New examples of defective secant varieties of Segre-Veronese - PowerPoint PPT Presentation

New examples of defective secant varieties of Segre-Veronese varieties (joint work with M. C. Brambilla) Hirotachi Abo Department of Mathematics University of Idaho abo@uidaho.edu http://www.uidaho.edu/˜abo October 15, 2011

Notation • V = ( n + 1)-dimensional vector space over C . • P V = projective space of V . • [ v ] ∈ P V = equivalence class containing v ∈ V \ { 0 } . • S d V = d th symmetric power of V . • � X � = linear span of X ⊆ P V .

Secant varieties • X = projective variety in P V . • Let p 1 , . . . , p s be generic points of X . Then � p 1 , . . . , p s � is called a secant ( s − 1) -plane to X . • The s th secant variety of X is defined to be the Zariski closure of the union of secant ( s − 1)-planes to X : � σ s ( X ) = � p 1 , . . . , p s � . p 1 , ··· ,p s ∈ X

Secant dimension and secant defectivity • A simple parameter count implies the following inequality holds: dim σ s ( X ) ≤ min { s · (dim X + 1) − 1 , dim P V } . • If equality holds, we say X has the expected dimension . • σ s ( X ) is said to be defective if it does not have the expected dimension. • X is said to be defective if σ s ( X ) is defective for some s .

The Alexander-Hrschowitz theorem • Let v d : P V → P S d V be the d th Veronese map, i.e., v d is the map given by v d ([ v ]) = [ v d ]. • Theorem (Alexander-Hirschowitz, 1995) σ s [ v d ( P V )] is non-defective except for the following cases: dim P V d s ≥ 2 2 2 ≤ s ≤ n 2 4 5 3 4 9 4 3 7 4 4 14

Secant varieties of Segre-Veronese varieties • n = ( n 1 , . . . , n k ), d = ( d 1 , . . . , d k ) ∈ N k . • V i = ( n i + 1)-dimensional vector space. �� k � • Seg : � k i =1 P V i → P i =1 V i = Segre map, i.e., the map given by Seg([ v 1 ] , . . . , [ v k ]) = [ v 1 ⊗ · · · ⊗ v k ]. �� k � �� k � i =1 S d i V i • X n , d := Seg i =1 v d i ( P V i ) ֒ → P is called a Segre-Veronese variety .

Conjecturally complete list of defective two factor cases n d s �� m + d � � m + d � � ( m, n ) with m ≥ 2 ( d, 1) − m < s < min n + 1 d d (2 , 2 k + 1) (1 , 2) 3 k + 2 (4 , 3) (1 , 2) 6 (1 , 2) (1 , 3) 5 (1 , n ) (2 , 2) n + 2 ≤ s ≤ 2 n + 1 (2 , 2) (2 , 2) 7, 8 � � 3 n 2 +9 n +5 (2 , n ) (2 , 2) ≤ s ≤ 3 n + 2 n +3 (3 , 3) (2 , 2) 14, 15 (3 , 4) (2 , 2) 19 ( n, 1) (2 , 2 k ) kn + k + 1 ≤ s ≤ kn + k + n

This conjecture is based on: • already existing results (by many people including E. Carlini and T. Geramita) and • computational experiments that employ the so-called “Terracini lemma”. • Remark. – Terracini’s lemma can be used to experimentally detect defective cases. – The result of a computation provides strong evidence, but it cannot be used as a rigorous proof of its deficiency. – Proving that experimentally determined defective secant varieties are actually defective requires more insight.

What about Segre-Veronese varieties with three or more factors? Let n = ( n 1 , · · · n k ) , d = ( d 1 , . . . , d k − 1 , 1) ∈ N k . Then ( n , d ) is said to be unbalanced if k − 1 k − 1 � n i + d i � � � n k ≥ − n i + 1 . d i i =1 i =1 Let ( n , d ) be unbalanced. Then σ s ( X n , d ) is defective if and only if s satisfies the following: k − 1 k − 1 k − 1 � �� � n i + d i � � n i + d i � � � − n i < s < min n k + 1 , d i d i i =1 i =1 i =1 (Catalisano-Geramita-Gimigliano, 2008).

