Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches...
Transcript of Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches...
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Moving Lines, Soccer, and Rees Algebras
Carlos D’Andrea
Ideals, Varieties, Applications – Amherst
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Back in 2012...
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Moving Lines, Soccer, and Rees Algebras
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David: Now that I am in Barcelona...
I want to visit the 2 churches
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David: Now that I am in Barcelona...I want to visit the 2 churches
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David: Now that I am in Barcelona...I want to visit the 2 churches
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Church # 1: Sagrada Família
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Church #2?
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Church #2: Camp Nou!!
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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“The” score of the (20th) Century
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Moving Lines, Soccer, and Rees Algebras
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Let us enjoy it
Diego’s goal
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David’s score of Last (and this) Century
Geometric Modelingvs
Algebraic Geometry
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David’s score of Last (and this) Century
Geometric Modelingvs
Algebraic Geometry
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David’s goal of Last (and this) Century
Geometric Modelingand
Algebraic Geometry
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Rolling the ball...
(Sederberg & Chen 1995)The implicit equation of a rational quartic can be
computed as a 2× 2 determinant.If the curve has a triple point, then one row is linear
and the other is cubic.Otherwise, both rows are quadratic.
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Rolling the ball...
(Sederberg & Chen 1995)
The implicit equation of a rational quartic can becomputed as a 2× 2 determinant.
If the curve has a triple point, then one row is linearand the other is cubic.
Otherwise, both rows are quadratic.
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Rolling the ball...
(Sederberg & Chen 1995)The implicit equation of a rational quartic can be
computed as a 2× 2 determinant.
If the curve has a triple point, then one row is linearand the other is cubic.
Otherwise, both rows are quadratic.
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 17: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/17.jpg)
Rolling the ball...
(Sederberg & Chen 1995)The implicit equation of a rational quartic can be
computed as a 2× 2 determinant.If the curve has a triple point, then one row is linear
and the other is cubic.
Otherwise, both rows are quadratic.
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 18: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/18.jpg)
Rolling the ball...
(Sederberg & Chen 1995)The implicit equation of a rational quartic can be
computed as a 2× 2 determinant.If the curve has a triple point, then one row is linear
and the other is cubic.Otherwise, both rows are quadratic.
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 19: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/19.jpg)
A quartic with a triple point
φ(t0, t1) = (t04 − t1
4 : −t02t12 : t0t1
3)F (X0,X1,X2) = X 4
2 − X 41 − X0X1X
22
L1,1(T ,X ) = T0X2 + T1X1L1,3(T ,X ) = T0(X
31 + X0X
22 ) + T1 X
32(
X2 X1X 3
1 + X0X22 X 3
2
)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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A quartic with a triple point
φ(t0, t1) = (t04 − t1
4 : −t02t12 : t0t1
3)F (X0,X1,X2) = X 4
2 − X 41 − X0X1X
22
L1,1(T ,X ) = T0X2 + T1X1L1,3(T ,X ) = T0(X
31 + X0X
22 ) + T1 X
32(
X2 X1X 3
1 + X0X22 X 3
2
)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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A quartic with a triple point
φ(t0, t1) = (t04 − t1
4 : −t02t12 : t0t1
3)F (X0,X1,X2) = X 4
