Geometric Chevalley-Warning conjecture
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1 Geometric Chevalley-Warning conjecture June Huh University of Michigan at Ann Arbor June 23, 2013 June Huh Geometric Chevalley-Warning conjecture 1 / 54
2 1. Chevalley-Warning type theorems June Huh Geometric Chevalley-Warning conjecture 2 / 54
3 Let k = F q and h = homogeneous polynomial of degree d in k [z 0; : : : ; z n ]. Write V (h) for the projective hypersurface of degree d in P n. Theorem (Chevalley-Warning) If d n, then # V (h) = 1 mod p: In particular, h = 0 has a nontrivial solution over k. June Huh Geometric Chevalley-Warning conjecture 3 / 54
4 Let k = F q and h = homogeneous polynomial of degree d in k [z 0; : : : ; z n ]. Theorem (Ax) If d n, then # V (h) = 1 mod q : In particular, h = 0 has a nontrivial solution over k. June Huh Geometric Chevalley-Warning conjecture 4 / 54
5 Let k = (function field of a curve over an algebraically closed field). Theorem (Tsen) If d n, then h = 0 has a nontrivial solution over k. (n = 2; d = 2) A conic bundle over a curve has a rational section. June Huh Geometric Chevalley-Warning conjecture 5 / 54
6 Let k = (algebraically closed field of characteristic 0) and suppose V (h) is smooth. Theorem (Roitman, Kollár-Miyaoka-Mori) V (h) is rationally connected if and only if d n. (A variety is rationally connected if two general points can be joined by a rational curve.) June Huh Geometric Chevalley-Warning conjecture 6 / 54
7 We have seen that, over various base fields k, small degree hypersurfaces (d n) tend to have rational points and rational curves. On the other hand, we know that such hypersurfaces need not be rational, even over excellent fields such as C. June Huh Geometric Chevalley-Warning conjecture 7 / 54
8 Let k = C. Theorem (Clemens-Griffiths) (d = 3; n = 4) Smooth cubic hypersurfaces in P 4 are not rational. Theorem (Iskovskikh-Manin) (d = 4; n = 4) Smooth quartic hypersurfaces in P 4 are not rational. June Huh Geometric Chevalley-Warning conjecture 8 / 54
9 Question Is there a geometric Chevalley-Warning theorem? (An answer should imply the classical Chevalley-Warning over finite fields, and capture near rationality of small degree hypersurfaces over algebraically closed fields.) One possible answer can be formulated at the level of Grothendieck ring of varieties, using its connection to stable birational equivalence between algebraic varieties. June Huh Geometric Chevalley-Warning conjecture 9 / 54
10 2. The geometric Chevalley-Warning conjecture June Huh Geometric Chevalley-Warning conjecture 10 / 54
11 Let k be a field. Definition Algebraic varieties X and Y are said to be stably birational over k if there is a k-birational equivalence X P m ' Y P n for some m ; n 0. If X and Y are birational over k, then X and Y are stably birational over k. If k is algebraically closed, the set of stable birational equivalence classes SB k is a monoid under the product. June Huh Geometric Chevalley-Warning conjecture 11 / 54
12 Let k = C. Theorem (Beauville-Colliot-Thélène-Sansuc-Swinnerton-Dyer) There is a smooth projective threefold X which is not rational but X P 3 is rational. June Huh Geometric Chevalley-Warning conjecture 12 / 54
13 Let k be a field. Definition The Grothendieck ring of k-varieties, denoted by K 0(Var k ), is defined as follows. It is the abelian group whose generators are the isomorphism classes of k-schemes, and relations are [X ny ] = [X ] [Y ] whenever Y is a closed subvariety of X. Multiplication is defined by [X ] [Y ] = [X Y ]: June Huh Geometric Chevalley-Warning conjecture 13 / 54
14 Definition A motivic measure is a ring homomorphism M : K 0(Var k )! R where R is a commutative ring with 1. June Huh Geometric Chevalley-Warning conjecture 14 / 54
