ED degree of smooth projective varieties

Transcription

1 ED degree of smooth projective varieties Paolo Aluffi and Corey Harris December 9, 2017 Florida State University Max Planck Institute for Mathematics in the Sciences, Leipzig

2 Euclidean distance degrees of affine and projective varieties The squared distance function on A n is n d u(x) = d(x, u) = (x i u i ) 2. For V A n a affine variety, the Euclidean distance degree of V is EDdeg(V ) := #{critical points of d u(x) on V \V sing }. i=1 1

3 Euclidean distance degrees of affine and projective varieties The squared distance function on A n is n d u(x) = d(x, u) = (x i u i ) 2. For V A n a affine variety, the Euclidean distance degree of V is EDdeg(V ) := #{critical points of d u(x) on V \V sing }. For X P n a projective variety, EDdeg(X ) := EDdeg( ˆX ) where ˆX A n+1 is the affine cone over X. i=1 1

4 Euclidean distance degrees of affine and projective varieties The squared distance function on A n is d u(x) = d(x, u) = n (x i u i ) 2. For V A n a affine variety, the Euclidean distance degree of V is EDdeg(V ) := #{critical points of d u(x) on V \V sing }. For X P n a projective variety, i=1 EDdeg(X ) := EDdeg( ˆX ) where ˆX A n+1 is the affine cone over X. Lemma (Draisma-Horobet-Ottaviani-Sturmfels-Thomas) EDdeg(X ) := #{critical points of d u(x) on V \(V sing Q) where Q = V ( x 2 i ) is the isotropic quadric. 1

5 Example 2

6 Euler characteristics The Euler characteristic of a CW-complex C is χ(c) = ( 1) i k i, where k i is the number of i-dimensional cells in C. 3

7 Euler characteristics The Euler characteristic of a CW-complex C is χ(c) = ( 1) i k i, where k i is the number of i-dimensional cells in C. If X is a projective variety, χ(x ) = χ(c X ) where C X ) is a CW-complex with equal support. 3

8 Euler characteristics The Euler characteristic of a CW-complex C is χ(c) = ( 1) i k i, where k i is the number of i-dimensional cells in C. If X is a projective variety, χ(x ) = χ(c X ) where C X ) is a CW-complex with equal support. If X is a smooth projective variety, its Euler characteristic is c(tx ) [X ]. 3

9 Euler characteristics The Euler characteristic of a CW-complex C is χ(c) = ( 1) i k i, where k i is the number of i-dimensional cells in C. If X is a projective variety, χ(x ) = χ(c X ) where C X ) is a CW-complex with equal support. If X is a smooth projective variety, its Euler characteristic is c(tx ) [X ]. If X is any projective variety, its Euler characteristic is c SM (TX ) [X ]. 3

10 ED degrees and Chern classes Let X P n be a projective variety. 4

11 ED degrees and Chern classes Let X P n be a projective variety. We define the generic ED degree to be the ED degree of X embedded generically with respect to the isotropic quadric Q (the hypersurface defined by x x 2 n = 0). 4

12 ED degrees and Chern classes Let X P n be a projective variety. We define the generic ED degree to be the ED degree of X embedded generically with respect to the isotropic quadric Q (the hypersurface defined by x x 2 n = 0). Draisma et al. show that if X is nonsingular, then dim X geddeg(x ) = ( 1) dim X +j (2 j+1 1)c j (X ) j=0 where c(x ) j is the degree of the j-dimensional part of c(t X ) [X ]. (If X is singular, replace c(x ) with c Ma (X ), the Chern-Mather class of X. 4

13 Nicest case Exercise For 0 j N, the coefficient of t N in the expansion of ( 1) j (2 j+1 1). t N j (1 + t)(1 + 2t) is 5

14 Nicest case Exercise For 0 j N, the coefficient of t N in the expansion of ( 1) j (2 j+1 1). If X is nonsingular, we have the following dim X geddeg(x ) = ( 1) dim X j=0 dim X = ( 1) dim X = ( 1) dim X j=0 c(x ) j ( 1) j (2 j+1 1) c(x ) j t N j (1 + t)(1 + 2t) is dim X j h (1 + h)(1 + 2h) hcodim X [P n ] 1 c(tx ) [X ] (1 + h)(1 + 2h) = ( 1) dim X ( 1 h 1 + h 2h 1 + 2h + h 2h (1 + h)(1 + 2h) ) c(tx ) [X ] 5