2 − X 41 − X0X1X
22
L1,1(T ,X ) = T0X2 + T1X1L1,3(T ,X ) = T0(X
31 + X0X
22 ) + T1 X
32(
X2 X1X 3
1 + X0X22 X 3
2
)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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A quartic with a triple point
φ(t0, t1) = (t04 − t1
4 : −t02t12 : t0t1
3)F (X0,X1,X2) = X 4
2 − X 41 − X0X1X
22
L1,1(T ,X ) = T0X2 + T1X1L1,3(T ,X ) = T0(X
31 + X0X
22 ) + T1 X
32(
X2 X1X 3
1 + X0X22 X 3
2
)Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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A quartic without triple points
φ(t0 : t1) = (t04 : 6t02t1
2 − 4t14 : 4t03t1 − 4t0t13)F (X ) = X 4
2 +4X0X31 +2X0X1X
22 −16X 2
0X21 −6X 2
0X22 +16X 3
0X1
L1,2(T ,X ) = T0(X1X2 − X0X2) + T1(−X 22 − 2X0X1 + 4X 2
0 )
L̃1,2(T ,X ) = T0(X21 + 1
2X22 − 2X0X1) + T1(X0X2 − X1X2)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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A quartic without triple points
φ(t0 : t1) = (t04 : 6t02t1
2 − 4t14 : 4t03t1 − 4t0t13)F (X ) = X 4
2 +4X0X31 +2X0X1X
22 −16X 2
0X21 −6X 2
0X22 +16X 3
0X1
L1,2(T ,X ) = T0(X1X2 − X0X2) + T1(−X 22 − 2X0X1 + 4X 2
0 )
L̃1,2(T ,X ) = T0(X21 + 1
2X22 − 2X0X1) + T1(X0X2 − X1X2)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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A quartic without triple points
φ(t0 : t1) = (t04 : 6t02t1
2 − 4t14 : 4t03t1 − 4t0t13)F (X ) = X 4
2 +4X0X31 +2X0X1X
22 −16X 2
0X21 −6X 2
0X22 +16X 3
0X1
L1,2(T ,X ) = T0(X1X2 − X0X2) + T1(−X 22 − 2X0X1 + 4X 2
0 )
L̃1,2(T ,X ) = T0(X21 + 1
2X22 − 2X0X1) + T1(X0X2 − X1X2)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Parametrizing the circle
φ : P1 −→ P2
(t0 : t1) 7−→(t0
2 + t12 : t0
2 − t12 : 2t0t1
)(a(t) : b(t) : c(t))
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Parametrizing the circle
φ : P1 −→ P2
(t0 : t1) 7−→(t0
2 + t12 : t0
2 − t12 : 2t0t1
)(a(t) : b(t) : c(t))
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Parametrizing the circle
φ : P1 −→ P2
(t0 : t1) 7−→(t0
2 + t12 : t0
2 − t12 : 2t0t1
)(a(t) : b(t) : c(t))
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Implicitization
X2a(T )− X0c(T ) = X2T20 − 2X0T0T1 + X2T
21
X2b(T )− X1c(T ) = X2T20 − 2X1T0T1 − X2T
21
ResT(X2 · a(T )− X0 · c(T ),X2 · b(T )− X1 · c(T )
)=
det
X2 −2X0 X2 00 X2 −2X0 X2
X2 −2X1 −X2 00 X2 −2X1 −X2
= 4X 22 (X
21 + X 2
2 − X 20 )
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Implicitization
X2a(T )− X0c(T ) = X2T20 − 2X0T0T1 + X2T
21
X2b(T )− X1c(T ) = X2T20 − 2X1T0T1 − X2T
21
ResT(X2 · a(T )− X0 · c(T ),X2 · b(T )− X1 · c(T )
)=
det
X2 −2X0 X2 00 X2 −2X0 X2
X2 −2X1 −X2 00 X2 −2X1 −X2
= 4X 22 (X
21 + X 2
2 − X 20 )
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Implicitization
X2a(T )− X0c(T ) = X2T20 − 2X0T0T1 + X2T
21
X2b(T )− X1c(T ) = X2T20 − 2X1T0T1 − X2T
21
ResT(X2 · a(T )− X0 · c(T ),X2 · b(T )− X1 · c(T )
)=
det
X2 −2X0 X2 00 X2 −2X0 X2
X2 −2X1 −X2 00 X2 −2X1 −X2
= 4X 22 (X
21 + X 2
2 − X 20 )
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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You can do it better!
ϕ1(T ,X ) = (−T1 : −T1 : T0)= X2T0 − (X0 + X1)T1
ψ1(T ,X ) = (−T0 : T0 : T1)(−X0 + X1)T0 + X2T1
det(
X2 −X0 − X1−X0 + X1 X2
)= X 2
1 + X 22 − X 2
0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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You can do it better!
ϕ1(T ,X ) = (−T1 : −T1 : T0)= X2T0 − (X0 + X1)T1
ψ1(T ,X ) = (−T0 : T0 : T1)
(−X0 + X1)T0 + X2T1
det(
X2 −X0 − X1−X0 + X1 X2
)= X 2
1 + X 22 − X 2
0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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You can do it better!
ϕ1(T ,X ) = (−T1 : −T1 : T0)= X2T0 − (X0 + X1)T1
ψ1(T ,X ) = (−T0 : T0 : T1)(−X0 + X1)T0 + X2T1
det(
X2 −X0 − X1−X0 + X1 X2
)= X 2
1 + X 22 − X 2
0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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You can do it better!
ϕ1(T ,X ) = (−T1 : −T1 : T0)= X2T0 − (X0 + X1)T1
ψ1(T ,X ) = (−T0 : T0 : T1)(−X0 + X1)T0 + X2T1
det(
X2 −X0 − X1−X0 + X1 X2
)= X 2
1 + X 22 − X 2
0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 36: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/36.jpg)
Moving Lines!
