Veronese variety

2009年2月6日

任意標数における Veronese variety _の

higher secant variety _{の定義方程式について}

基幹理工学研究科 数学応用数理専攻 修士課程2年 5107A007-1

伊藤 達哉

指導教員名

楫 元

1 Definition

k : 基礎体, k = k, ch(k) ≥ 0.

Veronese variety.

¶ ³

V₂^m := ν₂(P^m) : Veronese variety,

ν₂ : P^m −→ P^N:=(^m+2₂ )⁻¹ ; (x₀ : x₁ : · · · : x_m) 7−→ (x²₀ : x₀x₁ : x₀x₂ : · · · : x²_m).

µ ´

1 Definition

k : 基礎体, k = k, ch(k) ≥ 0.

Veronese variety.

¶ ³

V₂^m := ν₂(P^m) : Veronese variety,

ν₂ : P^m −→ P^N:=(^m+2₂ )⁻¹ ; (x₀ : x₁ : · · · : x_m) 7−→ (x²₀ : x₀x₁ : x₀x₂ : · · · : x²_m).

µ ´

Secant variety.

¶ ³

X ⊆ P^N : sm. proj. var.

Sec(X)^◦ := ∪

x,y∈X, x̸=y

xy, Sec(X) := Sec(X)^◦ ⊆ P^N.

µ ´

1 Definition

k : 基礎体, k = k, ch(k) ≥ 0.

Veronese variety.

¶ ³

V₂^m := ν₂(P^m) : Veronese variety,

ν₂ : P^m −→ P^N:=(^m+2₂ )⁻¹ ; (x₀ : x₁ : · · · : x_m) 7−→ (x²₀ : x₀x₁ : x₀x₂ : · · · : x²_m).

µ ´

Secant variety.

¶ ³

X ⊆ P^N : sm. proj. var.

Sec(X)^◦ := ∪

x,y∈X, x̸=y

xy, Sec(X) := Sec(X)^◦ ⊆ P^N.

µ ´

n-secant variety.

¶ ³

Secⁿ(X) := ∪

x₀, ... ,x_n∈X

⟨x₀, . . . , x_n⟩ ⊆ P^N.

⟨x₀, . . . , x_n⟩ : x₀, . . . , x_nで張られる線形部分空間.

µ ´

2 Secant variety _の応用例

P. Griffiths; J. Harris : Principles of algebraic geometry.





X ⊂ P^N : sm. proj. var. , P ∈ P^N \ X,

π_P : X → P^N−1 (P からの射影).

Question.

¶ ³

π_P : X → π_P(X)は同型になるか？

µ ´

2 Secant variety _の応用例

P. Griffiths; J. Harris : Principles of algebraic geometry.





X ⊂ P^N : sm. proj. var. , P ∈ P^N \ X,

π_P : X → P^N−1 (P からの射影).

Question.

¶ ³

π_P : X → π_P(X)は同型になるか？

µ ´

Proposition.

¶ ³

π_P : X → π_P(X) : 同型.

⇔ P ∈ P^N\Sec(X).

µ ´

3 Fact

Proposition 1.

¶ ³

ch(k) ≥ 0, V₂^m = V (I₂(Ω)).

µ ´

Z = (Z₀₀ : Z₀₁ : Z₀₂ : · · · : Z_m−1m : Z_mm) ∈ P^N:=(^m+2₂ )⁻¹,

Ω(Z) :=









Z₀₀ Z₀₁ Z₀₂ · · · Z_0m Z₁₀ Z₁₁ Z₁₂ · · · Z_1m Z₂₀ Z₂₁ Z₂₂ · · · Z_2m

... ... ... . .. ... Z_m0 Z_m1 Z_m2 · · · Z_mm









(但し, Z_ji = Z_ij).

I_t(Ω) := ⟨ Ωのt × t小行列式⟩.

3 Fact

Proposition 1.

¶ ³

ch(k) ≥ 0, V₂^m = V (I₂(Ω)).

µ ´

Proposition 2 (R. Gattazzo (1984)) .

¶ ³

(i) ch(k) ≥ 0, Sec(V₂^m) ⊆ V (I₃(Ω)).

(ii) ch(k) ̸= 2, Sec(V₂^m) = V (I₃(Ω)).

