Syzygies and singularities of tensor product
surfaces
Alexandra Seceleanu
joint with H. Schenck, J. Validashti
SIAM Applied AG 2013
Surface modeling
In Computer Aided Geometric Design
surface splines are made from patches defined parametrically by
rational maps
φ : R2 −→ R3
p1 (x, y ) p2 (x, y ) p3 (x, y )
φ(x, y ) =
,
,
p0 (x, y ) p0 (x, y ) p0 (x, y )
Tensor product surface
Instead of φ : R2 −→ R3 or φ : P2 −→ P3 (triangular surface),
a tensor product surface is the image of a bi-homogeneous
parametrization map
φ : P1 × P1 −→ P3
[(s : t), (u : v )] 7→ [p0 (s, t, u, v ) : p1 (s, t, u, v ) : p2 (s, t, u, v ) : p3 (s, t, u, v )]
with
deg(s) = deg(t) = (1, 0) and deg(u) = deg(v) = (0, 1).
Tensor product surface
Instead of φ : R2 −→ R3 or φ : P2 −→ P3 (triangular surface),
a tensor product surface is the image of a bi-homogeneous
parametrization map
φ : P1 × P1 −→ P3
[(s : t), (u : v )] 7→ [p0 (s, t, u, v ) : p1 (s, t, u, v ) : p2 (s, t, u, v ) : p3 (s, t, u, v )]
with
deg(s) = deg(t) = (1, 0) and deg(u) = deg(v) = (0, 1).
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This is a special case of toric parametrization.
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We restrict to the case of a bidegree (2, 1) parametrization
(yields a surface ruled by lines and quadrics) w/o base locus.
Example
φ : [(s : t), (u : v )] 7−→ [s 2 u : s 2 v : t 2 u : t 2 v + stv ]
Figure: Three double lines on a bidegree (2, 1) surface
Example - continued
Let I = (s 2 u, s 2 v , t 2 u, t 2 v + stv ) be the parametrization ideal.
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I has a linear syzygy of bidegree (0, 1)
v (s 2 u) − u(s 2 v ) = 0
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X = Im(φ) has the implicit equation:
X = V(x0 x12 x2 − x12 x22 + 2x0 x1 x2 x3 − x02 x32 ).
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the reduced codimension one singular locus of X is:
V(x0 , x2 ) ∪ V(x1 , x3 ) ∪ V(x0 , x1 ).
Example - continued
Let I = (s 2 u, s 2 v , t 2 u, t 2 v + stv ) be the parametrization ideal.
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I has a linear syzygy of bidegree (0, 1)
v (s 2 u) − u(s 2 v ) = 0 :
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X = Im(φ) has the implicit equation:
X = V(x0 x1 2 x2 − x12 x22 + 2x0 x1 x2 x3 − x02 x32 ).
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the reduced codimension one singular locus of X is:
V(x0 , x2 ) ∪ V(x1 , x3 ) ∪ V(x0 , x1 ).
Linear syzygies & the Segre-Veronese Σ2,1
Proposition (Schenck-S.-Validashti)
The ideal I = (p0 , p1 , p2 , p3 )
1. has a unique linear syzygy of bidegree (0, 1) iff
Php0 , p1 , p2 , p3 i ∩ Σ2,1 ,contains a P1 fiber of Σ2,1 .
e.g. Spanhs 2 u, s 2 v i is a P1 fiber
2. has a pair of linear syzygies of bidegree (0, 1) iff
Php0 , p1 , p2 , p3 i ∩ Σ2,1 = Σ1,1 .
3. has a unique linear syzygy of bidegree (1, 0) iff
Php0 , p1 , p2 , p3 i ∩ Q contains a P1 fiber of Q.
Main result
Theorem (Schenck-S.-Validashti)
There are exactly 6 (families of) resolutions for ideals generated by
four bidegree (2,1) forms without basepoints.
This uses
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the geometry of the Segre-Veronese variety (to determine how
many linear and quadratic syzygies may exist)
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the Buchsbaum-Eisenbud exactness criterion (to write down
an explicit resolution when linear syzygies exist)
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bigraded gins to determine the resolution in the generic case
Implicitization via the approximation complex
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Sederberg and Chen (1995) introduced for implicitization
purposes a method termed as moving curves and surfaces.
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Cox realized they were using syzygies with several coauthors
(Busé, Chen, D’Andrea, Goldman, Sederberg, Zhang).
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Jouanolou and Busé (2002) gave a sound theoretical basis for
the method of Sederberg-Chen via approximation
complexes, a tool in homological algebra developed by
Herzog–Simis–Vasconcelos.
Implicitization via the approximation complex
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Step 1: Find the syzygies on p0 , p1 , p2 , p3
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Step 2: Represent them as linear combinations
(j)
(j)
(j)
(j)
Lj = α0 x0 + α1 x1 + α2 x2 + α3 x3
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Step 3: Rewrite the syzygies of degree ν in terms of a
monomial basis {mβ }|β|=ν of k[s, t, u, v ]ν
Lj =
3 X
X
i=0 |β|=ν
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(j)
ci,β mβ xi =
X
3
X
|β|=ν
i=0
!
(j)
ci,β xi
mβ
Step 4: The implicit equation is the gcd of the maximal
P
(j)
minors of the matrix M = ( 3i=0 ci,β xi )β,j for well-chosen ν.
Implicitization example:
ν = (1, 1)
vx0 − ux1 = 0
s 2 x2 − t 2 x0
s x3 − (st + t 2 )x1
tux3 − (sv + tv )x2
sux3 − svx2 − tvx0
2

·
 ·

 ·

 x1

−x0

 ·

 ·

 ·

 ·
·
·
·
·
·
·
x1
−x0
·
·
·
·
·
·
·
·
·
·
x1
−x0
·
·
·
·
·
·
·
·
·
x1
−x0
x2
·
−x0
·
·
·
·
·
·
·
= 0
= 0
= 0
= 0
x3
−x1
−x1
·
·
·
·
·
·
·
·
·
·
·
−x2
x3
−x2
·
·
·

·
· 

· 

x3 

−x2 

· 

−x0 

· 

· 
·
Theorem (Schenck-S.-Validashti)
In fact the implicit equation is itself a smaller minor !
Relations between syzygies and singularity types
Type
1
2
3
4
5a
5b
6
Lin. Syz. Emb. Pri. Sing. Loc.
Example
none
m
T
(s 2 u+stv , t 2 u, s 2 v +stu, t 2 v +stv)
none
m, P1
C ∪ L1
(s 2 u, t 2 u, s 2 v + stu, t 2 v + stv )
1 type (1, 0)
m
L1
(s 2 u + stv , t 2 u, s 2 v , t 2 v + stu)
1 type (1, 0) m, P1
L1
(stv , t 2 v , s 2 v − t 2 u, s 2 u)
1 type (0, 1) P1 , P2 L1 ∪ L2 ∪ L3
(s 2 u, s 2 v , t 2 u, t 2 v + stv )
1 type (0, 1)
P1
L1 ∪ L2
(s 2 u, s 2 v , t 2 u, t 2 v + stu)
2 type (0, 1) none
∅
(s 2 u, s 2 v , t 2 u, t 2 v )
Table: The primary decomposition and singularities for the six Betti types
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T = twisted cubic curve, C = smooth plane conic Li = lines
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m = hs, t, u, v i, Pi = hli , s, ti, li = linear form of bidegree (0, 1)