15 (k = F q, R = Z, M = point counting) : There is a motivic measure M count : K 0(Var k )! Z such that [X ] 7! #(X (F q )): (k = C, R = Z, M = topological Euler characteristic) : There is a motivic measure M euler : K 0(Var k )! Z such that [X ] 7! (X ): June Huh Geometric Chevalley-Warning conjecture 15 / 54
16 Let k be an (algebraically closed) field of characteristic 0. (The condition on the characteristic of k is because theorems below use the weak factorization theorem in an essential way.) Theorem (Larsen-Lunts) There is a motivic measure M sb : K 0(Var k )! Z[SB k ] which maps a smooth projective variety to its stable birational equivalence class. June Huh Geometric Chevalley-Warning conjecture 16 / 54
17 The motivic measure M sb is surjective. What is the kernel of M sb? Example By definition, P 0 and P 1 are stably birational over k. Therefore [A 1 ] = [P 1 ] [P 0 ] maps to zero under the motivic measure M sb. June Huh Geometric Chevalley-Warning conjecture 17 / 54
18 Let k be an (algebraically closed) field of characteristic 0. A variety is said to be stably rational if it is stably birational to a point. Theorem (Larsen-Lunts) The kernel of M sb is the principal ideal generated by [A 1 ]. In particular, a smooth and projective X is stably rational if and only if [X ] = 1 mod [A 1 ]: This motivates the geometric Chevalley-Warning conjecture. June Huh Geometric Chevalley-Warning conjecture 18 / 54
19 Let k be an algebraically closed field and h be a homogeneous polynomial of degree d in k [z 0; : : : ; z n ]. Conjecture (Brown-Schnetz / Liao) If d n, then [V (h)] = 1 mod [A 1 ] in the Grothendieck ring of varieties over k. Hélène Esnault also suggested a stronger statement which is equivalent to the conjecture over an algebraically closed field. June Huh Geometric Chevalley-Warning conjecture 19 / 54
20 The conjecture holds for... (n = 1; d = 1) point. (n = 2; d = 1) line. (n = 2; d = 2) smooth conic, two lines meeting at a point, double line. (n = 3; d = 1) plane. (n = 3; d = 2) smooth quadric, two planes meeting along a line, double plane. (n = 3; d = 3) smooth cubic surface, various singular cubic surfaces. (n = 4; d = 1) space. (n = 4; d = 2) quadric 3-folds, smooth or singular. June Huh Geometric Chevalley-Warning conjecture 20 / 54
21 (n = 4; d = 3) All singular cubic threefolds satisfy the conjecture. Smooth cubic threefolds are first unknown case. (n = 4; d = 4) Some highly singular quartic threefolds are known to satisfy the conjecture. All graph hypersurfaces satisfy the conjecture (Paolo Aluffi and Matilde Marcolli). Other positive results are independently obtained by Emel Bilgin and Xia Liao. Over certain quasi-algebraically closed but non-algebraically closed fields of characteristic 0, Le Dang Thi Nguyen found a smooth cubic surface which disproves the conjecture. June Huh Geometric Chevalley-Warning conjecture 21 / 54
22 3. The main result June Huh Geometric Chevalley-Warning conjecture 22 / 54
23 If one wants to disprove the conjecture (over an algebraically closed field), one has to demonstrate that a variety X is not stably rational, that is, X P n is not rational for all n 0. There are (4 + ) known ways of showing nonrationality of a Fano variety. (Clemens-Griffiths) The method of intermediate Jacobian. (Iskovskikh-Manin) Birational automorphism group. (Artin-Mumford) Torsion in H 3 (X ; Z). (Kollár) Matsusaka s theorem on degeneration + characteristic p methods. June Huh Geometric Chevalley-Warning conjecture 23 / 54
24 (Clemens-Griffiths) The method of intermediate Jacobian: This seems to work well (only?) for certain 3 dimensional varieties. (Iskovskikh-Manin) Birational automorphism group: This seems to work well (only?) for certain Picard rank 1 varieties. Even if Bir(X ) is finite, Bir(X P m ) is very large. June Huh Geometric Chevalley-Warning conjecture 24 / 54