A moving line
L(T0,T1,X0,X1,X2) = v0(T )X0+v1(T )X1+v2(T )X2
follows the parametrization iff
L(T0,T1, a(T ), b(T ), c(T )) = 0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving Lines!
A moving line
L(T0,T1,X0,X1,X2) = v0(T )X0+v1(T )X1+v2(T )X2
follows the parametrization iff
L(T0,T1, a(T ), b(T ), c(T )) = 0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving Lines!
A moving line
L(T0,T1,X0,X1,X2) = v0(T )X0+v1(T )X1+v2(T )X2
follows the parametrization iff
L(T0,T1, a(T ), b(T ), c(T )) = 0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 39: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/39.jpg)
For the unit circle...
L1(T ,X ) = −2T02T1X0 + 0X1 + (T0
3 + T0T12)X2
L2(T ,X ) = −2T0T12X0 + 0X1 + (T0
2T1 + T13)X2
L3(T ,X ) = 0X0 − 2T02T1X1 + (T0
3 − T0T12)X2
L4(T ,X ) = 0X0 − 2T0T12X1 + (T0
2T1 − T13)X2
X2 −2X0 X2 00 X2 −2X0 X2
X2 −2X1 −X2 00 X2 −2X1 −X2
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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For the unit circle...
L1(T ,X ) = −2T02T1X0 + 0X1 + (T0
3 + T0T12)X2
L2(T ,X ) = −2T0T12X0 + 0X1 + (T0
2T1 + T13)X2
L3(T ,X ) = 0X0 − 2T02T1X1 + (T0
3 − T0T12)X2
L4(T ,X ) = 0X0 − 2T0T12X1 + (T0
2T1 − T13)X2
X2 −2X0 X2 00 X2 −2X0 X2
X2 −2X1 −X2 00 X2 −2X1 −X2
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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In general
The determinant of a “matrix ofmoving lines” is always a multiple of
the implicit equationL11(X ) L12(X ) . . . L1k(X )L21(X ) L22(X ) . . . L2k(X )
... ... ... ...Lk1(X ) Lk2(X ) . . . Lkk(X )
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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I dare you...
find a non-singular square matrix ofmoving linesfind one where the determinant isexactly the implicit equation
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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I dare you...
find a non-singular square matrix ofmoving lines
find one where the determinant isexactly the implicit equation
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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I dare you...
find a non-singular square matrix ofmoving linesfind one where the determinant isexactly the implicit equation
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David gets the ball...
Cox, Sederberg, ChenThe moving line ideal basis of planar rational curves
CAGD 98
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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David gets the ball...
Cox, Sederberg, ChenThe moving line ideal basis of planar rational curves
CAGD 98
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving Lines are Syzygies
The homogeneous idealI =
(a(T ), b(T ), c(T )
)⊂ K[T0,T1] has a
Hilbert-Burch resolution of the type
0→ K[T ]2(ϕµ,ψd−µ)
t
−→ K[T ]3(a,b,c)−→ K[T ]
The moving lines are a free K[T ]-module of rank 2The determinant of Sylv(ϕµ, ψd−µ) gives exactly the
implicit equation
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving Lines are Syzygies
The homogeneous idealI =
(a(T ), b(T ), c(T )
)⊂ K[T0,T1] has a
Hilbert-Burch resolution of the type
0→ K[T ]2(ϕµ,ψd−µ)
t
−→ K[T ]3(a,b,c)−→ K[T ]
The moving lines are a free K[T ]-module of rank 2The determinant of Sylv(ϕµ, ψd−µ) gives exactly the
implicit equation
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving Lines are Syzygies
The homogeneous idealI =
(a(T ), b(T ), c(T )
)⊂ K[T0,T1] has a
Hilbert-Burch resolution of the type
0→ K[T ]2(ϕµ,ψd−µ)
t
−→ K[T ]3(a,b,c)−→ K[T ]
The moving lines are a free K[T ]-module of rank 2The determinant of Sylv(ϕµ, ψd−µ) gives exactly the
implicit equation
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometrically...
There are two parametrizationsϕµ(t0 : t1), ψd−µ(t0 : t1) such that
φ(t0 : t1) = ϕµ(t0 : t1) ∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometrically...
