degree d with base locus of codim > 1

Denition Bir_d1,d2(P₃) =

φ∈Bir_d1(P₃),φ⁻¹∈Bir_d2(P₃)

Introduction ACM examples Other examples inBir4,4(P3)

φ∈Bir_d(P3), H hyperplane ofP3 then

φ⁻¹(H)is a rational surface of degree d

Bir_d(P₃) d≤3

6= Bir_d(P₃) d>3 so

Bir_4,4(P3)?

Introduction ACM examples Other examples inBir4,4(P3)

(book 1927) Hudson's Table of Bir

( P

)

Introduction ACM examples Other examples inBir4,4(P3)

Hudson's Table in bidegree (3,5)

D.p. of binode D. p.'s pt

of pt of ord. F -curves

contact osc. contact pts

· · · · · 2 ω₃(rational), l

· · 1 · · 2 ω₄≡O1(genus 1)

· · 1 · · 2 ω₃≡O1(rational), l≡O1

· 1 · · · 2 ω₄≡O₁²(2)

· 1 · · · 2 ω₂≡O1(1), l1≡O1(1), l2≡O1

1 · · · · 2 ω₃≡O₁², l1≡O1

1 · · · · 2 l≡O1(contact), l1≡O1, l2≡O1

· · 2 · · 2 ω₃≡O1O2 (rational), l≡O1O2

· 1 1 · · 2 ω₂≡O1(1)O2, l1≡O1O2, l2≡O1(1) 1 · 1 · · 2 l≡O1O2 (contact), l1≡O1, l2≡O1

1 1 · · · 2 l≡O1O2(1)(osculation), l1≡O1

· · · · 1 · ω₄(rational)

· · 1 · 1 · ω₄≡O₁²

· 1 · · 1 · ω₃≡O₁²(1), l≡O1(1)

1 · · · 1 · l1≡O1, l2≡O1, l3≡O1, l4≡O1

· · · · · 6 l²

Introduction ACM examples Other examples inBir4,4(P3)

Rough ideas about Bir

( P

)

The situation inBir₃(P₃)is poor with non normal surfaces Only one component ofBir_3,d(P₃)for each 2≤d≤5 But

The situation inBir₃(P3)is very rich with normal surfaces

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal cubo-cubic

0 → O

⁽⁻ 1 ) −→

O

^→ I

( 3 ) → 0 ,

Pf₃(C)is a complete intersection inP₃×P₃. (∼(1,1)³) The linear system|IC(3)|: P399K|IC(3)|^∨ is birational Denition (Determinantal cubo-cubic)

Let D₃,₃⊂Bir_3,3(P₃)be the set of maps dened by the maximal minors of a linear map O_P^⊕₃³⁽⁻1)→O_P^⊕₃⁴.

Remark

D₃,₃ is an irreducible component ofBir_3,3(P₃).

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal cubo-cubic

0 → O

⁽⁻ 1 ) −→

O

^→ I

( 3 ) → 0 ,

Pf₃(C)is a complete intersection inP₃×P₃. (∼(1,1)³)

The linear system|IC(3)|: P399K|IC(3)|^∨ is birational Denition (Determinantal cubo-cubic)

Let D₃,₃⊂Bir_3,3(P₃)be the set of maps dened by the maximal minors of a linear map O_P^⊕₃³⁽⁻1)→O_P^⊕₃⁴.

Remark

D₃,₃ is an irreducible component ofBir_3,3(P₃).

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal cubo-cubic

0 → O

⁽⁻ 1 ) −→

O

^→ I

( 3 ) → 0 ,

Pf₃(C)is a complete intersection inP₃×P₃. (∼(1,1)³) The linear system|IC(3)|: P399K|IC(3)|^∨ is birational

Denition (Determinantal cubo-cubic)

Let D₃,₃⊂Bir_3,3(P₃)be the set of maps dened by the maximal minors of a linear map O_P^⊕₃³⁽⁻1)→O_P^⊕₃⁴.

Remark

D₃,₃ is an irreducible component ofBir_3,3(P₃).

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal cubo-cubic

0 → O

⁽⁻ 1 ) −→

O

^→ I

( 3 ) → 0 ,

Pf₃(C)is a complete intersection inP₃×P₃. (∼(1,1)³) The linear system|IC(3)|: P399K|IC(3)|^∨ is birational Denition (Determinantal cubo-cubic)

Let D₃,₃⊂Bir_3,3(P₃)be the set of maps dened by the maximal minors of a linear map O_P^⊕₃³⁽⁻1)→O_P^⊕₃⁴.

