On the effective cone of P n blown-up at n + 3 points (and other positivity properties)
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1 On the effective cone of P n blown-up at n + 3 points (and other positivity properties) Elisa Postinghel KU Leuven joint with M.C. Brambilla (Ancona) and O. Dumitrescu (Hannover) Daejeon August 3-7, 2015 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
2 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
3 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { } L = L Pn d (m f C[x] 1,..., m s ) := d mult pi (f ) m i Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
4 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { } L = L Pn d (m f C[x] 1,..., m s ) := d mult pi (f ) m i Problem Compute dim(l). Unsolved in general. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
5 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { } L = L Pn d (m f C[x] 1,..., m s ) := d mult pi (f ) m i Problem Compute dim(l). Unsolved in general. Dimensionality ( n + d n ) s ( ) n + mi 1 =: vdim(l) n i=1 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
6 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { L = L Pn d (m 1,..., m s ) := Problem f C[x] d mult pi (f ) m i Compute dim(l). Unsolved in general. } { D Pic(Bls (P n )) D = dh s i=1 m ie i } Dimensionality ( n + d n ) s ( ) n + mi 1 =: vdim(l) n i=1 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
7 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { L = L Pn d (m 1,..., m s ) := Problem f C[x] d mult pi (f ) m i Compute dim(l). Unsolved in general. } { D Pic(Bls (P n )) D = dh s i=1 m ie i } Dimensionality ( n + d n ) s ( ) n + mi 1 =: vdim(l) = χ(bl s (P n ), O(D)) n i=1 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
8 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { L = L Pn d (m 1,..., m s ) := Problem f C[x] d mult pi (f ) m i Compute dim(l). Unsolved in general. } { D Pic(Bls (P n )) D = dh s i=1 m ie i } Dimensionality ( n + d n ) s ( ) n + mi 1 =: vdim(l) = χ(bl s (P n ), O(D)) n i=1 dim(l) = h 0 (Bl s (P n ), O(D)) χ(bl s (P n ), O(D)) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
9 Polynomial interpolation Choose p 1,..., p s P n C points in general position, and integers d, m i 0 { L = L Pn d (m 1,..., m s ) := Problem f C[x] d mult pi (f ) m i Compute dim(l). Unsolved in general. } { D Pic(Bls (P n )) D = dh s i=1 m ie i } Dimensionality ( n + d n ) s ( ) n + mi 1 =: vdim(l) = χ(bl s (P n ), O(D)) n i=1 dim(l) = h 0 (Bl s (P n ), O(D)) χ(bl s (P n ), O(D)) Examples: L P2 2 (2, 2) LP2 4 (2, 2, 2, 2, 2) dim vdim (L is special) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
10 Secant varieties of rational normal curves Theorem (Veronese 1881)! rational normal curve C P n of degree n through p 1,..., p n+3. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
11 Secant varieties of rational normal curves Theorem (Veronese 1881)! rational normal curve C P n of degree n through p 1,..., p n+3. Secant varieties: σ t (C) := x 1,..., x t x 1,...,x t C Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
12 Secant varieties of rational normal curves Theorem (Veronese 1881)! rational normal curve C P n of degree n through p 1,..., p n+3. Secant varieties: σ t (C) := x 1,..., x t (t n/2 ) dim σ t (C) = 2t 1 x 1,...,x t C Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
13 Secant varieties of rational normal curves Theorem (Veronese 1881)! rational normal curve C P n of degree n through p 1,..., p n+3. Secant varieties: σ t (C) := (t n/2 ) dim σ t (C) = 2t 1 x 1,...,x t C x 1,..., x t Cones: I {1,..., n + 3}, L I = p i : i I = P I 1 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