15 Nicest case If X is nonsingular, we have the following dim X geddeg(x ) = ( 1) dim X Lemma j=0 dim X = ( 1) dim X = ( 1) dim X j=0 c(x ) j ( 1) j (2 j+1 1) c(x ) j dim X j h (1 + h)(1 + 2h) hcodim X [P n ] 1 c(tx ) [X ] (1 + h)(1 + 2h) = ( 1) dim X ( 1 h 1 + h 2h 1 + 2h + h 2h (1 + h)(1 + 2h) If X is nonsingular and transverse to Q, the (generic) ED degree of X is ( 1) dim X ( c(tx ) [X ] c(t (X Q)) [X Q] + c(t (X H)) [X H] ) c(tx ) [X ] ) c(t (X Q H)) [X Q H]. where H P n is a general hyperplane. 5

16 Nicest case Lemma If X is nonsingular and transverse to Q, the (generic) ED degree of X is ( 1) dim X ( c(tx ) [X ] c(t (X Q)) [X Q] + c(t (X H)) [X H] ) c(t (X Q H)) [X Q H]. where H P n is a general hyperplane. Theorem If X is nonsingular and transverse to Q, then EDdeg(X ) = ( 1) dim X ( χ(x ) χ(x H) χ(x Q) + χ(x Q H) ) = ( 1) dim X χ(x \(H Q)). 5

17 Projective ED correspondence Let T P n be the projective cotangent space of P n realized as the set of pairs (p, H) with p H in P n ˇP n. 6

18 Projective ED correspondence Let T P n be the projective cotangent space of P n realized as the set of pairs (p, H) with p H in P n ˇP n. A subvariety X P n has a projective conormal space T X P n := {(p, H) p X ns and T px H} T P n. 6

19 Projective ED correspondence Let T P n be the projective cotangent space of P n realized as the set of pairs (p, H) with p H in P n ˇP n. A subvariety X P n has a projective conormal space T X P n := {(p, H) p X ns and T px H} T P n. The projective ED correspondence PE is the incidence correspondence PE := {([x], [x u], u) [x] is a critical point of d u X } T X P n C n+1. π 1 PE π 2 T X P n C n+1 6

20 Projective ED correspondence Let T P n be the projective cotangent space of P n realized as the set of pairs (p, H) with p H in P n ˇP n. A subvariety X P n has a projective conormal space T X P n := {(p, H) p X ns and T px H} T P n. The projective ED correspondence PE is the incidence correspondence PE := {([x], [x u], u) [x] is a critical point of d u X } T X P n C n+1. π 1 PE π 2 T X P n C n+1 The projection π 1 is generically finite-to-one. The ED degree of X is the degree of π 1. 6

21 An excess intersection Let and Z := {([x], [y], u) P n ˇP n C n+1 dim x, y, u 2}, Z u = {([x], [y]) P n 1 ˇP n 1 dim x, y, u 2}. 7

22 An excess intersection Let and Z := {([x], [y], u) P n ˇP n C n+1 dim x, y, u 2}, Z u = {([x], [y]) P n 1 ˇP n 1 dim x, y, u 2}. Lemma We have (T X P n 1 C n ) Z = PE Z the support of ( T X P n 1 ) C n., where the support of Z equals 7

23 An excess intersection Let and Z := {([x], [y], u) P n ˇP n C n+1 dim x, y, u 2}, Z u = {([x], [y]) P n 1 ˇP n 1 dim x, y, u 2}. Lemma We have (T X P n 1 C n ) Z = PE Z the support of ( T X P n 1 ) C n., where the support of Z equals Corollary For a general u C n, Z u T X P n 1 = {EDdeg(X ) simple points} Zu, where the support of Zu agrees with the support of T X P n 1. 7