25 (Artin-Mumford) Torsion in H 3 (X ; Z): This indeed shows that something is not stably rational (Künneth theorem). However, projective hypersurfaces tend to have no torsion in H 3. (Kollár) Matsusaka s theorem on degeneration + characteristic p methods: In fact, this method shows that a variety is not ruled (and there is no such thing as stable ruledness!). More seriously, we know very little about how rationality behaves in families. June Huh Geometric Chevalley-Warning conjecture 25 / 54
26 Theorem (H.) There is a quartic threefold V P 4 defined over Z such that [V ] 1 mod [A 1 ] in the Grothendieck ring of varieties over k, for all field k of characteristic 0. In fact, V is not stably rational over any field of characteristic 0. June Huh Geometric Chevalley-Warning conjecture 26 / 54
27 Theorem (H.) This quartic threefold V, viewed over C, has torsion in middle homology : Tor H 3(V ; Z) 0: It seems that this is the first example of an odd dimensional complete intersection with torsion in middle homology. Even dimensional complete intersections often have torsion in middle homology many examples were given by Alexandru Dimca. (I don t know much about middle cohomology of complete intersections.) June Huh Geometric Chevalley-Warning conjecture 27 / 54
28 The quartic threefold V, viewed over C, has the following properties: There is a resolution of singularities e V! V such that [ e V ] = [V ] mod [A 1]: Singularities of V are invisible (in a sense) from the point of view of stable birational geometry. Perhaps these singularities deserve a name. There is a torsion in H 3 ( e V ; Z), corresponding to torsion in H3(V ; Z). In particular, e V (and hence V ) is not stably rational: [ e V ] 1 mod [A 1]: June Huh Geometric Chevalley-Warning conjecture 28 / 54
29 4. Construction of V June Huh Geometric Chevalley-Warning conjecture 29 / 54
30 The quartic threefold V contains a quartic symmetroid as a hyperplane section; I will begin by explaining what a quartic symmetroid is. A quartic symmetroid is a surface in P 3 defined by the symmetric determinant 0 1 l 11 l 12 l 13 l 14 B l 12 l 22 l 23 l 24 C A = 0: l 13 l 23 l 33 l 34 l 14 l 24 l 34 l 44 Here l ij are linear forms in Q[z 0; z 1; z 2; z 3]. June Huh Geometric Chevalley-Warning conjecture 30 / 54
31 Quartic symmetroids have been introduced and studied by Arthur Cayley. All the properties of quartic symmetroids we need can be found in: A. Cayley. A Memoir on Quartic Surfaces. Proc. London Math. Soc. (1869) A. Cayley. Second Memoir on Quartic Surfaces. Proc. London Math. Soc. (1869) A. Cayley. Third Memoir on Quartic Surfaces. Proc. London Math. Soc. (1869) June Huh Geometric Chevalley-Warning conjecture 31 / 54
32 If the linear forms l ij are chosen sufficiently general, then the corresponding quartic symmetroid QS P 3 has 10 isolated nodes. The projection from any one of the nodes : QS P 2 is a double covering of P 2, ramified along a sextic curve defined over Q. The sextic curve is the union of two cubic curves E 1 [ E 2 P 2. (The factorization E 1 [ E 2 comes from a determinantal identity.) The cubic curves E 1 and E 2 are not defined over Q, but over Q(i). June Huh Geometric Chevalley-Warning conjecture 32 / 54
33 The cubic curves E 1 and E 2 are totally tangent to a conic curve A defined over Q. If l ij are sufficiently general, then 1. the cubic curves E 1 and E 2 are smooth and meet transversely at 9 points, 2. the conic curve A is smooth and tangent to each E i at 3 points distinct from the 9 points E 1 \ E 2. June Huh Geometric Chevalley-Warning conjecture 33 / 54
34 If coordinates are chosen so that the chosen node is (0 : 0 : 0 : 1), the defining equation of QS P 3 is (z 0; z 1; z 2)z (z 0; z 1; z 2)z 3 + (z 0; z 1; z 2) = 0: The leading coefficient defines the smooth conic A = (z 0; z 1; z 2) = 0 P 2 ; and the discriminant factors over Q(i): (z 0; z 1; z 2) 2 4(z 0; z 1; z 2) (z 0; z 1; z 2) = 1(z 0; z 1; z 2) 2(z 0; z 1; z 2): June Huh Geometric Chevalley-Warning conjecture 34 / 54