Remark

D₃,₃ is an irreducible component ofBir_3,3(P₃).

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal cubo-cubic

0 → O

⁽⁻ 1 ) −→

O

^→ I

( 3 ) → 0 ,

Pf₃(C)is a complete intersection inP₃×P₃. (∼(1,1)³) The linear system|IC(3)|: P399K|IC(3)|^∨ is birational Denition (Determinantal cubo-cubic)

Let D₃,₃⊂Bir_3,3(P₃)be the set of maps dened by the maximal minors of a linear map O_P^⊕₃³⁽⁻1)→O_P^⊕₃⁴.

Remark

D₃,₃ is an irreducible component ofBir_3,3(P₃).

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal quarto-quartic

Is there a similar construction with quartics ?

0→

O

O

O

Problem,if G is generic :P3 2:1

99KP3

Example (Déserti,−)

- LetIΓbe the ideal of a general trigonal curveΓ⊂P3of degree 8 and genus 5,

- I∆ be the ideal of ∆, the unique line 5-secant toΓ.

- 0→

O

O

O

- The linear system|I_∆²∩IΓ(4)|: P399K|I_∆²∩IΓ(4)|^∨ is birational

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal quarto-quartic

Is there a similar construction with quartics ?

0→

O

O

O

Problem,if G is generic :P3 2:1

99KP3

Example (Déserti,−)

- LetIΓbe the ideal of a general trigonal curveΓ⊂P3of degree 8 and genus 5,

- I∆ be the ideal of ∆, the unique line 5-secant toΓ.

- 0→

O

O

O

- The linear system|I_∆²∩IΓ(4)|: P399K|I_∆²∩IΓ(4)|^∨ is birational

Introduction ACM examples Other examples inBir4,4(P3)

Determinantal quarto-quartic

Is there a similar construction with quartics ?

0→

O

O

O

Problem,if G is generic :P3 2:1

99KP3

Example (Déserti,−)

- LetIΓ be the ideal of a general trigonal curveΓ⊂P3of degree 8 and genus 5,

- I∆ be the ideal of ∆, the unique line 5-secant toΓ.

- 0

O

O

O

Introduction ACM examples Other examples inBir4,4(P3)

Construction in f P

(∆)

Pf3(∆) =blow up ofP3 in a line∆. X ⊂ fP3(∆)×P3

∩

a complete intersection(1,0,1)·(0,1,1)·(1,1,1) X ⊂P1×P3×P3 a complete intersection(1,1,0)·(1,0,1)·(0,1,1)·(1,1,1)

P₃←X→P₃ are birational.

Denition (Determinantal quarto-quartic)

Bir_4,4(P₃)⊃D₄,₄={φ| ∃∆,X such that φ:P₃99KX→P₃}

Introduction ACM examples Other examples inBir4,4(P3)

Construction in f P

(∆)

Pf3(∆) =blow up ofP3 in a line∆. X ⊂ fP3(∆)×P3

∩

a complete intersection(1,0,1)·(0,1,1)·(1,1,1)

X ⊂P1×P3×P3 a complete intersection(1,1,0)·(1,0,1)·(0,1,1)·(1,1,1) P₃←X→P₃ are birational.

Denition (Determinantal quarto-quartic)

Bir_4,4(P₃)⊃D₄,₄={φ| ∃∆,X such that φ:P₃99KX→P₃}

Introduction ACM examples Other examples inBir4,4(P3)

as X⊂Pf3(∆)×P3 is a complete intersection(1,0,1)·(0,1,1)·(1,1,1) p:X →fP3(∆), p∗(OX(2,2,1))gives :

0→ O_Pf

3(∆)(−1,0)

⊕ O_Pf

3(∆)(0,−1)

⊕ O_Pf

3(∆)(−1,−1)

Ge

−→O^⊕4

Pf₃^(∆)→IZ(2,2)→0

Z has genus 5, degOZ(0,1) =8,|OZ(1,0)|is a g₃¹.

A general trigonal curve of degree 8 and genus 5 with∆as 5-secant line has the same resolution.