24 New definition for ED degree Let X be a subvariety of P n. Theorem EDdeg(X ) = geddeg(x ) (1 + H) n+1 s(zu, T X P n ) Here s(, ) is the Segre class. 8

25 New definition for ED degree Let X be a subvariety of P n. Theorem EDdeg(X ) = geddeg(x ) (1 + H) n+1 s(zu, T X P n ) Here s(, ) is the Segre class. Theorem Let X be a smooth subvariety of P n, and assume X Q. Then (1 + 2h) c(t X O(2h)) EDdeg(X ) = geddeg(x ) s(j(x Q), X ), 1 + h where J(X Q) is the Jacobian subscheme of X Q in X. 8

26 ED degree is an Euler characteristic Theorem Let X be a smooth subvariety of P n. Then EDdeg(X ) = ( 1) dim X j 0( 1) j (c(x ) j c SM (Q X ) j ). 9

27 ED degree is an Euler characteristic Theorem Let X be a smooth subvariety of P n. Then EDdeg(X ) = ( 1) dim X j 0( 1) j (c(x ) j c SM (Q X ) j ). Theorem Let X be a smooth variety in P n, with X Q. Then EDdeg(X ) = ( 1) dim X χ (X \(Q H)) where H P n is a general hyperplane. 9

28 Applications

29 Curves Let C be a smooth curve in P n with deg(c) = d. Then the main theorem gives EDdeg(C) = d + #(C Q) χ(c). Parametrize the isotropic quadric in P 2 by [s : t] [s 2 t 2 : 2st : i(s 2 + t 2 )]. Corollary Assume C is a nonsingular plane curve defined by F (x, y, z). Then EDdeg(C) = d(d 2) + R where R is the number of distinct factors of F (s 2 t 2, 2st, i(s 2 + t 2 )). 10

30 Hypersurfaces Let X be a smooth hypersurface in P n. By a result of Aluffi ( 00), if X is tangent to Q along a positive-dimensional locus, then deg(x ) = 2. Corollary Let X be a smooth hypersurface in P n with deg(x ) > 2. Then EDdeg(X ) = geddeg(x ) i µ(x i ) where {x i } is the set of isolated singularities of X Q and µ is the Milnor number. 11

31 Segre varieties Let X be the image of the usual Segre embedding of P m 1 P mp in P (m 1+1) (m p+1) 1. Then Q X = (Q 1 P m 2 P mp ) (P m 1 P m p 1 Q p) and we compute EDdeg(X ) = 1 (1 h1)m1+1 (1 h p) mp+1 [X ]. 1 h 1 h p (1 2h 1) (1 2h p) 12

32 Segre varieties Let X be the image of the usual Segre embedding of P m 1 P mp in P (m 1+1) (m p+1) 1. Then Q X = (Q 1 P m 2 P mp ) (P m 1 P m p 1 Q p) and we compute EDdeg(X ) = Theorem 1 (1 h1)m1+1 (1 h p) mp+1 [X ]. 1 h 1 h p (1 2h 1) (1 2h p) EDdeg(P m 1 P mp ) equals the coefficient of h m 1 1 h mp p in the expansion of 1 1 h 1 h p p i=1 (1 h i ) m i h i. 12

33 Segre varieties Theorem EDdeg(P m 1 P mp ) equals the coefficient of h m 1 1 h mp p in the expansion of 1 1 h 1 h p p i=1 (1 h i ) m i h i. Theorem (Friedland-Ottaviani) EDdeg(P m 1 P mp ) equals the coefficient of z m 1 1 z mp p in the expansion of p i=1 ẑ m i i z m i i, ẑ i z i where ẑ i = z z i 1 + z i+1 + z p. 12

34 Viel Dank! 12

35 References P. Aluffi and C. Harris. Euclidean distance degree of smooth complex projective varieties. arxiv: J. Draisma, E. Horobeţ, G. Ottaviani, B. Sturmfels, and R. R. Thomas. The Euclidean distance degree of an algebraic variety. Found. Comput. Math., 16(1):99 149, S. Friedland and G. Ottaviani. The number of singular vector tuples and uniqueness of best rank-one approximation of tensors. Found. Comput. Math., 14(6): ,