Implicitization of tensor product surfaces in the presence of a generic set of basepoints Eliana Duarte University of Illinois Urbana-Champaign April 16, 2016 Eliana Duarte (UIUC) Implicitization April 16, 2016 1 / 16
1 Motivation 2 Background and example 3 Main theorem and corollary Eliana Duarte (UIUC) Implicitization April 16, 2016 2 / 16
Algebraic surfaces in computer graphics Eliana Duarte (UIUC) Implicitization April 16, 2016 3 / 16
Different ways to describe algebraic surfaces r ( t , u ) = ( u , t 2 , t 2 u + tu ) X 2 Y − X 2 Y 2 + 2 XYZ − Z 2 = 0 Parametric For any pair ( s , t ) ∈ R 2 you associate a point r ( s , t ) ∈ R 3 that lies in the surface. Implicit The surface is described as the set of all points ( x , y , z ) ∈ R 3 that satisfy an equation. Eliana Duarte (UIUC) Implicitization April 16, 2016 4 / 16
Implicitization meets commutative algebra and algebraic geometry A 2 → A 3 � λ : P 1 × P 1 −→ P 3 A tensor product surface (TPS) is the closed image of a map λ : P 1 × P 1 −→ P 3 . Goal Given a generically finite rational map λ : P 1 × P 1 −→ P 3 with basepoints the goal is to understand the syzygies of the defining polynomials of λ to set up faster algorithms that compute the implicit equation of im λ . Eliana Duarte (UIUC) Implicitization April 16, 2016 5 / 16
Notation for TPS Example R = k [ s , t , u , v ] with grading f 0 = s 2 v deg( s , t ) = (1 , 0) and f 1 = stv deg( u , v ) = (0 , 1). f 2 = stu ∈ R (2 , 1) f 3 = t 2 u A map λ : P 1 × P 1 −→ P 3 is determined by four elements U := { f 0 , . . . , f 3 } f 0 , . . . , f 3 ∈ R ( a , b ) without U defines a rational map λ U : P 1 × P 1 −→ P 3 common factors. k [ X , Y , Z , W ] is the coordinate [ s , t , u , v ] �→ [ f 0 , f 1 , f 2 , f 3 ] . ring for P 3 . X U := λ U = V ( XW − YZ ) B = V ( f 0 , . . . , f 3 ) ⊂ P 1 × P 1 is the I U = � f 0 , . . . , f 3 � ⊂ R set of basepoints of U . Eliana Duarte (UIUC) Implicitization April 16, 2016 6 / 16
Rees-Algebra techniques to find implicit equations I U = � f 0 , . . . , f 3 � ⊂ R , we have a map β � R [ t ] R [ X , Y , Z , W ] Rees R ( I U ) = R [ X , Y , Z , W ] / ker β , (ker β ) 0 = ( H ). Rees R ( I U ) is in general hard to study so we look at Sym R ( I U ) α � Sym R ( I U ) R [ X , Y , Z , W ] Sym R ( I U ) = R [ X , Y , Z , W ] / Syz I U . If λ U : P 1 × P 1 −→ P 3 has finitely many local complete intersection basepoints, then Sym R ( I U ) and Rees R ( I U ) are “the same”. Thus we may find (ker β ) 0 by looking at Sym R ( I U ) which ultimately means studying Syz I U . Eliana Duarte (UIUC) Implicitization April 16, 2016 7 / 16
Implicit equations via syzygies Theorem[Botbol 2011] Let U = Span { f 0 , . . . , f 3 } ⊂ R ( a , b ) and λ U : P 1 × P 1 −→ P 3 the rational map defined by U . Then the syzygies of I U in degree ν = (2 a − 1 , b − 1) determine a complex Z ν such that det Z ν = H deg λ U where H denotes the irreducible implicit equation of X U . d 2 d 1 � ( Z 2 ) ν � ( Z 1 ) ν � ( Z 0 ) ν � 0 Z ν : 0 det Z ν = det d 1 det d 2 Eliana Duarte (UIUC) Implicitization April 16, 2016 8 / 16
Points in P 1 × P 1 STRATEGY: To understand λ U when U has basepoints we study the ideals of R corresponding to points in P 1 × P 1 . Example P = A × C ∈ P 1 × P 1 s t Take h ∈ R (1 , 0) , h ( A ) = 0, and u r q ∈ R (0 , 1) , q ( C ) = 0. Then v r I P = � h , q � I B = � s , u � ∩ � t , v � If B = { P 1 , . . . , P r } , then r � I B = I P i i =1 Eliana Duarte (UIUC) Implicitization April 16, 2016 9 / 16
