Cubic hypersurfaces over finite fields

Transcription

1 Cubic hypersurfaces over finite fields Olivier DEBARRE (joint with Antonio LAFACE and Xavier ROULLEAU) École normale supérieure, Paris, France Moscow, HSE, September 2015 Olivier DEBARRE Cubic hypersurfaces over finite fields 1 of 22

2 Theorem (Chevalley Warning) Any subscheme of P m F q defined by equations of degrees d 1,..., d s with d d s m has an F q -point. any cubic F q -hypersurface of dimension 2 contains an F q -point. What about F q -lines? Olivier DEBARRE Cubic hypersurfaces over finite fields 2 of 22

3 The scheme F (X ) X P n+1 k cubic of dimension n. F (X ) Gr(1, P n+1 k ) projective scheme of lines contained in X. F (X ) connected if n 3. Sing(X ) finite = F (X ) lci of dimension 2n 4 and ω F (X ) = O F (X ) (4 n). X smooth = F (X ) smooth. Olivier DEBARRE Cubic hypersurfaces over finite fields 3 of 22

4 High dimensions Introduction Theorem Any F q -cubic of dimension 5 contains an F q -line. Proof. Let x X (F q ). The scheme of lines through x contained in X is the intersection in P n of hyperplane, a quadric, and a cubic Chevalley Warning when n 6. When X is smooth, F (X ) is Fano when n 5 Esnault, and Fakhruddin Rajan to extend to all X. We now look at cubic surfaces, threefolds, and fourfolds. Olivier DEBARRE Cubic hypersurfaces over finite fields 4 of 22

5 Cubic surfaces Introduction The diagonal cubic surface x x x ax 3 4 = 0 contains no F q -lines when a F q is not a cube. a exists whenever q 1 (mod 3) There are smooth cubic F q -surfaces with no F q -lines for q arbitrarily large. Olivier DEBARRE Cubic hypersurfaces over finite fields 5 of 22

6 The Galkin Shinder beautiful formula X P n+1 F q F q -cubic F (X ) Gr(1, P n+1 F q ) scheme of lines contained in X N r (X ) := Card(X (F q r )) N r (F (X )) = N r (X ) 2 2(1 + q nr )N r (X ) + N 2r (X ) 2q 2r + q (n 2)r N r (Sing(X )). This formula comes from a relation between the classes of X and F (X ) in the Grothendieck ring of varieties. Olivier DEBARRE Cubic hypersurfaces over finite fields 6 of 22

7 Y smooth projective scheme defined over F q P i (Y, T ) := det(id TF, H i (Y, Q l )) =: b i (Y ) (1 T ω ij ) Z[T ] j=1 where ω ij = q i/2. The trace formula (n := dim(y )) N r (Y ) = 0 i 2n implies, for the zeta function, ( 1) i Tr(F r, H i (Y, Q l )) = ( Z(Y, T ) := exp N r (Y ) T r ) = r r 1 0 i 2n 0 i 2n b i (Y ) ( 1) i j=1 P i (Y, T ) ( 1)i+1. ω r ij Olivier DEBARRE Cubic hypersurfaces over finite fields 7 of 22

8 Zeta function of F (X ) X P 4 F q smooth cubic.the trace formula reads Z(X, T ) = 1 j 10 (1 qω jt ) (1 T )(1 qt )(1 q 2 T )(1 q 3 T ), with ω j algebraic integers and ω j = q 1/2. The Galkin Shinder formula implies Z(F (X ), T ) = 1 j 10 (1 ω jt ) 1 j 10 (1 qω jt ) (1 T )(1 q 2 T ) 1 j<k 10 (1 ω jω k T ) Olivier DEBARRE Cubic hypersurfaces over finite fields 8 of 22