Introduction ACM examples Other examples inBir4,4(P3)

as X⊂Pf3(∆)×P3 is a complete intersection(1,0,1)·(0,1,1)·(1,1,1) p:X →fP3(∆), p∗(OX(2,2,1))gives :

0→ O_Pf

3(∆)(−1,0)

⊕ O_Pf

3(∆)(0,−1)

⊕ O_Pf

3(∆)(−1,−1)

Ge

−→O^⊕4

Pf₃^(∆)→IZ(2,2)→0

Z has genus 5, degOZ(0,1) =8,|OZ(1,0)|is a g₃¹.

A general trigonal curve of degree 8 and genus 5 with∆as 5-secant line has the same resolution.

Introduction ACM examples Other examples inBir4,4(P3)

as X⊂Pf3(∆)×P3 is a complete intersection(1,0,1)·(0,1,1)·(1,1,1) p:X →fP3(∆), p∗(OX(2,2,1))gives :

0→ O_Pf

3(∆)(−1,0)

⊕ O_Pf

3(∆)(0,−1)

⊕ O_Pf

3(∆)(−1,−1)

Ge

−→O^⊕4

Pf₃^(∆)→IZ(2,2)→0

Z has genus 5, degOZ(0,1) =8,|OZ(1,0)|is a g₃¹.

A general trigonal curve of degree 8 and genus 5 with∆as 5-secant line has the same resolution.

Introduction ACM examples Other examples inBir4,4(P3)

Explicit constructions over P

L₂=H⁰(OP₁(1)), A₄=H⁰(OP₃(1)), A⁰₄=H⁰(O_P⁰₃⁽1)), B:L₂→^∼ L^∨₂ X ⊂P1×P3×P⁰₃a complete intersection(1,1,0)·(1,0,1)·(0,1,1)·(1,1,1)

∀z∈A^∨₄, g₁=N₁◦B◦^tN₀(z), g₂=M(z), g₃=T_B◦^t_N0(z)(z) g₁∧g₂∧g₃∈^V3A⁰₄=A^0∨₄, (g_i)gives the 3 columns of G

∀y∈A⁰₄^∨, g₁⁰=N₀◦^tB◦^tN₁(y), g₂⁰ =^tM(y), g₃⁰=^t(T_B◦^t_N1(y))(y) g₁⁰∧g₂⁰∧g₃⁰∈^V3A₄=A^∨₄, (g_i⁰)gives the 3 columns of G⁰ Minors of G and G⁰gives a birational map and its inverse.

Introduction ACM examples Other examples inBir4,4(P3)

Explicit constructions over P

L₂=H⁰(OP₁(1)), A₄=H⁰(OP₃(1)), A⁰₄=H⁰(O_P⁰₃⁽1)), B:L₂→^∼ L^∨₂

(1,1,0) L^∨₂^N→⁰A₄

(1,0,1) L^∨₂→^N1A⁰₄

(0,1,1) A^∨₄ →^M A⁰₄

(1,1,1)

T: L^∨₂ → Hom(A^∨₄,A⁰₄)

λ 7→ Tλ

N1◦B◦^tN0, M, Tλ: A^∨₄−→A⁰₄

∀z∈A^∨₄, g₁=N₁◦B◦^tN₀(z), g₂=M(z), g₃=T_B◦^t_N0(z)(z) g₁∧g₂∧g₃∈^V3A⁰₄=A^0∨₄, (g_i)gives the 3 columns of G

∀y∈A⁰₄^∨, g₁⁰=N₀◦^tB◦^tN₁(y), g₂⁰ =^tM(y), g₃⁰=^t(T_B◦^t_N1(y))(y) g₁⁰∧g₂⁰∧g₃⁰∈^V3A₄=A^∨₄, (g_i⁰)gives the 3 columns of G⁰ Minors of G and G⁰gives a birational map and its inverse.