14 Secant varieties of rational normal curves Theorem (Veronese 1881)! rational normal curve C P n of degree n through p 1,..., p n+3. Secant varieties: σ t (C) := (t n/2 ) dim σ t (C) = 2t 1 x 1,...,x t C x 1,..., x t Cones: I {1,..., n + 3}, L I = p i : i I = P I 1 Join(L I, σ t (C)) Figure: Join(p i, C) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
15 Secant varieties of rational normal curves Theorem (Veronese 1881)! rational normal curve C P n of degree n through p 1,..., p n+3. Secant varieties: σ t (C) := (t n/2 ) x 1,...,x t C x 1,..., x t dim σ t (C) = 2t 1 dim Join(L I, σ t (C)) = I + (2t 1) Cones: I {1,..., n + 3}, L I = p i : i I = P I 1 Join(L I, σ t (C)) Figure: Join(p i, C) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
16 A combinatorial tool: the Base Locus Lemma mult L{ij} (f ) max{m i + m j d, 0} =: k {ij} Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
17 A combinatorial tool: the Base Locus Lemma mult L{ij} (f ) max{m i + m j d, 0} =: k {ij} mult LI (f ) max{ i I m i ( I 1)d, 0} =: k I Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
18 A combinatorial tool: the Base Locus Lemma mult L{ij} (f ) max{m i + m j d, 0} =: k {ij} mult LI (f ) max{ i I m i ( I 1)d, 0} =: k I mult C (f ) max{ n+3 i=1 m i nd, 0} =: k C Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
19 A combinatorial tool: the Base Locus Lemma mult L{ij} (f ) max{m i + m j d, 0} =: k {ij} mult LI (f ) max{ i I m i ( I 1)d, 0} =: k I mult C (f ) max{ n+3 i=1 m i nd, 0} =: k C mult σt(c)(f ) max{tk C (t 1)d, 0} =: k σt Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
20 A combinatorial tool: the Base Locus Lemma mult L{ij} (f ) max{m i + m j d, 0} =: k {ij} mult LI (f ) max{ i I m i ( I 1)d, 0} =: k I mult C (f ) max{ n+3 i=1 m i nd, 0} =: k C mult σt(c)(f ) max{tk C (t 1)d, 0} =: k σt mult Join(LI,σ t(c))(f ) max{k σt + k I d, 0} =: k I,σt Lemma (Brambilla-Dumitrescu-P 15) k I,σt Join(L I, σ t ) Bs( D ) with that exact multiplicity. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
21 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
22 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Examples L P2 6 (5, 5) k 12 = 4, dim(l) = vdim(l) + 6 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
23 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Examples L P2 6 (5, 5) k 12 = 4, dim(l) = vdim(l) + 6 L P3 6 (5, 5, 5) k {ij} = 4 k {123} = 3 dim(l) = vdim(l) = 9 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
24 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Examples L P2 6 (5, 5) k 12 = 4, dim(l) = vdim(l) + 6 L P3 6 (5, 5, 5) k {ij} = 4 k {123} = 3 dim(l) = vdim(l) = 9 Lemma Contribution of L I is ( 1) I ( n I +k I n ). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
25 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Example: L P4 9 (67 ): vdim(l) = 167, dim(l) = 1. ( ) ( ) 7 5 dim(l) = vdim(l) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
26 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Example: L P4 9 (67 ): vdim(l) = 167, dim(l) = 1. ( ) ( ) ( ) dim(l) = vdim(l) k C = 6 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
27 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Example: L P4 9 (67 ): vdim(l) = 167, dim(l) = 1. ( ) ( ) ( ) dim(l) = vdim(l) ( 1) k C = 6 k Join(C,pi ) = 4 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
28 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Example: L P4 9 (67 ): vdim(l) = 167, dim(l) = 1. ( ) ( ) ( ) dim(l) = vdim(l) ( 1) k C = 6 k Join(C,pi ) = 4 k σ2 = 3 Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
29 Dimensionality problem for linear systems -Each subvariety contained with multiplicity in the base locus produces a jump between vdim(l) and dim(l). Can we identify it?- Example: L P4 9 (67 ): vdim(l) = 167, dim(l) = 1. ( ) ( ) ( ) dim(l) = vdim(l) ( 1) k C = 6 k Join(C,pi ) = 4 k σ2 = 3 Lemma Contribution of Join(L I, σ t ) is ( 1) I ( n ( I +2t)+k I,σ t n ). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