TPS with a generic set of basepoints Example The bigraded Hilbert function of B is defined by s t 0 1 2 3 0 1 2 2 2 H B ( i , j ) = dim k R ( i , j ) − dim k ( I B ) ( i , j ) u H X = 1 2 2 2 2 r v 2 2 2 2 2 r 3 2 2 2 2 B is said to be generic if I B = � s , u � ∩ � t , v � H B ( i , j ) = min { ( i + 1)( j + 1) , r } Generic set of points in P 1 × P 1 . f 0 = s 2 v f 1 = stv Take U to be a generic 4-dimensional f 2 = stu ∈ ( I B ) (2 , 1) vector subspace of ( I B ) ( a , 1) f 3 = t 2 u U = { f 0 , f 1 , f 2 , f 3 } Eliana Duarte (UIUC) Implicitization April 16, 2016 10 / 16
Example with two generic basepoints a = 2 , r = 2 , U = { f 0 , f 1 , f 2 , f 3 } f 0 = s 2 v f 2 = stu f 3 = t 2 u f 1 = stv Using B´ otbol’s results, find a basis for the syzygies in bidegree (2 a − 1 , b − 1) = (3 , 0), and use it to set up the complex Z ν . − t 0 0 0 − u s Syz I U = 0 t v 0 − s 0 We write − t · X + s · Y = 0 and use { s 2 , st , t 2 } to bump up, e.g − s 2 t · X + s 3 · Y = 0 Eliana Duarte (UIUC) Implicitization April 16, 2016 11 / 16
Example with two generic basepoints Fix R (3 , 0) = { s 3 , s 2 t , st 2 , t 3 } and write the coefficient vectors of the syzygies w.r.t this basis. Y 0 0 W 0 0 − X 0 − Z 0 Y W d 1 = 0 − X Y 0 − Z W − X − Z 0 0 0 0 − s 2 t · X + s 3 · Y = 0 Proceed in the same way for all syzygies in bidegree (3 , 0) to obtain the rest of the columns of M . d 2 = ker d 1 Eliana Duarte (UIUC) Implicitization April 16, 2016 12 / 16
We obtain the complex W 0 − Z W 0 0 0 0 0 − Z Y W 0 0 − Y 0 − X Y − Z W 0 0 X − Y − X Y − Z W 0 0 0 0 0 X − X − Z � ( Z 2 ) ν � ( Z 1 ) ν � ( Z 0 ) ν Z ν : 0 d 2 = ker d 1 , dim ker d 1 = 2 = # of basepoints. � � Y 0 0 W � � � − X Y 0 − Z � � � 0 − X Y 0 � � � � = − X 2 YZ + X 3 W 0 0 − X 0 � � det Z ν = = XW − YZ � � X 2 X − Y � � � � 0 X � � The Main theorem tells us that for any bidegree ( a , 1) and any set of r generic basepoints, then d 1 is always determined by two syzygies. Eliana Duarte (UIUC) Implicitization April 16, 2016 13 / 16
Main Theorem (-): Let ( I B ) ( a , 1) be the k -vector space of forms of bidegree ( a , 1) that vanish at a generic set B of r = 2 k + 1 points in P 1 × P 1 . Take U to be a generic 4-dimensional vector subspace of ( I B ) ( a , 1) and λ U : P 1 × P 1 −→ P 3 the rational map determined by U . Then the complex Z ν , ν = (2 a − 1 , 0) is determined by two syzygies S 1 , S 2 of ( f 0 , . . . , f 3 ). If r = 2 k then deg S 1 = deg S 2 = ( a − k , 0). If r = 2 k + 1 then deg S 1 = ( a − k , 0) deg S 2 = ( a − ( k + 1) , 0). The syzygies S 1 , S 2 span a free module. Eliana Duarte (UIUC) Implicitization April 16, 2016 14 / 16
Main Theorem (-): Let ( I B ) ( a , 1) be the k -vector space of forms of bidegree ( a , 1) that vanish at a generic set X of r = 2 k + 1 points in P 1 × P 1 . Take U to be a generic 4-dimensional vector subspace of ( I B ) ( a , 1) and λ U : P 1 × P 1 −→ P 3 the rational map determined by U . Then the complex Z ν , ν = (2 a − 1 , 0) is determined by two syzygies S 1 , S 2 of ( f 0 , . . . , f 3 ). Corollary If B = ∅ then d 1 is determined by two syzygies of ( f 0 , . . . , f 3 ) in bidegee ( a , 0). Eliana Duarte (UIUC) Implicitization April 16, 2016 15 / 16
Questions What happens for ( a , b ), b > 1? What if U is not generic? What if the basepoints have multiplicities? Eliana Duarte (UIUC) Implicitization April 16, 2016 16 / 16
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