µ ´

Z = (Z₀₀ : Z₀₁ : Z₀₂ : · · · : Z_m−1m : Z_mm) ∈ P^N:=(^m+2₂ )⁻¹,

Ω(Z) :=









Z₀₀ Z₀₁ Z₀₂ · · · Z_0m Z₁₀ Z₁₁ Z₁₂ · · · Z_1m Z₂₀ Z₂₁ Z₂₂ · · · Z_2m

... ... ... . .. ... Z_m0 Z_m1 Z_m2 · · · Z_mm









(但し, Z_ji = Z_ij).

I_t(Ω) := ⟨ Ωのt × t小行列式⟩.

4 ch(k) = 2 の現象の考察

• ch(k) = 2, V (I₃(Ω)) = Sec(V₂^m) ?

¨

§

¥ (R. Gattazzo) NOTE : ch(k) = 2, m = 2, ∃R₀ ∈ V (I₃(Ω))\Sec(V₂²). ¦

4 ch(k) = 2 の現象の考察

• ch(k) = 2, V (I₃(Ω)) = Sec(V₂^m) ?

¨

§

¥ (R. Gattazzo) NOTE : ch(k) = 2, m = 2, ∃R₀ ∈ V (I₃(Ω))\Sec(V₂²). ¦

¨ ⇓

§

¥ 実際は, ch(k) = 2, R₀ ∈ Sec(V₂²) \ Sec(V₂²)^◦. ¦^(注1)

(注1) Sec(X)^◦ := ∪

x,y∈X, x̸=y xy.

4 ch(k) = 2 の現象の考察

• ch(k) = 2, V (I₃(Ω)) = Sec(V₂^m) ?

¨

§

¥ (R. Gattazzo) NOTE : ch(k) = 2, m = 2, ∃R₀ ∈ V (I₃(Ω))\Sec(V₂²). ¦

¨ ⇓

§

¥ 実際は, ch(k) = 2, R₀ ∈ Sec(V₂²) \ Sec(V₂²)^◦. ¦^(注1)

¨

§

¥ R₀ ∈ Tan(V₂²) ⊆ Sec(V₂²). ¦^(注2)

(注1) Sec(X)^◦ := ∪

x,y∈X, x̸=y xy.

(注2) Tan(X) := ∪

P∈X T^PX, Sec(X) = Sec(X)^◦ ∪ Tan(X).

4 ch(k) = 2 の現象の考察

• ch(k) = 2, V (I₃(Ω)) = Sec(V₂^m) ?

¨

§

¥ (R. Gattazzo) NOTE : ch(k) = 2, m = 2, ∃R₀ ∈ V (I₃(Ω))\Sec(V₂²). ¦

¨ ⇓

§

¥ 実際は, ch(k) = 2, R₀ ∈ Sec(V₂²) \ Sec(V₂²)^◦. ¦^(注1)

¨

§

¥ R₀ ∈ Tan(V₂²) ⊆ Sec(V₂²). ¦^(注2) ch(k) = 2 の現象.

¶ ³

ch(k) ̸= 2 ⇒ Sec(V₂^m) = Sec(V₂^m)^◦. ch(k) = 2 ⇒ Sec(V₂^m) ̸= Sec(V₂^m)^◦.

µ ´

(注1) Sec(X)^◦ := ∪

x,y∈X, x̸=y xy.

(注2) Tan(X) := ∪

P∈X T^PX, Sec(X) = Sec(X)^◦ ∪ Tan(X).

5 Main Theorem

Main Theorem.

¶ ³

ch(k) ≥ 0, V₂^m ⊂ P^N:=(^m+2₂ )−1

: Veronese variety.

Secⁿ(V₂^m) = V (I_n+2(Ω)) (n ≥ 0).

µ ´

Z = (Z₀₀ : Z₀₁ : Z₀₂ : · · · : Z_m−1m : Z_mm) ∈ P^N,

Ω(Z) :=









Z₀₀ Z₀₁ Z₀₂ · · · Z_0m Z₁₀ Z₁₁ Z₁₂ · · · Z_1m Z₂₀ Z₂₁ Z₂₂ · · · Z_2m

... ... ... . .. ... Z_m0 Z_m1 Z_m2 · · · Z_mm









(但し, Z_ji = Z_ij).

I_t(Ω) := ⟨ Ωのt × t小行列式⟩.

5 Main Theorem

Main Theorem.

¶ ³

ch(k) ≥ 0, V₂^m ⊂ P^N:=(^m+2₂ )−1

: Veronese variety.

Secⁿ(V₂^m) = V (I_n+2(Ω)) (n ≥ 0).

