Symbolic Powers of Defining Ideals of Veronese Rings Fangu Chen - - PowerPoint PPT Presentation

Symbolic Powers of Defining Ideals of Veronese Rings

Fangu Chen joint work with Alan Tang REU summer 2019 at University of Michigan supervised by Professors Eric Canton, Elo´ ısa Grifo, Jack Jeffries

1

Overview

Ideals and Varieties Primary Decomposition Symbolic Powers Veronese Ring and Ideal Our Results

2

Ideals and Varieties

Variety

Example Let f ∈ R = R[x, y, z] defined by f (x, y, z) = 2x − y + 3z. Then V (f ) := {(x, y, z) ∈ R3|f (x, y, z) = 0} is a plane.

3

Variety

Example Let f ∈ R = R[x, y, z] defined by f (x, y, z) = 2x − y + 3z. Then V (f ) := {(x, y, z) ∈ R3|f (x, y, z) = 0} is a plane. Definition Let R = C[x1, . . . , xn] be a polynomial ring with C- coefficients and variables x1, . . . , xn. A variety is the set of common zeroes in Cn of a collection of polynomials fi ∈ R. The variety associated to the set {f1, . . . , fm} is written V (f1, . . . , fm).

3

Ideals and Varieties

If p is a point in the variety, then p is also a zero of any polynomial combination

m

• i=1

gi(x1, . . . , xn)fi(x1, . . . , xn)

4

Ideals and Varieties

If p is a point in the variety, then p is also a zero of any polynomial combination

m

• i=1

gi(x1, . . . , xn)fi(x1, . . . , xn) Observation The variety V (f1, . . . , fm) depends only on the ideal I generated by {f1, . . . , fm}.

4

Ideals and Varieties

Given a variety of an ideal J, we can form the set I(V (J)) of all polynomials vanishing on this variety. It is easy to check the set is an ideal.

5

Ideals and Varieties

Given a variety of an ideal J, we can form the set I(V (J)) of all polynomials vanishing on this variety. It is easy to check the set is an ideal. If f ∈ J, then by definition f (p) = 0 ∀p ∈ V (J). Hence, f ∈ I(V (J)), so J ⊆ I(V (J)).

5

Ideals and Varieties

Given a variety of an ideal J, we can form the set I(V (J)) of all polynomials vanishing on this variety. It is easy to check the set is an ideal. If f ∈ J, then by definition f (p) = 0 ∀p ∈ V (J). Hence, f ∈ I(V (J)), so J ⊆ I(V (J)). If J ⊆ C[x1, . . . , xn] is radical, then J = I(V (J)). Definition Given an ideal I in a ring R, the radical of I is √ I = {f ∈ R : f n ∈ I for some n ∈ N}. An ideal I is radical if √ I = I.

5

Primary Decomposition

Ideals and Varieties

If I, J in a polynomial ring R, then V (I ∩ J) = V (I) ∪ V (J).

6

Ideals and Varieties

If I, J in a polynomial ring R, then V (I ∩ J) = V (I) ∪ V (J). Example Consider the ideal I = (xz, yz) = (z) ∩ (x, y) in R[x, y, z]. In R3, V (z) corresponds to the xy-plane and V (x, y) corresponds to the z-axis, then V (I) = V (z) ∪ V (x, y).

6

Primary Decomposition

Given an ideal, we would like to decompose it as an intersection of simpler ideals.

7

Primary Decomposition

Given an ideal, we would like to decompose it as an intersection of simpler ideals. Example In the ring of integers Z, suppose a positive integer n has prime factorization n = pa1

1 . . . par r , then the ideal

(n) = (pa1

1 ) ∩ · · · ∩ (par r ).

For example, for 60 = 22 · 3 · 5, we have (60) = (4) ∩ (3) ∩ (5).

7

Primary Decomposition

Definition An ideal Q = (1) in a ring R is primary if fg ∈ Q implies either f ∈ Q or gm ∈ Q for some m ∈ N.

8

Primary Decomposition

Definition An ideal Q = (1) in a ring R is primary if fg ∈ Q implies either f ∈ Q or gm ∈ Q for some m ∈ N. Definition A primary decomposition of an ideal I consists of primary ideals Q1, . . . , Qn such that I = ∩n

i=1Qi. A primary decomposition

I = ∩n

i=1Qi is irredundant if ∩i=jQi = I for each j ∈ {1, . . . , n}

and √Qi =

• Qj for all i = j.

8

Primary Decomposition

Definition An ideal Q = (1) in a ring R is primary if fg ∈ Q implies either f ∈ Q or gm ∈ Q for some m ∈ N. Definition A primary decomposition of an ideal I consists of primary ideals Q1, . . . , Qn such that I = ∩n

i=1Qi. A primary decomposition

I = ∩n

i=1Qi is irredundant if ∩i=jQi = I for each j ∈ {1, . . . , n}

and √Qi =

• Qj for all i = j.