There are two parametrizationsϕµ(t0 : t1), ψd−µ(t0 : t1) such that
φ(t0 : t1) = ϕµ(t0 : t1) ∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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“Factorization” of the unit circle
ϕ1(t0 : t1) = (−t1 : −t1 : t0)ψ1(t0 : t1) = (−t0 : t0 : t1)∣∣∣∣∣∣
i j k−t1 −t1 t0−t0 t0 t1
∣∣∣∣∣∣ = (− t02 − t1
2 : t12 − t0
2 : −2t0t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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“Factorization” of the unit circle
ϕ1(t0 : t1) = (−t1 : −t1 : t0)ψ1(t0 : t1) = (−t0 : t0 : t1)
∣∣∣∣∣∣i j k−t1 −t1 t0−t0 t0 t1
∣∣∣∣∣∣ = (− t02 − t1
2 : t12 − t0
2 : −2t0t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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“Factorization” of the unit circle
ϕ1(t0 : t1) = (−t1 : −t1 : t0)ψ1(t0 : t1) = (−t0 : t0 : t1)∣∣∣∣∣∣
i j k−t1 −t1 t0−t0 t0 t1
∣∣∣∣∣∣ = (− t02 − t1
2 : t12 − t0
2 : −2t0t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry II
(µ ≤ d − µ)µ = 1 ⇐⇒ there is one point of multiplicityd − 1If there is a point of multiplicity > µ,then there isonly one and it has multiplicity d − µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry II
(µ ≤ d − µ)µ = 1 ⇐⇒ there is one point of multiplicityd − 1
If there is a point of multiplicity > µ,then there isonly one and it has multiplicity d − µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry II
(µ ≤ d − µ)µ = 1 ⇐⇒ there is one point of multiplicityd − 1If there is a point of multiplicity > µ,
then there isonly one and it has multiplicity d − µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry II
(µ ≤ d − µ)µ = 1 ⇐⇒ there is one point of multiplicityd − 1If there is a point of multiplicity > µ,then there isonly one and it has multiplicity d − µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry III
(Bernardi-Gimigliano-Idà 2015, Madsen 2015)The curve has a point of multiplicity
d − µ ⇐⇒ µ′ = 0φ(t0 : t1)
=ϕµ(t0 : t1) ∧ ψd−µ(t0 : t1)
=(ϕ′µ′(t0 : t1) ∧ ϕ′µ−µ′(t0 : t1)
)∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry III
(Bernardi-Gimigliano-Idà 2015, Madsen 2015)The curve has a point of multiplicity
d − µ ⇐⇒ µ′ = 0
φ(t0 : t1)=
ϕµ(t0 : t1) ∧ ψd−µ(t0 : t1)=(
ϕ′µ′(t0 : t1) ∧ ϕ′µ−µ′(t0 : t1))
∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry III
(Bernardi-Gimigliano-Idà 2015, Madsen 2015)The curve has a point of multiplicity
d − µ ⇐⇒ µ′ = 0φ(t0 : t1)
=ϕµ(t0 : t1) ∧ ψd−µ(t0 : t1)
=(ϕ′µ′(t0 : t1) ∧ ϕ′µ−µ′(t0 : t1)
)∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Geometry III
(Bernardi-Gimigliano-Idà 2015, Madsen 2015)The curve has a point of multiplicity
d − µ ⇐⇒ µ′ = 0φ(t0 : t1)
=ϕµ(t0 : t1) ∧ ψd−µ(t0 : t1)
=(ϕ′µ′(t0 : t1) ∧ ϕ′µ−µ′(t0 : t1)
)∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Conics instead of lines?
O(T )X 20 + P(T )X0X1 +Q(T )X0X2 +R(T )X 2
1 +S(T )X1X2 + T (T )X 2
2 ∈ K[T ,X ]is a moving conic following the parametrization ifO(T )a(T )2 + P(T )a(T )b(T ) +Q(T )a(T )c(T ) +R(T )b(T )2 + S(T )b(T )c(T ) + T (T )c(T )2 = 0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Conics instead of lines?
O(T )X 20 + P(T )X0X1 +Q(T )X0X2 +R(T )X 2
1 +S(T )X1X2 + T (T )X 2
2 ∈ K[T ,X ]is a moving conic following the parametrization if
O(T )a(T )2 + P(T )a(T )b(T ) +Q(T )a(T )c(T ) +R(T )b(T )2 + S(T )b(T )c(T ) + T (T )c(T )2 = 0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Conics instead of lines?
O(T )X 20 + P(T )X0X1 +Q(T )X0X2 +R(T )X 2
1 +S(T )X1X2 + T (T )X 2
2 ∈ K[T ,X ]is a moving conic following the parametrization ifO(T )a(T )2 + P(T )a(T )b(T ) +Q(T )a(T )c(T ) +R(T )b(T )2 + S(T )b(T )c(T ) + T (T )c(T )2 = 0
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving lines, conics, cubics,...