µ ´

Z = (Z₀₀ : Z₀₁ : Z₀₂ : · · · : Z_m−1m : Z_mm) ∈ P^N,

Ω(Z) :=









Z₀₀ Z₀₁ Z₀₂ · · · Z_0m Z₁₀ Z₁₁ Z₁₂ · · · Z_1m Z₂₀ Z₂₁ Z₂₂ · · · Z_2m

... ... ... . .. ... Z_m0 Z_m1 Z_m2 · · · Z_mm









(但し, Z_ji = Z_ij).

I_t(Ω) := ⟨ Ωのt × t小行列式⟩.

6 Application(Rank と Betti 数の関係について )

X ⊆ P^N : sm. proj. var. , P ∈ P^N \ X.

Rank.

¶ ³

rk_X(P) := min{n | P ∈ Secⁿ(X)}.

µ ´

X ⊆ Sec(X) ⊆ Sec²(X) ⊆ · · · ⊆ Secⁿ(X) ⊆ · · · ⊆ P^N.

π_P : X → P^N−1.

6 Application(Rank と Betti 数の関係について )

Proposition (E. Park (2007)).

¶ ³

ch(k) ≥ 0, P₁, P₂ ∈ P^d \ Sec(X), X := V_d¹,

rk_X(P₁) = rk_X(P₂) ⇐⇒ B(π_P1(X)) = B(π_P2(X)).

µ ´

V_d¹ := ν_d(P¹), ν_d : P¹ → P^d; (x₀ : x₁) 7→ (x^d₀ : x^d₀⁻¹x₁ : · · · : x^d₁).

次数付きBetti数.

¶ ³

B(X) := (β_ij(X)).

µ ´

F_i = ⊕

j R(−j)^βij(X), · · · −→ F_i −→ · · · −→ F₀ −→ I_X −→ 0.

rk_X(P) := min{n | P ∈ Secⁿ(X)}.

6 Application(Rank と Betti 数の関係について )

Proposition (E. Park (2007)).

¶ ³

ch(k) ≥ 0, P₁, P₂ ∈ P^d \ Sec(X), X := V_d¹,

rk_X(P₁) = rk_X(P₂) ⇐⇒ B(π_P1(X)) = B(π_P2(X)).

µ ´

;

Question.

¶ ³

ch(k) ≥ 0, P₁, P₂ ∈ P^d \ Sec(X), X := V₂^m,

rk_X(P₁) = rk_X(P₂) ⇐⇒ B(π_P1(X)) = B(π_P2(X)). ?

µ ´

V_d¹ := ν_d(P¹), ν_d : P¹ → P^d; (x₀ : x₁) 7→ (x^d₀ : x^d₀⁻¹x₁ : · · · : x^d₁).

V₂^m := ν₂(P^m), ν₂ : P^m → P^N; (x₀ : · · · : x_m) 7→ (x²₀ : x₀x₁ · · · : x²_m).

6 Application(Rank と Betti 数の関係について )

Example.

¶ ³

X := V₂⁴, ∃P₁, P₂ ∈ P¹⁴ \ Sec(X) s.t. rk_X(P₁) = rk_X(P₂) = 3, (i) ch(k) = 2, B(π_P1(X)) ̸= B(π_P2(X)).

(ii) ch(k) = 0,3, B(π_P1(X)) = B(π_P2(X)).

µ ´

Question.

¶ ³

ch(k) ≥ 0, P₁, P₂ ∈ P^d \ Sec(X), X := V₂^m,

rk_X(P₁) = rk_X(P₂) ⇐⇒ B(π_P1(X)) = B(π_P2(X)). ?

µ ´

rk_X(P) := min{n | P ∈ Secⁿ(X)}.

X ⊆ Sec(X) ⊆ Sec²(X) ⊆ · · · ⊆ Secⁿ(X) ⊆ · · · ⊆ P^N.

7 _まとめ

まとめ.

¶ ³

Secⁿ(V₂^m) = V (I_n+2(Ω)) (n ≥ 0).

• Main Theorem.

未解決だったch(k) = 2の部分を証明した.

• Application.

ch(k) = 2において, 1次元の場合に成り立つrankと次数付きBetti数の関係が, 高次元で成り立たない例を与えた.

µ ´

7 _まとめ

まとめ.

¶ ³

Secⁿ(V₂^m) = V (I_n+2(Ω)) (n ≥ 0).

• Main Theorem.

未解決だったch(k) = 2の部分を証明した.