30 Fröberg-Iarrobino conjecture dim(l) apolarity Hilbert function of ideals generated by powers of linear forms Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
31 Fröberg-Iarrobino conjecture dim(l) apolarity Hilbert function of ideals generated by powers of linear forms Conjecture (Fröberg 85, Iarrobino 97) dim(l) = vdim(l) + I ( 1) I ( n I + ki n expect in a list of cases, among which s = n + 3. ) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
32 Fröberg-Iarrobino conjecture dim(l) apolarity Hilbert function of ideals generated by powers of linear forms Conjecture (Fröberg 85, Iarrobino 97) dim(l) = vdim(l) + I ( 1) I ( n I + ki n expect in a list of cases, among which s = n + 3. ) Conjecture (Brambilla-Dumitrescu-P 15) dim(l) = vdim(l) + I,t if s = n + 3. ( 1) I ( n ( I + 2t) + ki,σt n ) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
33 Motivation II: Bl n+3 (P n ) is... [Mukai 05] Bl n+3 (P n ) = U( a), the moduli space of rank-2 parabolic vector bundles over (n + 3)-pointed P 1, where a = (a,..., a) [0, 1] n+3 is a parabolic weight. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
34 Motivation II: Bl n+3 (P n ) is... [Mukai 05] Bl n+3 (P n ) = U( a), the moduli space of rank-2 parabolic vector bundles over (n + 3)-pointed P 1, where a = (a,..., a) [0, 1] n+3 is a parabolic weight. [Mukai 01] Cox(Bl n+3 (P n )) = R G, the Cox-Nagata ring, ring of polynomials in C[x 1,..., x n+3, y 1,..., y n+3 ] fixed by Nagata action of G 2 a (Hilbert s 14th problem). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
35 Motivation II: Bl n+3 (P n ) is... [Mukai 05] Bl n+3 (P n ) = U( a), the moduli space of rank-2 parabolic vector bundles over (n + 3)-pointed P 1, where a = (a,..., a) [0, 1] n+3 is a parabolic weight. [Mukai 01] Cox(Bl n+3 (P n )) = R G, the Cox-Nagata ring, ring of polynomials in C[x 1,..., x n+3, y 1,..., y n+3 ] fixed by Nagata action of G 2 a (Hilbert s 14th problem). [Dolgachev 81; Castravet-Tevelev 06; Sturmfels-Velasco 10] Generators of Cox(Bl n+3 (P n )) 2 n+2 weights of the half-spin representation of so 2(n+3). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
36 Birational geometry of Bl n+3 (P n ) [Castravet-Tevelev 06] Bl n+3 (P n ) is a Mori Dream Space. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
37 Birational geometry of Bl n+3 (P n ) [Castravet-Tevelev 06] Bl n+3 (P n ) is a Mori Dream Space. The birational geometry of a MDS can be encoded in some finite data: the polyhedral cones Eff(Bl n+3 (P n )) Mov(Bl n+3 (P n )) together with their Mori/nef chamber decompositions. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
38 Birational geometry of Bl n+3 (P n ) [Castravet-Tevelev 06] Bl n+3 (P n ) is a Mori Dream Space. The birational geometry of a MDS can be encoded in some finite data: the polyhedral cones Eff(Bl n+3 (P n )) Mov(Bl n+3 (P n )) together with their Mori/nef chamber decompositions. MDSs behave particularly well from the MMP point of view. Can we find a good minimal model, i.e. one for which the abundance conjecture holds? Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
39 Effective cone, movable cone and more Theorem (Brambilla-Dumitrescu-P 15) Eff(Bl n+3 (P n )) is given by inequalities: {m i d, k I,σt 0, I, σ t : dim Join(L I, σ t ) = n}. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
40 Effective cone, movable cone and more Theorem (Brambilla-Dumitrescu-P 15) Eff(Bl n+3 (P n )) is given by inequalities: {m i d, k I,σt 0, I, σ t : dim Join(L I, σ t ) = n}. Mov(Bl n+3 (P n )) is given by inequalities: {m i d, k I,σt 0, I, σ t : dim Join(L I, σ t ) = n, n 1}. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