Syz(a, b, c) moving linesSyz(a2, b2, c2, ab, ac, bc) moving conicsSyz(a3, b3, c3, a2b, . . .) moving cubics
... ...
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving lines, conics, cubics,...
Syz(a, b, c) moving lines
Syz(a2, b2, c2, ab, ac, bc) moving conicsSyz(a3, b3, c3, a2b, . . .) moving cubics
... ...
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving lines, conics, cubics,...
Syz(a, b, c) moving linesSyz(a2, b2, c2, ab, ac, bc) moving conics
Syz(a3, b3, c3, a2b, . . .) moving cubics... ...
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Moving lines, conics, cubics,...
Syz(a, b, c) moving linesSyz(a2, b2, c2, ab, ac, bc) moving conicsSyz(a3, b3, c3, a2b, . . .) moving cubics
... ...
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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The method of moving curves
The implicit equation can be computed as thedeterminant of a small matrix with entries∣∣∣∣∣∣∣∣
some moving linessome moving conicssome moving cubics
· · ·
∣∣∣∣∣∣∣∣The more singular the curve, the simpler the
description of the determinant
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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The method of moving curves
The implicit equation can be computed as thedeterminant of a small matrix with entries
∣∣∣∣∣∣∣∣some moving linessome moving conicssome moving cubics
· · ·
∣∣∣∣∣∣∣∣The more singular the curve, the simpler the
description of the determinant
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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The method of moving curves
The implicit equation can be computed as thedeterminant of a small matrix with entries∣∣∣∣∣∣∣∣
some moving linessome moving conicssome moving cubics
· · ·
∣∣∣∣∣∣∣∣
The more singular the curve, the simpler thedescription of the determinant
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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The method of moving curves
The implicit equation can be computed as thedeterminant of a small matrix with entries∣∣∣∣∣∣∣∣
some moving linessome moving conicssome moving cubics
· · ·
∣∣∣∣∣∣∣∣The more singular the curve, the simpler the
description of the determinant
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Extreme case
If the curve has a point of multiplicity d − 1the implicit equation is always a 2× 2 determinant∣∣∣∣ L1,1(X ) L′1,1(X )
L1,d−1(X ) L′1,d−1(X )
∣∣∣∣
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Extreme case
If the curve has a point of multiplicity d − 1
the implicit equation is always a 2× 2 determinant∣∣∣∣ L1,1(X ) L′1,1(X )
L1,d−1(X ) L′1,d−1(X )
∣∣∣∣
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Extreme case
If the curve has a point of multiplicity d − 1the implicit equation is always a 2× 2 determinant∣∣∣∣ L1,1(X ) L′1,1(X )
L1,d−1(X ) L′1,d−1(X )
∣∣∣∣Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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In general, we do not know..
∣∣∣∣∣∣∣∣which moving lines?which moving conics?which moving cubics?
· · ·
∣∣∣∣∣∣∣∣
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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In general, we do not know..
∣∣∣∣∣∣∣∣which moving lines?which moving conics?which moving cubics?
· · ·
∣∣∣∣∣∣∣∣
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
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Spatial Curves/Surfaces
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 80: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/80.jpg)
Rational Surfaces
φS : P2 99K P3
t = (t0 : t1 : t2) 7−→(a(t) : b(t) : c(t) : d(t)
)
There are base points!
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 81: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/81.jpg)
Rational Surfaces
φS : P2 99K P3
t = (t0 : t1 : t2) 7−→(a(t) : b(t) : c(t) : d(t)
)
There are base points!
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 82: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/82.jpg)
And David goes...
Cox, Goldman, ZhangOn the validity of implicitization by moving quadrics
of rational surfaces with no base pointsJSC 2000
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 83: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/83.jpg)
And David goes...