Defective cases for Segre-Veronese with k ≥ 3 known before 2010 (modulo the unbalanced case) P n s d P 1 × P 1 × P 1 (1 , 1 , 2 n ) 2 n + 1 P 1 × P 1 × P n (1 , 1 , 2) 2 n + 1 P n × P 1 × P 1 (1 , 1 , n + 1) 2 n + 1 � � P n × P n × P 1 (2 d +1)( n +1) (1 , 1 , 2 d ) ≤ s ≤ dn + n + d 2 P 2 × P 2 × P 3 (1 , 1 , 2) 11 P n × P n × P 2 (1 , 1 , 2) 3 n + 2 P 1 × P 1 × P 1 (2 , 2 , 2) 7 P 1 × P 1 × P 2 (2 , 2 , 2) 11 P 1 × P 1 × P 3 (2 , 2 , 2) 15 P 2 n +1 × P 1 × P 1 × P 1 (1 , 1 , 1 , n + 1) 4 n + 3 P 2 × P 5 × P 1 × P 1 (1 , 1 , 1 , 2) 11

The main theorem (rough version) • Theorem (A-Brambilla, 2010) Let k ∈ { 3 , 4 } , let n = ( n 1 , . . . , n k ) and let d = (1 , . . . , 1 , 2) . Then there exist infinitely many defective secant varieties of X n , d , which were previously not known. • Remark. The family we discovered includes some of the defective secant varieties listed one slide ago as special cases.

Defective cases known before 2010 revisited P n d s P 1 × P 1 × P 1 (1 , 1 , 2 n ) 2 n + 1 P 1 × P 1 × P n (1 , 1 , 2) 2 n + 1 P n × P 1 × P 1 (1 , 1 , n + 1) 2 n + 1 � � P n × P n × P 1 (2 d +1)( n +1) (1 , 1 , 2 d ) ≤ s ≤ dn + n + d 2 P 2 × P 2 × P 3 (1 , 1 , 2) 11 P n × P n × P 2 (1 , 1 , 2) 3 n + 2 P 1 × P 1 × P 1 (2 , 2 , 2) 7 P 1 × P 1 × P 2 (2 , 2 , 2) 11 P 1 × P 1 × P 3 (2 , 2 , 2) 15 P 2 n +1 × P 1 × P 1 × P 1 (1 , 1 , 1 , n + 1) 4 n + 3 P 2 × P 5 × P 1 × P 1 (1 , 1 , 1 , 2) 11

Outline of the proof • Step 1. Find a non-singular subvariety C of X n , d passing through s generic points. • Step 2. Use C to provide an upper bound of dim σ s ( X n , d ): dim σ s ( X n , d ) ≤ s · (dim X n , d − dim C ) + dim � C � . • Step 3. Find ( n , d , s ) satisfying � k �� � n i + d i � s · (dim X n , d − dim C )+dim � C � < min s · (dim X n , d + 1) , − 1 . n i i =1

Example • Let n, d ∈ N , a ∈ { 0 , · · · , ⌈ n/d ⌉ − 1 } ; • n = ( n, n + a, 1), d = (1 , 1 , 2 d ) ∈ N 3 , and • s = ( n + a + 1) d + k for ∀ k ∈ { 1 , . . . , n − ad } . • Then σ s ( X n , d ) is defective. • Remark. This includes the following previously known example as a special case: � � P n × P n × P 1 (2 d +1)( n +1) (1 , 1 , 2 d ) ≤ s ≤ dn + n + d 2 The theorem now implies P n × P n × P 1 (1 , 1 , 2 d ) d ( n + 1) + 1 ≤ s ≤ dn + n + d

Thank you very much for your attention!