35 These factors define elliptic curves E 1 = 1(z 0; z 1; z 2) = 0 P 2 and E 2 = 2(z 0; z 1; z 2) = 0 P 2 : Choose a sufficiently general quadratic form (z 0; z 1; z 2) over Q, and define D = (z 0; z 1; z 2) = 0 P 2 : D is smooth and intersects A and E 1; E 2 transversely in the expected number of points. June Huh Geometric Chevalley-Warning conjecture 35 / 54
36 Definition Let V P 4 be the quartic threefold defined by (z 0; z 1; z 2)z (z 0; z 1; z 2)z 3 + (z 0; z 1; z 2) + (z 0; z 1; z 2)z 2 4 = 0: June Huh Geometric Chevalley-Warning conjecture 36 / 54
37 5. Analysis of V June Huh Geometric Chevalley-Warning conjecture 37 / 54
38 We work over the field of complex numbers C. Recall that V is defined by (z 0; z 1; z 2)z (z 0; z 1; z 2)z 3 + (z 0; z 1; z 2) + (z 0; z 1; z 2)z 2 4 = 0: By staring at the equation for a while, one realizes the following: 1. V is singular along the line L = fz 0 = z 1 = z 2 = 0g: 2. V has 9 isolated nodes (A 1-singularities), corresponding to 9 points E 1 \ E V has no other singularities. June Huh Geometric Chevalley-Warning conjecture 38 / 54
39 How to resolve singularities of V? First we blowup P 4 along L. The blowup b P4 resolves the rational map bp 4 P 4 (z0:z1:z2) P 2 If we write b V for the strict transform of V, the above diagram restricts to bv V (z0:z1:z2) P 2 June Huh Geometric Chevalley-Warning conjecture 39 / 54
40 Lemma bv is smooth over L. This is not trivial, but elementary. (Here one uses that (z 0; z 1; z 2) is chosen generically.) June Huh Geometric Chevalley-Warning conjecture 40 / 54
41 Let S be the exceptional surface of the blowup b V! V. Lemma S is smooth and rational over C. In particular, [S ] = 1 mod [A 1 ]: Equivalently, [ b V ] = [V ] mod [A 1]: June Huh Geometric Chevalley-Warning conjecture 41 / 54
42 S is the biprojective hypersurface in L P 2 S = (z 3 : z 4) (w 0 : w 1 : w 2) j (w 0; w 1; w 2)z (w 0; w 1; w 2)z 2 4 = 0 S is a conic bundle over L, which is rational by Tsen s theorem. : (There are a few places where Chevalley-Warning type theorems are used to disprove the Chevalley-Warning conjecture!) June Huh Geometric Chevalley-Warning conjecture 42 / 54
43 Let e V! b V be the blowup of the 9 singular points of b V. This replaces the 9 points of b V by 9 smooth quadric surfaces in e V. (This is the definition of node of a threefold hypersurface.) Since smooth quadric surfaces are rational, we have [ e V ] = [ b V ] mod [A 1]: June Huh Geometric Chevalley-Warning conjecture 43 / 54
44 We show Tor H 3 ( e V ; Z) ' Tor H 4( e V ; Z) 0 by (what Artin and Mumford call) a brutal procedure of constructing cycles. We construct a 2-dimensional oriented cycle 2 and a 3-dimensional oriented chain 3 such 3 = 2 2. j 2j and j 3j will be (real analytic) oriented pseudomanifolds (with boundary). June Huh Geometric Chevalley-Warning conjecture 44 / 54
45 How do we show that 2 is not homologous to zero? We construct a 3-dimensional oriented cycle 3 and a 4-dimensional oriented chain 4 such 4 is homologous to 2 3. j 3j will be a real analytic manifold intersecting j 3j transversely at one point. This shows that 2 pairs nontrivially with 3 under the torsion linking form: Tor H 4 ( V e ; Z) 3 (3 Tor H ( V e 3) ; Z)! Q=Z; ( 2; 3) 7! = : June Huh Geometric Chevalley-Warning conjecture 45 / 54
46 Recall that V is defined by (z 0; z 1; z 2)z (z 0; z 1; z 2)z 3 + (z 0; z 1; z 2) + (z 0; z 1; z 2)z 2 4 = 0; where (z 0; z 1; z 2) 2 4(z 0; z 1; z 2) (z 0; z 1; z 2) = 1(z 0; z 1; z 2) 2(z 0; z 1; z 2): This shows that, over a point of E 1 [ E 2 outside A [ D [ (E 1 \ E 2), the fiber of V e! P 2 consists of two P 1 intersecting at one point: r r z i 2 z4 z 3 + i 2 z4 = 0: June Huh Geometric Chevalley-Warning conjecture 46 / 54