Cox, Goldman, ZhangOn the validity of implicitization by moving quadrics
of rational surfaces with no base pointsJSC 2000
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 84: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/84.jpg)
Main resultThe “method of moving quadrics”works without base points, for properparametrizations with the “right num-ber” of moving planes of degree d − 1
Proof uses Commutative Algebra(Cohen-Macaulay rings) and Algebraic
Geometry (sheaf cohomology)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 85: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/85.jpg)
Main resultThe “method of moving quadrics”works without base points, for properparametrizations with the “right num-ber” of moving planes of degree d − 1
Proof uses Commutative Algebra(Cohen-Macaulay rings) and Algebraic
Geometry (sheaf cohomology)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 86: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/86.jpg)
Main resultThe “method of moving quadrics”works without base points, for properparametrizations with the “right num-ber” of moving planes of degree d − 1
Proof uses Commutative Algebra(Cohen-Macaulay rings) and Algebraic
Geometry (sheaf cohomology)Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 87: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/87.jpg)
Got the community’s attention
Laurent BuséMarc ChardinDWilliam HoffmanJean-Pierre JouanolouAmit KhetanHao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 88: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/88.jpg)
Got the community’s attention
Laurent Busé
Marc ChardinDWilliam HoffmanJean-Pierre JouanolouAmit KhetanHao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 89: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/89.jpg)
Got the community’s attention
Laurent BuséMarc Chardin
DWilliam HoffmanJean-Pierre JouanolouAmit KhetanHao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 90: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/90.jpg)
Got the community’s attention
Laurent BuséMarc ChardinD
William HoffmanJean-Pierre JouanolouAmit KhetanHao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 91: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/91.jpg)
Got the community’s attention
Laurent BuséMarc ChardinDWilliam Hoffman
Jean-Pierre JouanolouAmit KhetanHao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 92: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/92.jpg)
Got the community’s attention
Laurent BuséMarc ChardinDWilliam HoffmanJean-Pierre Jouanolou
Amit KhetanHao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 93: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/93.jpg)
Got the community’s attention
Laurent BuséMarc ChardinDWilliam HoffmanJean-Pierre JouanolouAmit Khetan
Hao Hao Wang. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 94: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/94.jpg)
Got the community’s attention
Laurent BuséMarc ChardinDWilliam HoffmanJean-Pierre JouanolouAmit KhetanHao Hao Wang
. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 95: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/95.jpg)
Got the community’s attention
Laurent BuséMarc ChardinDWilliam HoffmanJean-Pierre JouanolouAmit KhetanHao Hao Wang. . .Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 96: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/96.jpg)
And David goes...
Chen, Cox, LiuThe µ-basis and implicitization of a rational
parametric surfaceJSC 2005
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 97: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/97.jpg)
And David goes...
Chen, Cox, LiuThe µ-basis and implicitization of a rational
parametric surfaceJSC 2005
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 98: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/98.jpg)
Main resultEvery parametrization of a rational sur-face has a µ-basis.
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 99: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/99.jpg)
Nicolás BotbolYairon Cid RuizAlicia DickensteinEliana DuarteHoon HongIrina KoganHal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 100: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/100.jpg)
Nicolás Botbol
Yairon Cid RuizAlicia DickensteinEliana DuarteHoon HongIrina KoganHal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 101: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/101.jpg)
Nicolás BotbolYairon Cid Ruiz
Alicia DickensteinEliana DuarteHoon HongIrina KoganHal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 102: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/102.jpg)
Nicolás BotbolYairon Cid RuizAlicia Dickenstein
Eliana DuarteHoon HongIrina KoganHal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 103: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/103.jpg)
Nicolás BotbolYairon Cid RuizAlicia DickensteinEliana Duarte
Hoon HongIrina KoganHal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 104: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/104.jpg)
Nicolás BotbolYairon Cid RuizAlicia DickensteinEliana DuarteHoon Hong
Irina KoganHal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 105: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/105.jpg)
Nicolás BotbolYairon Cid RuizAlicia DickensteinEliana DuarteHoon HongIrina Kogan
Hal Schenck. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 106: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/106.jpg)
Nicolás BotbolYairon Cid RuizAlicia DickensteinEliana DuarteHoon HongIrina KoganHal Schenck
. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 107: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/107.jpg)
Nicolás BotbolYairon Cid RuizAlicia DickensteinEliana DuarteHoon HongIrina KoganHal Schenck. . .Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 108: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/108.jpg)
And David goes...
Cox, Kustin, Polini, UlrichA study of singularities on rational curves via syzygies
MAMS 2013
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 109: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/109.jpg)
And David goes...