Theorem Any ideal in a Noetherian ring has a primary decomposition.

8

Symbolic Powers

Symbolic Powers

C[x]

• f vanishes at p
• f vanishes to
• rder ≥ k at p
• (x − p) | f or f ∈ (x − p)

x = p is a root of f

• (x − p)k | f or f ∈ (x − p)k

x = p is a root of f , f ′, . . . , f (k−1)

9

Symbolic Powers

C[x]

• f vanishes at p
• f vanishes to
• rder ≥ k at p
• (x − p) | f or f ∈ (x − p)

x = p is a root of f

• (x − p)k | f or f ∈ (x − p)k

x = p is a root of f , f ′, . . . , f (k−1) C[x1, . . . , xn]

• f vanishes to
• rder ≥ k at

(p1, . . . , pn)

• f ∈ (x1 − p1, . . . , xn − pn)k

x = (p1, . . . , pn) is a root of

∂d1 ∂xd1

1

. . . ∂dn

∂xdn

n f for all d1 + · · · + dn < k

9

Symbolic Powers

Definition (Zariski-Nagata) Let R = C[x1, . . . , xn] and I a radical ideal in R. Then the k-th symbolic power of I is I (k) = {f ∈ R|f vanishes to order ≥ k at every x ∈ V (I)}.

10

Symbolic Powers

Definition (Zariski-Nagata) Let R = C[x1, . . . , xn] and I a radical ideal in R. Then the k-th symbolic power of I is I (k) = {f ∈ R|f vanishes to order ≥ k at every x ∈ V (I)}. I (k) = {f ∈ R| ∂d1

∂xd1

1

. . . ∂dn

∂xdn

n f ∈ I for all d1 + · · · + dn < k}

10

Symbolic Powers

Definition (Zariski-Nagata) Let R = C[x1, . . . , xn] and I a radical ideal in R. Then the k-th symbolic power of I is I (k) = {f ∈ R|f vanishes to order ≥ k at every x ∈ V (I)}. I (k) = {f ∈ R| ∂d1

∂xd1

1

. . . ∂dn

∂xdn

n f ∈ I for all d1 + · · · + dn < k}

• C[x1, . . . , xn]
• radical ideal I
• symbolic power I (k)
• Cn
• vanish on variety V (I)
• vanish to order k over variety

10

Veronese Ring and Ideal

Veronese Ring and Ideal

Example Let S2 = C[x1, x2].

11

Veronese Ring and Ideal

Example Let S2 = C[x1, x2]. The 3-rd Veronese subring S2,3 := (C[x1, x2])3 = C[x3

1, x2 1x2, x1x2 2, x3 2]. 11

Veronese Ring and Ideal

Example Let S2 = C[x1, x2]. The 3-rd Veronese subring S2,3 := (C[x1, x2])3 = C[x3

1, x2 1x2, x1x2 2, x3 2].

Let R = C[t1, t2, t3, t4].

11

Veronese Ring and Ideal

Example Let S2 = C[x1, x2]. The 3-rd Veronese subring S2,3 := (C[x1, x2])3 = C[x3

1, x2 1x2, x1x2 2, x3 2].

Let R = C[t1, t2, t3, t4]. There is a surjective ring homomorphism π : R → S2,3 defined by t1 → x3

1, t2 → x2 1x2, t3 → x1x2 2, t4 → x3 2, and I2,3 := ker(π). 11

Veronese Ring and Ideal

Example Let S2 = C[x1, x2]. The 3-rd Veronese subring S2,3 := (C[x1, x2])3 = C[x3

1, x2 1x2, x1x2 2, x3 2].

Let R = C[t1, t2, t3, t4]. There is a surjective ring homomorphism π : R → S2,3 defined by t1 → x3

1, t2 → x2 1x2, t3 → x1x2 2, t4 → x3 2, and I2,3 := ker(π).

Note that π(t1t3 − t2

2) = x4 1x2 2 − x4 1x2 2 = 0

π(t1t4 − t2t3) = x3

1x3 2 − x3 1x3 2 = 0

π(t2t4 − t2

3) = x2 1x4 2 − x2 1x4 2 = 0 11

Veronese Ring and Ideal

Example Let S2 = C[x1, x2]. The 3-rd Veronese subring S2,3 := (C[x1, x2])3 = C[x3

1, x2 1x2, x1x2 2, x3 2].

Let R = C[t1, t2, t3, t4]. There is a surjective ring homomorphism π : R → S2,3 defined by t1 → x3

1, t2 → x2 1x2, t3 → x1x2 2, t4 → x3 2, and I2,3 := ker(π).