Introduction ACM examples Other examples inBir4,4(P3)

Explicit constructions over P

L₂=H⁰(OP₁(1)), A₄=H⁰(OP₃(1)), A⁰₄=H⁰(O_P⁰₃⁽1)), B:L₂→^∼ L^∨₂

(1,1,0) L^∨₂^N→⁰A₄

(1,0,1) L^∨₂→^N1A⁰₄

(0,1,1) A^∨₄ →^M A⁰₄

(1,1,1)

T: L^∨₂ → Hom(A^∨₄,A⁰₄)

λ 7→ Tλ

N1◦B◦^tN0, M, Tλ: A^∨₄−→A⁰₄

∀z∈A^∨₄, g₁=N₁◦B◦^tN₀(z), g₂=M(z), g₃=T_B◦^t_N0(z)(z) g₁∧g₂∧g₃∈^V3A⁰₄=A^0∨₄, (g_i)gives the 3 columns of G

∀y∈A⁰₄^∨, g₁⁰=N₀◦^tB◦^tN₁(y), g₂⁰ =^tM(y), g₃⁰=^t(T_B◦^t_N1(y))(y) g₁⁰∧g₂⁰∧g₃⁰∈^V3A₄=A^∨₄, (g_i⁰)gives the 3 columns of G⁰ Minors of G and G⁰gives a birational map and its inverse.

Introduction ACM examples Other examples inBir4,4(P3)

Explicit constructions over P

L₂=H⁰(OP₁(1)), A₄=H⁰(OP₃(1)), A⁰₄=H⁰(O_P⁰₃⁽1)), B:L₂→^∼ L^∨₂

(1,1,0) L^∨₂^N→⁰A₄

(1,0,1) L^∨₂→^N1A⁰₄

(0,1,1) A^∨₄ →^M A⁰₄

(1,1,1)

T: L^∨₂ → Hom(A^∨₄,A⁰₄)

λ 7→ Tλ

N1◦B◦^tN0, M, Tλ: A^∨₄−→A⁰₄

∀z∈A^∨₄, g₁=N₁◦B◦^tN₀(z), g₂=M(z), g₃=T_B◦^t_N0(z)(z) g₁∧g₂∧g₃∈^V3A⁰₄=A^0∨₄, (g_i)gives the 3 columns of G

∀y∈A⁰₄^∨, g₁⁰=N0◦^tB◦^tN1(y), g₂⁰ =^tM(y), g₃⁰=^t(T_B◦^t_N1(y))(y) g₁⁰∧g₂⁰∧g₃⁰∈^V3A4=A^∨₄, (g_i⁰)gives the 3 columns of G⁰ Minors of G and G⁰gives a birational map and its inverse.

Introduction ACM examples Other examples inBir4,4(P3)

Proposition

D4,4 is an irreducible component ofBir_4,4(P3).

Proposition

Let φbe a general element ofD4,4, andΓbe the associated trigonal curve of genus 5 embedded inP3byOΓ(H), then

1 φ⁻¹∈D₄,₄ is also constructed fromΓbut embedded in P₃byOΓ(H⁰)

2 OΓ(H⁰) =ω^⊗_Γ²(−H)

Introduction ACM examples Other examples inBir4,4(P3)

Proposition

D4,4 is an irreducible component ofBir_4,4(P3). Proposition

Let φbe a general element ofD4,4, andΓbe the associated trigonal curve of genus 5 embedded inP3byOΓ(H), then

1 φ⁻¹∈D4,₄ is also constructed fromΓbut embedded inP3byO^Γ(H⁰)

2 OΓ(H⁰) =ω^⊗_Γ²(−H)

Introduction ACM examples Other examples inBir4,4(P3)

Contracted locus

φ:P399KP3,φ∈D4,4, thenφcontracts

1 A ruled surface of degree 9 (triangles with vertices elements of the g₃¹ofΓ)

2 The cubic surface containingΓ.

Introduction ACM examples Other examples inBir4,4(P3)

Contracted locus

φ:P399KP3,φ∈D4,4, thenφcontracts

1 A ruled surface of degree 9 (triangles with vertices elements of the g₃¹ofΓ)

2 The cubic surface containingΓ.

Introduction ACM examples Other examples inBir4,4(P3)

Classical examples

J_d,_d⊂Bir_d,d(P₃)(lift an automorphism ofP₂ with monoids) (z₀:z₁:z₂:z₃)7→

z₀:z₁:z₂: z3Pd−₁(z0,z1,z2) +Pd(z0,z1,z2) z₃Q_d−₂(z₀,z₁,z₂) +Q_d−₁(z₀,z₁,z₂)

Rd,_d⊂Bir_d,d(P3), ruled surfaces with a line of multiplicity d−1 + d−1 base rules(φfactors through a threefold of degree d inP_d+2) + d−1 base points

G. Loria's example (1890) in Bir_4,4(P3). (Steiner quartics + 3 base points)