Cox, Kustin, Polini, UlrichA study of singularities on rational curves via syzygies
MAMS 2013
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 110: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/110.jpg)
Main results
Study of the singularities of arational curve via the Hilbert-BurchmatrixClassification of curves with pointsof “highest” multiplicity. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 111: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/111.jpg)
Main results
Study of the singularities of arational curve via the Hilbert-Burchmatrix
Classification of curves with pointsof “highest” multiplicity. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 112: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/112.jpg)
Main results
Study of the singularities of arational curve via the Hilbert-BurchmatrixClassification of curves with pointsof “highest” multiplicity
. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 113: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/113.jpg)
Main results
Study of the singularities of arational curve via the Hilbert-BurchmatrixClassification of curves with pointsof “highest” multiplicity. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 114: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/114.jpg)
Alessandra Bernardi
Eduardo Casas-AlveroLouiza FouliTony IarrobinoKuei-Nuan LinAron SimisWolmer Vasconcelos. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 115: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/115.jpg)
Alessandra BernardiEduardo Casas-Alvero
Louiza FouliTony IarrobinoKuei-Nuan LinAron SimisWolmer Vasconcelos. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 116: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/116.jpg)
Alessandra BernardiEduardo Casas-AlveroLouiza Fouli
Tony IarrobinoKuei-Nuan LinAron SimisWolmer Vasconcelos. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 117: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/117.jpg)
Alessandra BernardiEduardo Casas-AlveroLouiza FouliTony Iarrobino
Kuei-Nuan LinAron SimisWolmer Vasconcelos. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 118: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/118.jpg)
Alessandra BernardiEduardo Casas-AlveroLouiza FouliTony IarrobinoKuei-Nuan Lin
Aron SimisWolmer Vasconcelos. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 119: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/119.jpg)
Alessandra BernardiEduardo Casas-AlveroLouiza FouliTony IarrobinoKuei-Nuan LinAron Simis
Wolmer Vasconcelos. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 120: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/120.jpg)
Alessandra BernardiEduardo Casas-AlveroLouiza FouliTony IarrobinoKuei-Nuan LinAron SimisWolmer Vasconcelos
. . .
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 121: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/121.jpg)
Alessandra BernardiEduardo Casas-AlveroLouiza FouliTony IarrobinoKuei-Nuan LinAron SimisWolmer Vasconcelos. . .Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 122: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/122.jpg)
And David scores
CoxThe moving curve ideal and the Rees algebra
TCS 2008
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 123: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/123.jpg)
And David scores
CoxThe moving curve ideal and the Rees algebra
TCS 2008
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 124: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/124.jpg)
What did we learn there?
Kφ := {Moving curves following φ} =homogeneous elements in the kernel of
K[T0,T1,X0,X1,X2] → K[T0,T1, s]Ti 7→ Ti
X0 7→ a(T0,T1)sX1 7→ b(T0,T1)sX2 7→ c(T0,T1)s
“The ideal of moving curves following φ”
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 125: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/125.jpg)
What did we learn there?
Kφ := {Moving curves following φ} =homogeneous elements in the kernel of
K[T0,T1,X0,X1,X2] → K[T0,T1, s]Ti 7→ Ti
X0 7→ a(T0,T1)sX1 7→ b(T0,T1)sX2 7→ c(T0,T1)s
“The ideal of moving curves following φ”
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 126: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/126.jpg)
Method of moving curves revisited
The implicit equation should be obtained as thedeterminant of a matrix with∣∣∣∣∣∣∣∣
· · ·some minimal generators of Kφ
and relations among them· · ·
∣∣∣∣∣∣∣∣The more singular the curve, the simpler the
description of Kφ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 127: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/127.jpg)
Method of moving curves revisited
The implicit equation should be obtained as thedeterminant of a matrix with
∣∣∣∣∣∣∣∣· · ·
some minimal generators of Kφand relations among them
· · ·
∣∣∣∣∣∣∣∣The more singular the curve, the simpler the
description of Kφ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 128: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/128.jpg)
Method of moving curves revisited
The implicit equation should be obtained as thedeterminant of a matrix with∣∣∣∣∣∣∣∣
· · ·some minimal generators of Kφ
and relations among them· · ·
∣∣∣∣∣∣∣∣
The more singular the curve, the simpler thedescription of Kφ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 129: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/129.jpg)
Method of moving curves revisited
The implicit equation should be obtained as thedeterminant of a matrix with∣∣∣∣∣∣∣∣
· · ·some minimal generators of Kφ
and relations among them· · ·
∣∣∣∣∣∣∣∣The more singular the curve, the simpler the
description of KφCarlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 130: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/130.jpg)
Goal!
