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On the Chow-Witt rings of toric surfaces
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On the Chow-Witt rings of toric surfaces

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Content ON THE CHOW-WITT RINGS OF TORIC SURFACES by Wenhan Jiang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) May 2024 Copyright 2024 Wenhan Jiang Dedication I dedicate this thesis to my advisor, Prof. Aravind Asok, who told me that it is normal to get stuck in research and one should never feel frustrated and lose hope in solving problems. ii Acknowledgements I would like to thank my advisor for giving me this interesting thesis topic and for fruitful discussions about the classification of toric surfaces, without whom the thesis would not exist. I would like to thank the USC Math Department for six years of financial support. I also thank my committee members, all of my friends, and my family for their support. iii Table of Contents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Milnor-Witt K Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Chow-Witt Groups and Chow-Witt Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Chow Ring of Toric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Chapter 3: Chow-Witt Rings of Real Toric Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.1 Outline for the Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Computation of Hn (X,I n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.3 The Additive Structure of CHf i (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 The Multiplicative Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Chapter 4: Chow-Witt Rings of Toric Surfaces Over a More General Field . . . . . . . . . . . . . . 36 4.1 The Computation of Hi (X,W) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 The Computation of Hi (X,I i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Geometric Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 iv Abstract We compute the Chow-Witt rings of smooth projective toric surfaces based on joint work by HornbostelWendt-Xie-Zibrowius. By considering the real cycle class map, one can turn the real case of the problem into a problem in algebraic topology and interpret the I j -cohomology as an analogue of the singular cohomology with the integer coefficients. v Chapter 1 Introduction Let X be a smooth scheme with dimension n over a field k with characteristic not equal to 2 and let E be a rank n vector bundle over X. In 2000, Jean Barge and Fabien Morel introduced Chow-Witt groups CHf i (X) in [6] and defined the Euler class e(E) ∈ CHf n (X), for solving the following problem: for X affine, is the vanishing of the Euler class the only obstruction for E to have a free direct summand of rank one. In their definition, the i-th Chow-Witt group is the i-th cohomology of the Gersten complex C ∗ (X, KMW j ) = C ∗ (X, KM j )×C∗(X,Ij/Ij+1,OX)C ∗ (X, Ij , OX)(see definition 1.2 in [6]). Roughly speaking, an element in CHf i (X) is an element in the Chow group CHi (X) together with a quadratic form on it. One may expect the Chow-Witt groups reveal more information than the Chow groups in enumerative geometry. In fact, Jesse Kass and Kirsten Wickelgren showed that for a smooth cubic surface V ⊂ P 3 , there is an equality in the Grothendieck-Witt group (See [17, Theorem 2]) X lines TrL/K⟨α⟩ = 15 · ⟨1⟩ + 12 · ⟨−1⟩, where ⟨a⟩ ∈ GW(F) denotes the one-dimensional symmetric bilinear form (x, y) 7→ axy, TrL/k denotes the trace GW(L) → GW(k) and α is the associated arithmetic type of the line in L ∗/(L ∗ ) 2 . This signed count of the number of lines on V is a refinement of the classical result that there are 27 lines on a smooth 1 cubic surface over C and signifies the power of Morel’s degree map in A 1 -homotopy theory. Up to now, the computation of Chow-Witt rings is still far from finished. In 2013, Fasel computed the Chow-Witt groups of projective spaces P n and proposed the projective bundle theorem in I j -cohomologies. Later, Matthias Wendt