47 Let ee 1! E 1; e E2! E 2 be the ramified double coverings obtained by adjoining r to the function fields of E 1 and E 2 respectively. These coverings are possibly ramified at (A [ D) \ E i. June Huh Geometric Chevalley-Warning conjecture 47 / 54
48 The fundamental group of E in(a [ D) is free, and consists of two types of cycles: those with trivial monodromy under the double covering (in green), and those with nontrivial monodromy under the double covering (in red). June Huh Geometric Chevalley-Warning conjecture 48 / 54
49 Construction of 2: Choose a point in E i not in A [ D [ (E 1 \ E 2). Choose a P 1 over the chosen point and call it i. 2 is the algebraic cycle equipped with the canonical orientation 2 = 1 2: June Huh Geometric Chevalley-Warning conjecture 49 / 54
50 Construction of 3: The oriented chain 3 is the following family obtained by joining a cycle in E 1 (with nontrivial monodromy) with a cycle in E 2 (with nontrivial monodromy). The orientation is given by that of the base and the canonical orientation on fibers. June Huh Geometric Chevalley-Warning conjecture 50 / 54
51 Construction of 3: Choose the other P 1 over the chosen point in E 1 and call it 0 1. The oriented cycle 3 is the following family a cycle in E 1 (with trivial monodromy). The orientation is given by that of the base and the canonical orientation on fibers. June Huh Geometric Chevalley-Warning conjecture 51 / 54
52 Construction of 4: 4 is the family of fibers of e V! P 2 over the green 4 is the family of fibers of e V! P 2 over the green circle. Moving 3 by dragging the green circle along the red circle, using appropriate limit cycles over points in E 1 \ (E 2 [ A [ D), we see that 2 3 is homologous 4. June Huh Geometric Chevalley-Warning conjecture 52 / 54
53 It appears to the speaker that, assuming weak factorization over characteristic p, the quartic threefold V we constructed satisfies [V ] 1 mod [A 1 ] over all fields with sufficiently large characteristic, while # V (F q ) = 1 mod # A 1 (F q ) for all finite fields F q. Question Is it true that [V ] 1 mod [A 1 ] over all fields with sufficiently large characteristic? June Huh Geometric Chevalley-Warning conjecture 53 / 54
54 P. Aluffi and M. Marcolli. Graph hypersurfaces and a dichotomy in the Grothendieck ring. Lett. Math. Phys. 95 (2011), M. Artin and D. Mumford. Some elementary examples of unirational varieties which are not rational. Proc. London Math. Soc. (3) 25 (1972), A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, and P. Swinnerton-Dyer. Variétés stablement rationnelles non rationnelles. Ann. of Math. (2) 121 (1985), no. 2, P. Belkale and P. Brosnan. Matroids, motives, and a conjecture of Kontsevich. Duke Math. J. 116 (2003), E. Bilgin. On the Classes of Hypersurfaces of Low Degree in the Grothendieck Ring of Varieties, Int. Math. Res. Not. IMRN, to appear. F. Brown and O. Schnetz. A K3 in 4. Duke Math. J. 161 (2012), no. 10, A. Cayley. A Memoir on Quartic Surfaces. Proc. London Math. Soc. (1869) s1-3 (1): A. Cayley. Second Memoir on Quartic Surfaces. Proc. London Math. Soc. (1869) s1-3 (1): A. Cayley. Third Memoir on Quartic Surfaces. Proc. London Math. Soc. (1869) s1-3 (1): A. Dimca. On the homology and cohomology of complete intersections with isolated singularities. Compositio Math. 58 (1986), no. 3, X. Liao. Stable birational equivalence and geometric Chevalley-Warning. Proc. Amer. Math. Soc., to appear. L. D. T. Nguyen. Unramified cohomology, A 1 -connectedness, and the ChevalleyWarning problem in Grothendieck ring. C. R. Acad. Sci. Paris, Ser. I 350 (2012) H. Seifert and W. Threlfall, A Textbook of Topology. Pure and Applied Mathematics, 89. Academic Press, Inc., New York-London, June Huh Geometric Chevalley-Warning conjecture 54 / 54
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