Note that π(t1t3 − t2

2) = x4 1x2 2 − x4 1x2 2 = 0

π(t1t4 − t2t3) = x3

1x3 2 − x3 1x3 2 = 0

π(t2t4 − t2

3) = x2 1x4 2 − x2 1x4 2 = 0

In fact, I2,3 = ker(π) = (t1t3 − t2

2, t1t4 − t2t3, t2t4 − t2 3), and

I2,3 ← → V (t1t3 − t2

2, t1t4 − t2t3, t2t4 − t2 3) ⊂ C4. 11

Veronese Ring and Ideal

Definition Let Sn = C[x1, . . . , xn]. The d-th Veronese in n variables Sn,d := (C[x1, . . . , xn])d = C[xd

1 , xd−1 1

x2, . . . , xd

n ]. Let

R = k[t1, t2, . . . , t(n+d−1

d

)]. There is a surjective ring homomorphism R − → Sn,d t1 − → xd

1

t2 − → xd−1

1

x2 . . . t(n+d−1

d

) − → xd

n

We write In,d for the kernel of this map.

12

Our Results

The ideal In,d (n, d ≥ 2) is generated by 2-minors of a n × n+d−2

d−1

• matrix.

13

The ideal In,d (n, d ≥ 2) is generated by 2-minors of a n × n+d−2

d−1

• matrix.

Example I2,3 = I2

• t(3,0)

t(2,1) t(1,2) t(2,1) t(1,2) t(0,3)

• where t(3,0) → x3

1, t(2,1) → x2 1x2,

t(1,2) → x1x2

2, t(0,3) → x3 2. 13

Example I2,4 = I2

• t(4,0)

t(3,1) t(2,2) t(1,3) t(3,1) t(2,2) t(1,3) t(0,4)

• .

14

Example I2,4 = I2

• t(4,0)

t(3,1) t(2,2) t(1,3) t(3,1) t(2,2) t(1,3) t(0,4)

• .

Example I3,3 = I2    t(3,0,0) t(2,1,0) t(2,0,1) t(1,2,0) t(1,1,1) t(1,0,2) t(2,1,0) t(1,2,0) t(1,1,1) t(0,3,0) t(0,2,1) t(0,1,2) t(2,0,1) t(1,1,1) t(1,0,2) t(0,2,1) t(0,1,2) t(0,0,3)   .

14

Macaulay2 Computations

We know I b

n,d ⊆ I (b) n,d for all b ∈ N.

We use Macaulay2 to find the smallest a such that I (a)

n,d ⊆ I b n,d for some n, d, b.

n d b a n d b a 2 4 2 3 3 2 2 3 3 4 3 4 4 5 4 5 5 7 5 7 6 8 6 8 7 9 7 9 2 5 2 3 8 11 3 4 3 3 2 3 4 5 3 4 2 6 2 3 4 2 2 3 3 4 3 4 2 7 2 3 5 2 2 3 3 4 2 8 2 3 15

I (2b−1)

2,d

⊆ I b

2,d for all b ∈ N. 16

I (2b−1)

2,d

⊆ I b

2,d for all b ∈ N.

in<(I (2b−1)

n,d

) ⊆ in<(I b

n,d) for all b ∈ N. 16

• For d = 1, 2, 3, I a

2,d = I (a) 2,d .

• For d ≥ 4 and a ≥ 2, an irredundant primary decomposition

for I a

2,d is

I a

2,d = I (a) 2,d ∩ m2a

where m = (t(d,0), t(d−1,1), t(d−2,2), . . . , t(1,d−1), t(0,d)) is the ideal of all the t variables.

17

For d ≥ 2, the least degree of an element in I (a)

2,d is

• ⌈ (d+2)a

d

⌉ if d is even;

• ⌈ (d+1)a

d−1 ⌉ if d is odd. 18

Example For d ≥ 4, the minimal degree of an element in the I (2)

2,d is 3. 19

Example For d ≥ 4, the minimal degree of an element in the I (2)

2,d is 3.

In particular, for d = 4, [I (2)

2,4 ]3 ⊆ I3

   t(4,0) t(3,1) t(2,2) t(3,1) t(2,2) t(1,3) t(2,2) t(1,3) t(0,4)   

19

Example I2,4 = I2

• t(4,0)

t(3,1) t(2,2) t(1,3) t(3,1) t(2,2) t(1,3) t(0,4)

• = I2
• t1

t2 t3 t4 t2 t3 t4 t5

• .

Let f = det    t1 t2 t3 t2 t3 t4 t3 t4 t5   .

• 1. I (2)

2,4 = I 2 2,4 + I3

   t1 t2 t3 t2 t3 t4 t3 t4 t5    = I 2

2,4 + (f ).

• 2. I (2k+1)

2,4

= I (2k)

2,4 I2,4 for all k 1.

• 3. I (2k+2)

2,4

= (I (2)

2,4 )k+1 for all k 1. 20

Thank You!