computed the Chow-Witt rings of Grassmannians and other classifying spaces, including BSLn and BSp2n, based on the fundamental square CHf n (X) ker ∂ Hn (X,I n ) Chn (X). For smooth schemes over R, the joint work of Hornbostel-Wendt-Xie-Zibrowius showed that the computation of Chow-Witt rings of a cellular space can be reduced to a purely algebraic topology problem. In this paper, we will first compute the Chow-Witt ring of real toric surfaces using real cycle class maps as in [16] and then apply Wendt’s method to compute the Chow-Witt ring of toric surfaces over a perfect field with characteristic not 2. 2 Chapter 2 Preliminaries 2.1 Milnor-Witt K Theory Throughout this chapter, every X will denote a smooth variety over a field F with characteristic not equal to 2. We want to define the Chow-Witt rings by means of Gersten resolutions. First of all, one has the following Milnor-Witt K-theory for a field F. Definition 2.1.1. Let F be a field. The Milnor-Witt K-theory of F is the graded associative ring KMW ∗ (F) generated by symbols [u], for each unit u ∈ F ×, of degree +1, and one symbol η of degree −1 subject to the following relations: • (1) (Steinberg relation) For each a ∈ F × − {1}, [a].[1 − a] = 0 • (2) For each pair (a, b) ∈ (F ×) 2 : [ab] = [a] + [b] + η.[a].[b] • (3) For each u ∈ F ×: η.[u] = [u].η • (4) η.h = 0, where h = 2 + η.[−1] is the hyperbolic form. We also have the following Milnor K-theory of F obtained by "mod η". Definition 2.1.2. For a field F, KM ∗ (F) := KMW ∗ (F)/η is called the Milnor K-theory of F. In particular, KM 0 (F) = Z, KM 1 (F) = F ×. 3 To understand KMW 0 (F) we have the following proposition due to Morel. Proposition 2.1.1. (See [20, Lemma 3.9].) We have KMW 0 (F) ∼= GW(F), where GW(F)is the Grothendieck group of the monoid of stable isomorphism classes of nondegenerate symmetric bilinear forms over F. The isomorphism is given by GW(F) → KMW 0 (F) ⟨a⟩ 7→ 1 + η[a], where ⟨a⟩ ∈ GW(F) denotes the one dimensional symmetric bilinear form (x, y) 7→ axy. Remark. GW(F) admits a ring structure where the multiplication is determined by the tensor product of symmetric bilinear forms. The following proposition is due to Milnor-Husemoller and gives the generators of GW(F). Proposition 2.1.2. (See [19, Lemma (1.1), Chap. IV]. ) The group GW(F) is generated by the elements < u >, u ∈ F ×, and the following relations give a presentation of GW(F): 1) ⟨uv2 ⟩ = ⟨u⟩; 2) ⟨u⟩ + ⟨−u⟩ = ⟨1⟩ + ⟨−1⟩; 3) ⟨u⟩ + ⟨v⟩ = ⟨u + v⟩ + ⟨(u + v)uv⟩ if (u + v) ̸= 0. Definition 2.1.3. The Witt ring of a field F is W(F) := GW(F)/h, where h = ⟨1⟩ + ⟨−1⟩ is called the hyperbolic form. Set ϵ = −⟨−1⟩ ∈ KMW 0 (F). Then the last relation in Milnor-Witt K theory η.h = 0 is the same as ϵ.η = η. Moreover, we have the following properties that describe the graded commutative ring structure of KMW ∗ (F). 4 Proposition 2.1.3. (See [20, Lemma 3.7]. ) 1) For a ∈ F × one has: [a][−a] = 0 and < a > + < −a >= h; 2) For a ∈ F × one has: [a].[a] = [a].[−1] = [−1].[a]; 3) For a ∈ F × one has: [a].[b] = ϵ.[b].[a]; 4) For a ∈ F × one has: < a2 >= 1. Theorem 2.1.1. (See [20, Lemma 3.8]. ) We have a Cartesian square: GW(F) Z W(F) Z/2 rank mod h mod 2 rank (2.1) Definition 2.1.4. The fundamental ideal I(F) is the kernel of the rank map. GW(F) Z rank Let I n (F) denote the n-th power of I and let ¯I n (f) = I n (F)/In+1(F). Note that I(F) is generated by Pfister forms << a >>:=< 1, −a >. Similarly, the Pfister forms << a1, · · · , an >>:=<< a1 >> ⊗ · · · ⊗ << an >> generate I n (F). The proof can be found in [18, Page 329]. Note that we have a map sn : KM n (F) → ¯I n (F) given by {a1, · · · , an} 7→<< a1, · · · , an >> . Let ρ be the quotient map I n (F) → ¯I(F). In fact, the Cartesian square in (2.1) is a special case of the following one. 5 Theorem 2.1.2. (See [21, Théorème 5.3].) For a general n, KMW n (F) fits into a Cartesian square: KMW n (F) KM n (F) I n (F) ¯I n (F) sn ρ Remark. It worth mentioning that the map sn : KM n (F) → ¯I n (F) factors through a morphism s¯n : KM n (F)/2 → ¯I n (F) and the Milnor conjecture