9 Cohomology of F (X ) This formula gives the Betti numbers of F (X ). Actually, the full Galkin Shinder relation gives isomorphisms of Gal(F q /F q )-modules H 3 (X, Q l ) H 1 (F (X ), Q l (1)) 2 H 1 (F (X ), Q l ) H 2 (F (X ), Q l ) 2 H 1 (A(F (X )), Q l ) H 2 (A(F (X )), Q l ) The first one can also be obtained using the incidence correspondence. The second one can also be deduced by smooth and proper base change from the statement in char. 0. Olivier DEBARRE Cubic hypersurfaces over finite fields 9 of 22

10 Existence of lines Theorem Any smooth F q -cubic threefold contains at least 10 F q -lines if q 11. Proof. Write the Frobenius eigenvalues as ω 1,..., ω 5, ω 1,..., ω 5. Set r j := ω j + ω j [ 2 q, 2 q]. Use the trace formula N 1 (F (X )) = 1 r j qr j + q 2 1 j 5 + 5q + 1 j 5 1 j<k 5 = 1 + 5q + q 2 (q + 1) (ω j ω k + ω j ω k + ω j ω k + ω j ω k ) 1 j 5 r j + and study the minimum of this real function... 1 j<k 5 r j r k Olivier DEBARRE Cubic hypersurfaces over finite fields 10 of 22

11 Examples Introduction We found smooth cubic threefolds over F 2, F 3, F 4, and F 5 with no lines. The cubic threefold in P 4 F 5 with equation x x2 3 + x2 2 x 3 + 3x 1 x3 2 + x1 2 x 4 + x 1 x 2 x 4 + x 1 x 3 x 4 + 3x 2 x 3 x 4 + 4x3 2 x 4 + x 2 x x 3 x x2 2 x 5 + x 1 x 3 x 5 + 3x 2 x 3 x 5 + 3x 1 x 4 x 5 + 3x4 2 x 5 + x 2 x x5 3 is smooth and contains no F 5 -lines and 126 F 5 -points. Remains F 7, F 8, and F 9... Olivier DEBARRE Cubic hypersurfaces over finite fields 11 of 22

12 Average numbers of lines Average numbers of lines computed on random samples of 10 5 F 2 -cubic threefolds all cubics; average smooth cubics; average Olivier DEBARRE Cubic hypersurfaces over finite fields 12 of 22

13 Singular threefolds Let X P 4 F q F q -cubic with a single singular point, of type A 1 or A 2. The curve C of lines in X through the singular point is smooth of genus 4 and canonically embedded in P 3 F q. It has two pencils g3 1 and h3 1, used to embed C in C (2) by x g 1 3 x Clemens Griffiths, Kouvidakis van der Geer F (X ) is the non-normal surface obtained by gluing in C (2) these two copies of C. Olivier DEBARRE Cubic hypersurfaces over finite fields 13 of 22

14 Lines on singular threefolds Theorem For any r 1, set n r := Card(C(F q r )). We have Card(F (X )(F q )) = 1 2 (n2 1 + n 2) n 1 if g3 1 h1 3 are defined over F q ; 1 2 (n2 1 + n 2) + n 1 if g3 1 h1 3 are not defined over F q ; 1 2 (n2 1 + n 2) if g3 1 = h1 3. Again, this can also be obtained with the Galkin Shinder method. Olivier DEBARRE Cubic hypersurfaces over finite fields 14 of 22

15 Lines on singular threefolds Corollary When q 4, any cubic threefold X P 4 F q defined over F q with a single singular point, of type A 1 or A 2, contains an F q -line. Proof. We need to exclude n 1 = n 2 = 1 and g 1 3 h1 3 defined over F q. Write C(F q ) = C(F q 2) = {x}, and g 1 3 x + x + x. g 1 3 is defined over F q x + x defined over F q x, x are defined over F q 2 x = x = x g 1 3 3x h 1 3 Contradiction. Olivier DEBARRE Cubic hypersurfaces over finite fields 15 of 22