41 Effective cone, movable cone and more Theorem (Brambilla-Dumitrescu-P 15) Eff(Bl n+3 (P n )) is given by inequalities: {m i d, k I,σt 0, I, σ t : dim Join(L I, σ t ) = n}. Mov(Bl n+3 (P n )) is given by inequalities: {m i d, k I,σt 0, I, σ t : dim Join(L I, σ t ) = n, n 1}. Theorem (Mukai 05; Araujo-Massarenti 15) The Mori chamber decomposition of Eff is given by the hyperplanes k I,σt = 0 : dim Join(L I, σ t } = n r, r 2, I = r (mod 2) Walls corresponds to flips of a P n r in a P r+1. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
42 Mori chamber decomposition N 1 (X) Eff Mov Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
43 Semiample, ample and nef cones Theorem (Dumitrescu-P 15) If s 2n, D on Bl s (P n ) is basepoint free iff m i 0, k {i,j} 0 (A) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
44 Semiample, ample and nef cones Theorem (Dumitrescu-P 15) If s 2n, D on Bl s (P n ) is basepoint free iff m i 0, k {i,j} 0 (A) If s 2n, D on Bl s (P n ) is very ample iff m i 1, k {i,j} 1 (B) Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
45 Semiample, ample and nef cones Theorem (Dumitrescu-P 15) If s 2n, D on Bl s (P n ) is basepoint free iff m i 0, k {i,j} 0 (A) If s 2n, D on Bl s (P n ) is very ample iff m i 1, k {i,j} 1 (B) Corollary For s 2n, the semi-ample cone of Bl s (P n ) is given by (A). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
46 Semiample, ample and nef cones Theorem (Dumitrescu-P 15) If s 2n, D on Bl s (P n ) is basepoint free iff m i 0, k {i,j} 0 (A) If s 2n, D on Bl s (P n ) is very ample iff m i 1, k {i,j} 1 (B) Corollary For s 2n, the semi-ample cone of Bl s (P n ) is given by (A). For s 2n, the ample cone of Bl s (P n ) is given by (B). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
47 Semiample, ample and nef cones Theorem (Dumitrescu-P 15) If s 2n, D on Bl s (P n ) is basepoint free iff m i 0, k {i,j} 0 (A) If s 2n, D on Bl s (P n ) is very ample iff m i 1, k {i,j} 1 (B) Corollary For s 2n, the semi-ample cone of Bl s (P n ) is given by (A). For s 2n, the ample cone of Bl s (P n ) is given by (B). Corollary The nef cone coincides with the semi-ample cone Bl s (P n ), for s 2n. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
48 Good minimal models for Bl n+3 (P n ) Conjecture ((Log) abundance conjecture) If (Bl n+3 (P n ), ), 0 is a log canonical Q-divisor, then K X + is nef iff it is semi-ample. Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
49 Good minimal models for Bl n+3 (P n ) Conjecture ((Log) abundance conjecture) If (Bl n+3 (P n ), ), 0 is a log canonical Q-divisor, then K X + is nef iff it is semi-ample. Theorem (Dumitrescu-P 15) X = Bl n+3 (P n ), = ɛd, with 0 ɛ << 1 and D Eff(Bl n+3 (P n )) line bundle. Assume K X + is nef (equiv. semi-ample), i.e. ɛm i n 1, ɛk {i,j} n 3. Then (X, ) admits a log resolution such that (X, ) is log canonical. This gives good minimal models for the pairs (Bl n+3 (P n ), ). Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
50 Bibliography [1] C. Araujo, A. Massarenti, Explicit log Fano structures on blow-ups of projective spaces, preprint ArXiv: (2015) [2] M. C. Brambilla, O. Dumitrescu and E. Postinghel, On the effective cone of P n blown-up at n + 3 points, preprint ArXiv: (2015) [3] A. M. Castravet and J. Tevelev, Hilbert s 14th problem and Cox rings, Compos. Math. 142 (2006), no. 6, [4] O. Dumitrescu and E. Postinghel, Positivity of divisors on blown-up projective spaces, preprint ArXiv: (2015) [5] A. Iarrobino, Inverse system of symbolic power III. Thin algebras and fat points, Compositio Math. 108 (1997), no. 3, [6] S. Mukai, Finite generation of the Nagata invariant rings in A-D-E cases, RIMS Preprint n (2005) [7] B. Sturmfels, M. Velasco Blow-ups of P n3 at n points and spinor varieties. J. Commut. Algebra 2 (2010), no. 2, THANKS!! Elisa Postinghel (KU Leuven) () Effective cone of Bl n+3 (P n ) SIAM AG / 14
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