on quadratic forms is the assertion that s¯n is an isomorphism for any field F (irrespective of characteristic). For any field with characteristic zero, the conjecture was later proved by the joint work of Orlov-Vishik-Voevodsky (See [24, Theorem 4.1]). Under this isomorphism, sn can be identified with KM n (F) mod 2 −−−→ KM n (F)/2. For a valuation v : F ∗ → Z, we denote by Oν, mν, κ(ν) its ring, maximal ideal and residue field. Let π be a uniformizing parameter. We have the following theorem due to Morel. Theorem 2.1.3. (See [20, Theorem 3.15].) There exists a unique homomorphism of graded groups ∂ π v : KMW ∗ (F) → KMW ∗−1 (κ(v)) commuting with the product by η and such that ∂ π v ([π, u2, · · · , un]) = [¯u2, · · · , u¯n] and ∂ π v ([u1, · · · , un]) = 0 for any units u1, · · · , un ∈ O× v . It worth mentioning that ∂ π v depends on both uniformizing parameter π and valuation v (See [8, Remark 1.8]). There is also a specialization map s π v : KMW ∗ (F) → KMW ∗ (k(v)) that is a graded ring homomorphism and is given by s π v (α) = ∂ π v ([π]α) − [−1]∂ π v (α). 6 Proposition 2.1.4. (See [5, Chapter 2, Lemma 1.1.3].) Both the kernel of ∂ π v and the restriction of s π v to this kernel are independent of the choice of the uniformizing parameter π. Let Smk denote the category of smooth schemes over k. We are now ready to define a Nisnevich sheaf KMW n on Smk for each n ∈ Z. Definition 2.1.5. K MW n (X) := Ker (KMW n (k(X)) → M x∈X(1) KMW n−1 (k(x))), where the map is induced by the residue homomorphism ∂ πx vx . To see X 7→ KMW n (X) defines a presheaf on Smk, let f : X → Y be any morphism. Note that f has a standard factorization X i −→ X ×k Y π−→ Y , where i is a closed immersion and π is a smooth projection. Note that each smooth morphism h : S → T induces a pull-back map h ∗ : KMW n (T) → KMW n (S)(See [11, Proposition 6.6] ). On the other hand, one can define j ∗ : KMW n (X′ ) → KMW n (X) for a closed immersion j : X → X′ as follows. Since a closed immersion of smooth schemes is always a regular embedding, one can find a Zariski open covering of X′ by open subsets U such that for each U, X ∩ U → U admits a factorization of codimension 1 closed immersions X ∩ U = U0 j1 −→ U1 j2 −→ · · · jd −→ Ud = U. Let vi be the valuation on k(Ui) determined by ji with uniformizing parameter π. We then have k(Ui−1) = k(vi) and the specialization map s π vi : KMW n (k(Ui)) → KMW n (k(Ui−1)) 7 restricts to a map KMW n (Ui) → KMW n (k(Ui−1)). One can also show that s π vi (KMW n (Ui)) ⊂ KMW n (Ui−1) (See [20, Lemma 2.12]) and denote j ∗ U,i : KMW n (Ui) → KMW n (Ui−1) to be the restriction of s π vi . Furthermore, by [20, Lemma 2.13], the composition j ∗ U = j ∗ U,n ◦ · · · ◦ j ∗ U,1 is independent of the factorization of X ∩ U → U as a sequence of codimension 1 closed immersions. Therefore, we have a well-defined pull-back j ∗ U : KMW n (U) → KMW n (X ∩ U) for any closed immersion j : X → V . Furthermore, by the proof of [20, Lemma 2.12], j ∗ U glues to a morphism j ∗ : KMW n (X′ ) → KMW n (X). Finally, we have f ∗ : KMW n (Y ) → KMW n (X) as the composition K MW n (Y ) π ∗ −→ K MW n (X ×k Y ) i ∗ −→ K MW n (X). Proposition 2.1.5. (See [3, Theorem 4.1.3]. ) For any smooth scheme X, the sheaves KMW n admit a resolution of the form K MW n |X → M x∈X(0) K MW n (κx) → · · · → M x∈X(i) K MW n−i (κx) → · · · . We write C ∗ (X, KMW n ) for this complex. Remark. This is a flasque resolution of the KMW n on the small Zariski site of X. It is a fact that it also computes Nisnevich cohomology of X. One can similarly define the Nisnevich sheaves KM n , I n , I n/I n+1 and W on the category Smk and we have the following analogous Gersten resolutions. Definition 2.1.6 (Gersten resolution). The Gersten resolution of the sheaf KM n is of the form K M n |X → M x∈X(0) K M n (κx) → · · · → M x∈X(i) K M n−i (κx) → · · · . We write C ∗ (X, KM n ) for this complex. 