• Application.

ch(k) = 2において, 1次元の場合に成り立つrankと次数付きBetti数の関係が, 高次元で成り立たない例を与えた.

µ ´

以上です. ありがとうございました.

References

[1] D. Eisenbud : The geometry of syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics, 229. Springer-Verlag, New York, 2005.

[2] R. Gattazzo : In characteristic p = 2 the Veronese variety V ^m ⊂ P^m(m+3)/2 and each of its generic projection is set-theoretic complete intersection, Complete intersections (Acireale, 1983), 221–228, Lecture Notes in Math., 1092, Springer, Berlin, 1984.

[3] P. Griffiths; J. Harris : Principles of algebraic geometry, Pure and Applied Mathematics.

Wiley-Interscience [John Wiley & Sons], New York, 1978.

[4] J. Harris : Algebraic Geometry: A First Course, Graduate Texts in Mathematics, 133.

Springer-Verlag, New York, 1992.

[5] R. Hartshorne : Algebraic Geometry, Graduate Texts in Mathematics, 52. Springer-Verlag, New York-Heidelberg, 1977.

[6] H. Kaji : 随伴多様体の射影幾何的魅力 , 数理解析研究所講究録 1460巻 2005年 23-32.

[7] V. Kanev : Chordal varieties of Veronese varieties and catalecticant matrices. Algebraic geometry, 9. J. Math. Sci. (New York) 94 (1999), no. 1, 1114–1125.

[8] A. Micali, O. E. Villamayor : Sur les alg`ebres de Clifford, Ann. Sci. ´Ecole Norm. Sup. (4) 1 (1968), 271-304

[9] E. Park: Projective curves of degree = codimension+2. Math. Z. 256 (2007), no. 3, 685–697.

[10] M. Pucci : The Veronese variety and catalecticant matrices. J. Algebra 202 (1998), no. 1, 72–95.

[11] F. L. Zak : Tangents and Secants of Algebraic Varieties, Translated from the Russian manuscript by the author. Translations of Mathematical Monographs, 127. American Mathematical Society, Providence, RI, 1993.

8 _{定理の証明に関して}

Lemma 1.

¶ ³

Secⁿ(V₂^m) ⊆ V (I_n+2(Ω)) (n ≥ 0).

µ ´

Lemma 2.

¶ ³

R = (r_ij) ∈ V (I₃(Ω)) ⊂ P^N,

(i) ∃i, r_ii ̸= 0 ⇒ ∃P ∈ V₂^m,∃Q ∈ V₂^m s.t. R ∈ ⟨P, Q⟩.

(ii) ∀i, r_ii = 0 ⇒ ∃P ∈ P^m s.t. R ∈ T^PV₂^m ⊆ Tan(V₂^m) = Sec(V₂^m).

µ ´

Lemma 3.

¶ ³

R = (r_ij) ∈ V (I_n+2(Ω)) ⊂ P^N, n ≥ 2,

(i) ∃i, r_ii ̸= 0 ⇒ ∃P ∈ V₂^m , ∃Q ∈ Secⁿ⁻¹(V₂^m) s.t. R ∈ ⟨P, Q⟩.

(ii) ∀i, r_ii = 0 ⇒ ∃P ∈ Tan(V₂^m) , ∃Q ∈ Secⁿ⁻²(V₂^m) s.t. R ∈ ⟨P, Q⟩.

µ ´

9 d = 3 _{についての予想}

Conjecture.

¶ ³

Secⁿ(V₃^m) = V (I_n+2(Cat(1, 3 − 1; m + 1))).

但し, Cat(1, 3 − 1; m + 1)はCatalecticant matricesのこと.

µ ´

ch(k) = 0, n = 1ならこれは正しいことが知られている.

Cat(1, 3 − 1; 2 + 1) :=



 Z₃₀₀ Z₂₁₀ Z₂₀₁ Z₁₂₀ Z₁₁₁ Z₁₀₂ Z₂₁₀ Z₁₂₀ Z₁₁₁ Z₀₃₀ Z₀₂₁ Z₀₁₂ Z₂₀₁ Z₁₁₁ Z₁₀₂ Z₀₂₁ Z₀₁₂ Z₀₀₃



 ,

Z := (Z₃₀₀ : Z₂₁₀ : Z₂₀₁ : Z₁₂₀ : Z₁₁₁ : Z₁₀₂ : Z₀₃₀ : Z₀₂₁ : Z₀₁₂ : Z₀₀₃) ∈ P⁹.