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 131: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/131.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ for any φ
Known forµ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 132: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/132.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ
for any φKnown for
µ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 133: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/133.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ for any φ
Known forµ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 134: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/134.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ for any φ
Known forµ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)
µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 135: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/135.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ for any φ
Known forµ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)
(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 136: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/136.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ for any φ
Known forµ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 137: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/137.jpg)
New Problem(s)
Compute a minimal system ofgenerators of Kφ for any φ
Known forµ = 1 (Hong-Simis-Vasconcelos,Cox-Hoffmann-Wang, Busé, Cortadellas-D)µ = 2 (Busé, Cortadellas-D, Kustin-Polini-Ulrich)(Kφ)(1,2) 6= 0 (Cortadellas- D)
Monomial Parametrizations (Cortadellas-D)Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 138: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/138.jpg)
A coarser problem
Compute n0(Kφ), the number ofminimal generators of Kφ
Show that if φ is “more singular” thanφ′ then n0(Kφ) ≤ n0(Kφ′)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 139: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/139.jpg)
A coarser problem
Compute n0(Kφ), the number ofminimal generators of Kφ
Show that if φ is “more singular” thanφ′ then n0(Kφ) ≤ n0(Kφ′)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 140: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/140.jpg)
A coarser problem
Compute n0(Kφ), the number ofminimal generators of Kφ
Show that if φ is “more singular” thanφ′ then n0(Kφ) ≤ n0(Kφ′)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 141: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/141.jpg)
Example: µ = 2
The curve has eitherone point of multiplicity d − 2n0 = O
(d2
)(Cortadellas-D, Kustin-Polini-Ulrich)or only double pointsn0 = O
(d2
2
)(Busé)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 142: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/142.jpg)
Example: µ = 2
The curve has eitherone point of multiplicity d − 2n0 = O
(d2
)(Cortadellas-D, Kustin-Polini-Ulrich)
or only double pointsn0 = O
(d2
2
)(Busé)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 143: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/143.jpg)
Example: µ = 2
The curve has eitherone point of multiplicity d − 2n0 = O
(d2
)(Cortadellas-D, Kustin-Polini-Ulrich)or only double pointsn0 = O
(d2
2
)(Busé)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 144: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/144.jpg)
Last Progress
Madsen, arXiv:1511.04073Cortadellas–Cox–D, JPAA toappear
Describe the bi-degrees of all minimalgenerators in T -degree larger than or
equal to µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 145: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/145.jpg)
Last Progress
Madsen, arXiv:1511.04073Cortadellas–Cox–D, JPAA toappear
Describe the bi-degrees of all minimalgenerators in T -degree larger than or
equal to µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 146: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/146.jpg)
Last Progress
Madsen, arXiv:1511.04073Cortadellas–Cox–D, JPAA toappear
Describe the bi-degrees of all minimalgenerators in T -degree larger than or
equal to µ
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 147: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/147.jpg)
We “got” this region
φ(t0 : t1)=(
ϕ′µ1(t0 : t1) ∧ ψ′µ2
(t0 : t1))∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 148: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/148.jpg)
We “got” this region
φ(t0 : t1)=(
ϕ′µ1(t0 : t1) ∧ ψ′µ2
(t0 : t1))∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 149: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/149.jpg)
We “got” this region
φ(t0 : t1)=(
ϕ′µ1(t0 : t1) ∧ ψ′µ2
(t0 : t1))∧ ψd−µ(t0 : t1)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 150: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/150.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 151: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/151.jpg)
Further results
Implicitization
Quadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 152: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/152.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)
Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 153: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/153.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)
Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 154: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/154.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 155: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/155.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras
“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 156: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/156.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)
de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 157: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/157.jpg)
Further results
ImplicitizationQuadratic and cubic surfaces (Chen-Shen-Deng)Steiner surfaces (Wang-Chen)Revolution surfaces (Shi-Goldman). . .
Rees Algebras“Monoid” Surfaces (Cortadellas - D)de Jonquières surfaces (Hassanzadeh- Simis)D-modules approach (Cid Ruiz)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 158: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/158.jpg)
This was quite a journey...
(Do not clap yet!)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 159: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/159.jpg)
This was quite a journey...
(Do not clap yet!)
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 160: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/160.jpg)
A gift for David
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 161: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/161.jpg)
A gift for David
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 162: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/162.jpg)
Happy Retirement, David!
\end{talk}
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 163: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/163.jpg)
Happy Retirement, David!
\end{talk}
Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras
![Page 164: Moving Lines, Soccer, and Rees AlgebrasDavid:NowthatIaminBarcelona... Iwanttovisitthe2churches Carlos D’Andrea Moving Lines, Soccer, and Rees Algebras](https://reader034.fdocuments.net/reader034/viewer/2022050112/5f49758179a6cc2793298b51/html5/thumbnails/164.jpg)
Happy Retirement, David!
\end{talk}Carlos D’Andrea
Moving Lines, Soccer, and Rees Algebras