8 • For the sheaf I n , we have C ∗ (X,I n ) : I n (k(X)) → M x1∈X(1) I n−1 (OX,x1 ) → · · · → M xi∈X(i) I n−i (OX,xi ) → · · · and C ∗ (X,I n/I n+1) : I n /In+1(k(X)) → · · · → M xi∈X(i) I n−i /In+1−i (OX,xi ) → · · · • For the sheaf W, we have the Gersten-Witt complex 0 → W(k(X)) → M x1∈X(1) W(k(x1)) → · · · → M xd∈X(d) W(k(xd)), where dim X = d. The map sn : Kn (M)(F) → I n/In+1(F) induces a morphism of Gersten complexes C ∗ (X, KM n ) → C ∗ (X,I n/I n+1) and the reduction map ρ : I n (F) → I n (F)/In+1(F)induces a morphism C ∗ (X,I n ) → C ∗ (X,I n/I n+1). Theorem 2.1.4. (See [2, Proposition 2.3.1].) Let k be a perfect field. For any smooth scheme X, there is a (functorial in X) quasi-isomorphism of complexes C ∗ (−, K MW j ) ≃ C ∗ (−, K M j ) ×C∗(−,I j/I j+1) C ∗ (−,I j ). 2.2 Chow-Witt Groups and Chow-Witt Rings We are now ready to define Chow-Witt groups. Definition 2.2.1. Let X be a smooth scheme over a field k. Its i-th Chow-Witt group is CHfi (X) := Hi Zar(X, K MW i ). 9 Remark. By [20, Corollary 5.43], for any strongly A 1 -invariant sheaf of abelian groups M, and for any essentially smooth scheme X, H∗ Zar(X, M) ∼= H∗ N is(X, M), i.e., the Gersten complex of KMW ∗ computes the cohomology in both Zariski cohomology and the Nisnevich cohomology. Note that KMW n , KM n , I n , I n/I n+1 , W are all strongly A 1 invariant sheaves. Because of this reason, we will omit the topology of cohomologies throughout the paper. One can also study Chow groups via Gersten resolutions. Indeed, consider the Gersten complex K M n |X → M x∈X(0) KM n (k(x)) → · · · → M x∈X(n−1) KM 1 (k(x)) → M x∈X(n) KM 0 (k(x)) → 0. Since KM 0 (F) = Z, KM 1 (F) = F × for any field F, we have Hn (X, K M n ) ∼= Coker( L x∈X(n−1) k(x) ∗ L x∈X(n) Z ∂ (n−1) ) ∼= CHn (X), where the right hand side is the n-th Chow group of X and ∂ (n−1) is the divisor map. Now let CHf • = Ln i=0 CHf i (X). The ring structure of CHf • (X) can be found in [8]. We list a few important definitions and results here. In [8], Rost defines a product ⊙ : C i (X, KM r ) × C j (Y, KM s ) → C i+j (X × Y, KM r+s ) in the following way: 10 Let X, Y be smooth schemes and let p1 : X × Y → X, p2 : X × Y → Y be projection maps. Let u ∈ (X × Y ) (i+j) , x ∈ X(i) , y ∈ Y (j) be such that p1(u) = x, p2(u) = y. Let ρ = {a1, · · · , ar−i} ∈ KM r−i (k(x)) and µ = {b1, · · · , bs−j} ∈ KM s−j (k(y)). Then (ρ ⊙ µ)u = l((k(x) ⊗ k(y))u){(a1)u, · · · ,(ar−i)u,(b1)u, · · · ,(bs−j )u} where the (am)u and (bn)u denote the images of am and bn under the inclusion maps k(x) → k(u) and k(y) → k(u), and l((k(x) ⊗ k(y))u) is the length of the module k(x) ⊗k k(y) localized in u. The product ⊙ has the following property. Proposition 2.2.1. (Rost, 1996) For any ρ ∈ C i (X, KM r ) and µ ∈ C j (Y, KM s ) we have d(ρ ⊙ µ) = d(ρ) ⊙ µ + (−1)j ρ ⊙ d(µ). The proof can be found in [22][Paragraph 14.4]. Fasel also defined a product ⋆ : C i (X,I r )×C j (Y,I s ) → C i+j (X ×Y,I r+s ) together with the product ⋄ : C i (X, K MW r ) × C j (Y, K MW s ) → C i+j (X × Y, K MW r+s ) in [9]. To begin with, let X be a connected scheme and let G(X) be the category of graded line bundles whose objects are pairs (L, a) where L is an invertible OX-module and a ∈ Z and whose morphisms are of the form HomG(X) ((L, a),(L ′ , a′ )) =    ∅ if a ̸= a ′ Isom(L, L′ ) if a = a ′ . 11 Note that all morphisms in this category are isomorphisms. One can endow G(X) with a symmetric monoidal structure by (L, a) ⊗ (L ′ , a′ ) := (L ⊗ L ′ , a + a ′ ). One can also define the twisted version of Chow-Witt rings as follows. This is done by Fasel in [9]. Definition 2.2.2. We define the n-th Milnor-Witt group of F twisted by (i, L) as KMW n (F,(i, L)) = KMW n (F) O Z[F ×] Z[L ×]. and K MW n (X,(i, L)) = K MW n (X) O Z[Gm] Z[L ×]. To make sense of the definition, the (multiplicative) action of units on KMW n (F) comes from the fact that each u ∈ F × defines an element of KMW 0 (F), namely ⟨u⟩. Let V denote the category of vector bundles on X in which morphisms are vector bundle isomorphisms. Let D : V(X) → G(X) be the functor given by V 7→ (det V,rank (V )). Let X, Y be essentially smooth schemes and let x ∈ X(i) and y ∈ Y (i ′ ) . Since k is a perfect field, k(x) ⊗ k(y) = k(u1) × · · · × k(un) for some u1, · · · , un ∈ (X × Y ) (i+i ′ ) . Definition 2.2.3. Let x ∈ X(i) , y ∈ Y (i ′ ) and u ∈ (X × Y ) (i+i ′ ) such that u = ui for some i. Then we define µ(x, y; u) : KMW i (k(x), D(mx/m 2 x ) −1 ) × KMW i ′ (k(y), D(my/m 2 y ) −1 ) → KMW i+i ′ (k(u), D(mu/m 2 u ) −1 ) 12 as follows. Explicitly, let α ∈ KMW i (k(x)), x1, · · · , xi be generators of mx/m2 x , β ∈ KMW i ′ (k(x)), and y1 , · · · , yi ′ be generators of my/m2 y , then µ(x, y; u)(α ⊗ x ∗ 1 ∧ · · · ∧ x ∗ i , β ⊗ y ∗ 1 ∧ · · · ∧ y ∗ i ′) = α · β ⊗ x ∗ 1 ∧ · · · ∧ x ∗ i ∧ y ∗ 1 ∧ · · · ∧ y ∗ i ′ Definition 2.2.4. [5] For any scheme X, any graded line bundle (i,L) over X, any closed subset Z ⊂ X and any n ∈ N, we define the n-th Chow-Witt group twisted by (i,L) by CHfn (X,(i,L)) := Hn (X, KMW n (i,L)). The ring structure is induced by C i (X, K MW r ,(j,L)) × C i ′ (Y, K MW s ,(j ′ ,L ′ )) → C i+i ′ (X × Y, K MW r+s ,(j + j ′ ,L ⊗ L′ )). 2.3 The Chow Ring of Toric Varieties In this section, we are working with varieties over a field. We are interested in "split" toric varieties, i.e. , the equivariant compactifications of the split group scheme G×n m . To be more precise, we first recall a few definitions. We use the same notations as [7] or [12] for toric varieties. Definition 2.3.1. A toric variety over a field k is an irreducible variety X containing a k-split torus TN ∼= G×n m as a Zariski open subset such that the action of TN on itself extends to an algebraic action of TN on X. As many invariants of toric varieties can be computed in terms of naturally attached combinatorial data, it suffices to define them in terms of cones and fans. To begin with, say we have a lattice N, isomorphic to Z n . Let M = Hom(N, Z) denote the dual lattice. Let V be the vector space NR := N ⊗Z R, with dual space V ∗ = MR = M ⊗Z R. 13 Definition 2.3.2. A convex polyhedral cone is a set σ = {r1v1 + · · · + rsvs ∈ V : ri ≥ 0} generated by any finite set of vectors v1, · · · , vs in V . Definition 2.3.3. The dual cone of σ is the set of equations of supporting hyperplanes, i.e., σ ∨ = {u ∈ V ∗ : ⟨u, v⟩ ≥ 0 for all v ∈ σ} Note that σ ∨ determines a semigroup Sσ = σ ∨ ∩ M = {u ∈ M : ⟨u, v⟩ ≥ 0 for all v ∈ σ }. A character of a torus T is a morphism χ : T → Gm. Explicitly, for each m = (a1, · · · , an) ∈ Z n , let χ m be the character given by χ m(t1, · · · , tn) = t a1 1 · · ·t an n Definition 2.3.4. The affine toric variety Uσ associated to the cone σ is Spec k[Sσ], where k[Sσ] = {Σm∈Sσ cmχ m | cm ∈ k and cm = 0 for all but finitely many m}. Example 2.3.1. Let N ∼= Z n . Then • σ = span{e1, · · · , en} gives k[Sσ] = k[x1, · · · , xn], Uσ ∼= A n . • σ = {0} gives k[Sσ] = k[x1, x−1 1 , · · · , xn, x−1 n ], Uσ ∼= G×n m . Definition 2.3.5. A cone σ is smooth if it can be generated by a subset of a basis of the lattice. 14 Proposition 2.3.1. Let σ ⊆ NR be a strongly convex rational polyhedral fan. Then Uσ is smooth if and only if σ is smooth. Now, let Σ be a fan. Let σ1, σ2 be two cones in Σ with τ = σ1 ∩ σ2. Then one can glue Uσ1 with Uσ2 via an isomorphism gσ2,σ1 : (Uσ1 )χm ≃ (Uσ2 )χ−m. Let X be a nonsingular toric surface associated to a fan Σ. Suppose Σ is given by vectors v0, · · · , vd−1, vd = v0 in the counterclockwise order such that for all i, vi and vi+1 give a basis of Z 2 . Lemma 2.3.1. Let σ be a strongly convex rational polyhedral cone in NR. Let N(σ) = N/σ ∩N. Then there is a natural isomorphism HomZ(σ ⊥ ∩ M, k×) ∼= TN(σ) , where TN(σ) = N(σ) ⊗Z k × is the torus associated to N(σ). We have the Orbit-Cone Correspondence as follows. Proposition 2.3.2. (See [7, Theorem 3.2.6].) Let XΣ be the toric variety of the fan Σ in NR. Then: (a) There is a bijective correspondence {cones σ in Σ } ←→ {TN -orbits in XΣ} σ ←→ O(σ) ≃ HomZ(σ ⊥ ∩ M, k×). (b) Let n = dimNR. For each cone σ ∈ Σ, dim O(σ) = n − dim σ. 15 (c) Uσ = ∪τ⪯σO(τ ). (d) Let O(τ ) denote the Zariski closure of O(τ ). Then τ ⪯ σ if and only if O(σ) ⊆ O(τ ), and O(τ ) = ∪τ⪯σO(σ). Remark: We say a fan Σ ′ refines Σ if every cone of Σ ′ is contained in a cone of Σ. Note