16 Lines on singular threefolds n 1 = n 2 = 0. Then q 7 (Howe Lauter Top). Weil conjectures for C: Frobenius roots ω 1,..., ω 4, ω 1,..., ω 4, with ω j = q; H monic with (real) roots r j := ω j + ω j with r j 2 q has integral coefficients; 1 j 4 r j = 1 j 8 ω j = q + 1 n 1 = q + 1; 1 j 4 r j 2 = 1 j 8 (ω2 j +2q) = q2 +8q+1 n 2 = q 2 +8q+1. Hence H(T ) = T 4 (q + 1)T 3 3qT 2 + at + b, with b = r 1 r 2 r 3 r 4 16q 2 and a = 4 j=1 b/r j 32q 3/2 integral. Computer search: such polynomials with 4 such real roots and q {2, 3, 4, 5, 7} only exist for q 3. Olivier DEBARRE Cubic hypersurfaces over finite fields 16 of 22

17 Examples Introduction We found nodal cubics threefolds over F 2 and F 3 with no lines. Over F 2 : x x 2 2 x 3 + x x 1 x 2 x 4 + x 2 3 x 4 + x x 2 1 x 5 + x 1 x 3 x 5 + x 2 x 4 x 5 contains no F 2 -lines and Over F 3 : H(T ) = T 4 3T 3 6T T 15. 2x x 2 1 x 2 + x 1 x x 2 x x 1 x 2 x 4 + x 2 x 3 x 4 contains no F 3 -lines and + x 1 x x x 2 x 3 x 5 + 2x 2 3 x 5 + x 2 x x 3 5 H(T ) = T 4 4T 3 9T T 32. Olivier DEBARRE Cubic hypersurfaces over finite fields 17 of 22

18 Zeta function of F (X ) X P 5 F q smooth cubic.the trace formula reads Z(X, T ) = 1 (1 T )(1 qt )(1 q 3 T )(1 q 4 T ) 23 j=1 (1 qω j T ), with ω j algebraic integers, ω j = q, and ω 23 = q. The Galkin Shinder formula implies Z(F (X ), T ) = 1 (1 T )(1 q 4 T ) j ((1 ω j T )(1 q 2 ω j T )) j k (1 ω j ω k T ). Olivier DEBARRE Cubic hypersurfaces over finite fields 18 of 22

19 Cohomology of F (X ) Again, the Galkin Shinder relation implies that there are isomorphisms of Gal(F q /F q )-modules H 4 (X, Q l ) H 2 (F (X ), Q l (1)) Sym 2 H 2 (F (X ), Q l ) H 4 (F (X ), Q l ) The first also follows using the incidence correspondence. The second one can also be deduced by smooth and proper base change from statement in char. 0 (Beauville Donagi, Bogomolov). Olivier DEBARRE Cubic hypersurfaces over finite fields 19 of 22

20 Existence of lines Theorem Any smooth F q -cubic fourfold contains at least 26 F q -lines if q 5. One can use another trace formula (Katz). If Sing(X ) finite, the cohomology of O F (X ) is very simple (Altman Kleiman): dim Fq H j (F (X ), O F (X ) ) = 1 for j {0, 2, 4}, the others are 0, and the multiplication H 2 (F (X ), O F (X ) ) H 2 (F (X ), O F (X ) ) H 4 (F (X ), O F (X ) ) is an isomorphism of Galois modules (Serre duality). Olivier DEBARRE Cubic hypersurfaces over finite fields 20 of 22

21 The Katz trace formula 4 N 1 (F (X )) ( 1) j Tr(F, H j (F (X ), O F (X ) )) (mod p) j=0 1 + t + t 2 (mod p) Corollary Assume q 2 (mod 3). Any F q -cubic fourfold with finite singular set contains an F q -line. This applies to q = 2 and leaves only the cases q {3, 4} open for the existence of a line on a smooth cubic fourfold. Olivier DEBARRE Cubic hypersurfaces over finite fields 21 of 22

22 Final question Introduction The computer found a smooth cubic fourfold over F 2 with a single line. Question We found no cubic fourfolds without lines. Do they exist? Olivier DEBARRE Cubic hypersurfaces over finite fields 22 of 22