that toric blowups are closed related to subdivisions of fans. Definition 2.3.6. Let Σ be a fan in NR ≃ R n and let σ ∈ Σ be a smooth cone such that v1, · · · , vn is a basis for N and σ = Cone(v1, · · · , vn). Let v0 = v1 + · · · + vn and let Σ ′ (σ) = {Cone(A) : A ⊆ {u0, · · · , un} and A ⊉ {u1, · · · , un}}. Then Σ ∗ (σ) = (Σ\{σ}) ∪ Σ ′ (σ) is called the star subdivision of Σ along σ. Example 2.3.2. Let Σ be a fan in NR ≃ R 2 that is generated by a smooth cone σ = Cone(v1, v2) together with its faces. Then the star subdivision Σ ∗ (σ) consists of two cones Cone(v1, v1 + v2), Cone(v2, v1 + v2) together with their faces. Observe that XΣ∗(σ) ≃ Bl0A 2 . Indeed, up to linear isomorphism, we may assume v1 = e1, v2 = e2 so that σ1 = Cone(e1, e1 + e2) and σ2 = Cone(e2, e1 + e2) generate Σ ∗ (σ). We have Xσ1 = Spec k[xy−1 , y], Xσ2 = Spec k[x −1y, x], and the gluing is given by the identification of localizations k[xy−1 , y]xy−1 ∼= k[x −1 y, x]x−1y . 16 On the other hand, Bl0A 2 , the subvariety given by {((x1, x2), [y1, y2]) : x1y2 = x2y1} ⊆ A 2 ×P 1 , is covered by U1 = Spec k[y2/y1, x1] and U2 = Spec k[y1/y2, x2]. Thus, x = x1, y = x2 gives an explicit identifications Xσ1 ∼= U1, Xσ2 ∼= U2. In higher dimensions, it is still true that Bl0A n is isomorphic to XΣ∗(σ) , where σ = Cone(e1, · · · , en) and Σ is generated by σ. More generally, X∗ Σ (σ) is exactly the blowup of XΣ at the toric invariant point O(σ). The details of the proof can be found in [7][Prop.3.3.15]. One may wish to classify smooth projective toric surfaces up to blowups. Let Fa denote the Hirzebruch surface P(OP1 ⊕ OP1 (a)). We have the following result due to Fulton. Proposition 2.3.3. (See [7, Theorem 10.4.3].) Every smooth complete toric surface XΣ is obtained from either P 2 , P 1 × P 1 or Fr(r ≥ 2) by a finite sequence of successive blow-ups at fixed points of the torus action. This follows from the following claims. See [7], Theorem 10.4.3 for more details. Claim. • (1) If d ≥ 5, there must be some j, 1 ≤ j ≤ d, such that vj−1 and vj+1 generate a strongly convex cone, and vj = vj−1 + vj+1. • (2) For d = 3, the resulting toric variety must be P 2 . For d = 4, the resulting toric surface must be a Hirzebruch surface Fa for some a ∈ Z. Recall: The Chow ring CH• (X) of an arbitrary toric variety X = XΣ is generated by the classes Di , where Di ’s are the irreducible T-divisors that corresponds to the minimal lattice points v1, · · · , vd 17 along the edges (See [12], Section 5.2). More precisely, the following proposition gives the relation of the generators. Proposition 2.3.4. [12, Propostion in Section 5.2]. Let XΣ be a smooth projective toric variety. The Chow ring of XΣ is given by Z[D1, · · · , Dd]/I, where I is the ideal generated by all (1) Di1 · · · Dik for vi1 , · · · , vik not in a cone of ∆. (2) Pd i=1 ⟨u, vi⟩ Di for all u in M. Example 2.3.3. The Chow ring of Blx(P 1 × P 1 ) and BlyF2 Let X = Blx(P 1 × P 1 ) = XΣ, Y = BlyF2 = YΓ, where Σ is generated by u1 = e1, u2 = e2, u3 = −e1 + e2, u4 = −e1, u5 = −e2 and Γ is generated by v1 = e1, v2 = e2, v3 = −e1 + 3e2, v4 = −e1 + 2e2, v5 = −e2 By Proposition 2.3.4, CH• (Y ) ∼= Z[D1, D2, · · · , D5]/I, where I =< Σ 5 i=1 ⟨ej , ui⟩ Di ,(j = 1, 2), D1D3, D1D4, D2D4, D2D5, D3D5 > =< D1 − D3 − D4, D2 + 3D3 + 2D4 − D5, D1D3, D1D4, D2D4, D2D5, D3D5 > 18 This gives CH• (Y ) ∼= Z[D1, D2, D3]/ < D1D3, D2 1 , D1D2 − D2D3, D2 2 + 3D2D3, D2D3 + D2 2 > . Similarly, direct computation gives CH• (X) = Z[D1, D2, D3]/ < D1D3, D2 1 , D1D2 − D2D3, D2 2 + D2D3, D2D3 + D2 2 > One can verify that D2 7→ D2 + D1 gives a ring isomorphism from CH• (X) to CH• (Y ). Finally, we briefly recall the cellular decomposition of toric surfaces. The fact that toric surfaces are cellular follows from various lemmas in [12, section 5.2]. Indeed, one can show that there exists a filtration of closed subschemes X = Z1 ⊃ Z2 ⊃ · · · ⊃ Zm, with Zi\Zi+1 = Yi = ∪τi⊂γ⊂σiOγ = V (τi) ∩ Uσi . In particular, if X is nonsingular, we have Yi ∼= A n−ki , where ki = dim(τi) and τi , σi are cones defining the toric variety X. 19 Chapter 3 Chow-Witt Rings of Real Toric Surfaces 3.1 Outline for the Computation Throughout this chapter, all the toric surfaces are real toric surfaces. By [16], we have a Cartesian diagram CHf n (X) ker ∂n Hn (X,I n ) Chn (X) (3.1) that comes from the following key diagram (See [15], Section 2.4) CHn (X) CHn (X) Hn (X,I n+1) CHf n (X) CHn (X) Hn+1(X,I n+1) Hn (X,I n+1) Hn (X,I n ) Chn (X) Hn+1(X,I n+1) 0 0 Chn+1(X). = 2 = ∂n mod 2 = η ρ β Sq2 ρ (3.2) The second row is the long exact sequence induced by the short exact sequence 0 → I n+1 → KMW n → KM n → 0 and ∂n : CHn (X) → Hn+1(X,I n+1) is its connecting homomorphism. The third row is the long exact sequence induced by 0 → I n+1 → I n → I n/I n+1 → 0. Totaro has established that Sq2 = β◦ρ in [23, 20 Theorem 0.1], where ρ : Hn+1(X,I n+1) → Chn+1(X) and β : Chn (X) → Hn+1(X,I n+1). Furthermore, the map CHf n (X) → Hn (X, I n ) ×Chn(X) ker ∂n is surjective, and is injective when the 2-torsion is trivial in CHn (X). This is done by Hornbostol-Wendt in [15, Proposition 2.11]. Since the map Ker ∂n → Chn (X) is given by reduction modulo 2, it suffices to understand the map Hn (X, I n ) → Chn (X) and Ker ∂n ⊆ CHn (X). Now, let X be a smooth projective toric surface. By Proposition 2.3.3, we can classify X as a successive blowups from P 2 , P 1 × P 1 or Fr. In terms of A 1 -homotopy, one has a stronger result that is established by Asok-Morel. Proposition 3.1.1. (See [4, Proposition 3.2.10].) Fr and F ′ r are A 1 -weakly equivalent if and only if r = r ′ mod 2. Indeed, there is an A 1 -h-cobordism between the bundles OP1 (−r+ 1)⊕OP1 (−1) and OP1 (−r)⊕OP1 via the family of transition functions   t a xt 0 1   . Taking projective bundles, we have Hr−2 ≃ Hr. For more details, see [4, Proposition 3.2.10]. Note that we have H0 ∼= P 1 × P 1 and H1 ∼= BlxP 2 . To compute a A 1 -invariant cohomology theory of a toric surface X (e.g., CH• (X), CHf • (X)), it suffices to compute that of blow-ups from P 2 and P 1 × P 1 . Since blowing up a point on P 1 × P 1 is isomorphic to blowing up two points on P 2 , blowups on P 1 × P 1 can also be identified with blowups on P 2 . Therefore, it suffices to compute the following two cases: • (a) CHf • (Xm), where Xm = Blxm(· · · Blx1 (P 2 )) be a successive blow-up of P 2 . • (b) CHf • (P 1 × P 1 ). 21 3.2 Computation of Hn (X,I n ) To compute Hn (X,I n ), one may wish to use the following result established by Horbonstol-Wendt-XieZibrowius for cellular varieties in [16, Theorem 5.7]. Proposition 3.2.1. Let X be a smooth cellular variety over R. Consider the real cycle class maps Hi (X,I j ) → Hi (X(R), Z). For j ≥ i, the real cycle class map is a group isomorphism. In particular, we have a graded commutative ring isomorphism L i Hi (X,I i ) ∼= L i Hi (X(R), Z). Recall: Note that Xm(R) is nonorientable. In fact, (BlxX)(R) ∼= X(R)#RP2 , where # denotes taking connected sums. In particular, for Xm = Blxm(· · · Blx1 (P 2 )), Xm(R) ∼= (RP2 ) #(m+1) is represented by the polygon D E1 D E1 E2 E2 where D denotes the real realization of the hyperplane class of P 2 and Ei denotes the real realization of the exceptional divisors. The singular homology of Xm(R) is computed by the complex Z (2,2,··· ,2)T −−−−−−→ Z ⊕(m+1) 0 −→ Z. 22 and the Z/2 cohomology is computed by the cochain complex Z/2 0 −→ Z/2 ⊕(m+1) 0 −→ Z. It follows that Hi (Xm(R), Z/2) ∼=    Z/2 if i = 0 (Z/2)⊕m+1 if i = 1 Z/2 if i = 2. and the generators of H1 (Xm(R), Z/2) are precisely D¯, E¯ i , 1 ≤ i ≤ m. Let E¯ 0 = D¯. Hatcher has computed that E¯ i∪E¯ j = 0 for all i ̸= j and E¯ i∪E¯ i is the nontrivial element of H2 (Xm(R), Z/2) in [14, Example 3.8]. This gives the following result on the Z/2-cohomology ring of Xm(R). Proposition 3.2.2. H• (Xm(R), Z/2) ∼= Z/2[E¯ 0, E¯ 1, · · · , E¯m]/ ¯I, where ¯I is generated by E¯2 i − E¯2 j , E¯ iE¯ j , E¯3 i , 0 ≤ i, j ≤ m and i ̸= j. One can similarly work out the singular cohomology of Xm(R) with Z coefficients. Lemma 3.2.1. Hi (Xm(R), Z) ∼=    Z if i = 0 Z ⊕m if i = 1 Z/2 if i = 2. Proof. Note that Xm(R) ∼= Xm−1(R)#RP2 , Xm(R) admits an open covering given by Xm−1(R)\pt and RP2 \pt. Consider the Mayor-Vietoris long exact sequence with Z coefficients 0 → H1 (Xm(R)) → H1 (Xm−1(R)\x) ⊕ H1 (RP2 \pt) → H1 (S 1 ) → H2 (Xm(R)) → 0. 23 Up to homotopy, RP2 \pt ≃ S 1 and Xm−1\x ≃ Wm i=1 S 1 , the map H1 (Xm−1(R)\x) ⊕ H1 (RP2 \pt) → H1 (S 1 ) is exactly (2, 2, · · · , 2) : Z m+1 → Z. Let d and ei denote the generators of H1 (Xm(R)) corresponding to D and Ei in the polygon representation of Xm(R). Then the kernel of(2, 2, · · · , 2) gives H1 (Xm(R)) ∼= Z m, which is generated by d−e1, · · · , d− em and the cokernel of (2, 2, · · · , 2) gives H2 (Xm(R)) ∼= Z/2. We need to understand the reduction map ρ : Hi (Xm(R), Z) → Hi (Xm(R), Z/2). Lemma 3.2.2. Let vi = d − ei be the generators of H1 (Xm(R), Z). The reduction map ρ : H1 (Xm(R), Z) → H1 (Xm(R), Z/2) sends v1, · · · , vm to D¯ − E¯ 1, · · · , D¯ − E¯m. Proof. By universal coefficient theorem, there is a natural isomorphism Hi (X(R), A) ∼= Exti (Hi−1(X(R)), A) ⊕ Hom(Hi(X(R)), A) for any abelian group A. In particular, when i = 1, we have a commutative diagram H1 (X(R), Z) Hom(H1(X(R)), Z) H1 (X(R), Z/2) Hom(H1(X(R)), Z/2) ∼= mod 2 ∼= that identifies ρ as modulo 2. To be more precise, the lemma follows from comparing Mayer-Vietoris long exact sequences 24 0 H1 (Xm(R), Z) H1 (Xm−1(R)\pt, Z) ⊕ H1 (RP2 \pt, Z) H1 (S 1 , Z) 0 H1 (Xm(R), Z/2) H1 (Xm−1(R)\pt, Z/2) ⊕ H1 (RP2 \pt, Z/2) H1 (S 1 , Z/2) mod 2 (2,2,· · · ,2) mod 2 mod 2 0 since the middle vertical map sends d to D¯ and ei to E¯ i . Lemma 3.2.3. The reduction map ρ : H2 (Xm(R), Z) → H2 (Xm(R), Z/2) is an isomorphism. Proof. Note that H1 (Xm(R), Z/2) H2 (Xm(R), Z) ∼= Z/2 H2 (Xm(R), Z/2) ∼= Z/2 δ1 β ρ is commutative, where β is the Bockstein homomorphism. Again, by Hatcher’s computation in [14, Example 3.8], β is nontrivial. It follows that ρ is an isomorphism. We are now ready to describe the multiplicative structure of H• (Xm(R), Z). Corollary 3.2.1. The integral cohomology ring of Xm(R) is Z[v1, · · · , vm, ¯θ]/J, ¯ where deg(vi) = 1, deg( ¯θ) = 2, and J¯ is generated by • v 2 i , vi ¯θ, for all i = 1, · · · , m • vivj − ¯θ, for all i ̸= j, • ¯θ 2 , 2 ¯θ. 25 Proof. Consider the commutative diagram H1 (X(R), Z) × H1 (X(R), Z) H2 (X(R), Z) ∼= Z/2 H1 (X(R), Z/2) × H1 (X(R), Z/2) H2 (X(R), Z/2) ∼= Z/2. ∪ ρ×ρ ≃ ∪ By lemma 3.2.2, ρ(vi) = D¯ − E¯ i . This gives v 2 i = ρ(vi) ∪ ρ(vi) = (D¯ − E¯ i) 2 = D¯ 2 + E¯2 i = 2D¯ 2 = 0. vivj = ρ(vi) ∪ ρ(vj ) = (D¯ − E¯ i)(D¯ − E¯ j ) = D¯ 2 − D¯E¯ i + D¯E¯ j + E¯ iE¯ j = D¯ 2 = ¯θ. Other relations follows from the fact Hi (X(R), Z) = 0 for all i ≥ 3. Recall: Note that Hn (X(R), Z/2Z) can be computed in terms of a Gersten resolution 0 → M x∈X(0) H0 (x, ρ∗Z/2Z) → M y∈X(1) H0 (y, ρ∗Z/2Z) → M z∈X(2) H0 (z, ρ∗Z/2Z) → · · · where ρ is defined as follows. For a commutative ring A, the real spectrum sper k is the topological space built upon pairs (x, <x), where x ∈ Spec A is a point and <x is an ordering of the residue field κ(x). The topology is generated by basis open sets D(a) = {(x, >x) ∈ Sper A | a >x 0 in κ(x)} for each a ∈ A. Gluing these data gives the real spectrum Xr of a scheme X. There is a continuous map supp : Xr → X (called the support map), forgetting the orderings of the residue fields. On the other hand, we have the inclusion map ι : X(R) → Xr, sending x to (x, R≥0). Define ρ = supp ◦ ι : X(R) → X. For the integral coefficient case, we have Hs (X, ρ∗Z(L)) ≃−→ Hs (X(R), Z(L)). 26 It can be shown that Hq (X(R), Z(L)) is isomorphic to the q-th cohomology of the Gersten complex 0 → M x∈X(0) H0 x (X(R), Z(L)) → M x∈X(1) H1 x (X(R), Z(L)) → M x∈X(2) H2 x (X(R), Z(L)) → · · · , where H∗ x (X(R), Z(L)) := colimH∗ (¯x∩U)(R) (U(R), Z(L)) with the colimit taken over the Zariski open neighborhoods U of X. 3.3 The Additive Structure of CHgi (X) Note that Chn (X) Chn+1(X) Hn+1(X,I n+1) Hn (X(R), Z/2) Hn+1(X(R), Z/2) Hn+1(X(R), Z) . Sq2 β ρ β¯ Sq1 ρ¯ where the vertical maps are cycle class maps. Since Pic(X) ∼= CH1 (X), by [13, Chapter V, prop 3.12], we have Proposition 3.3.1. Let π : X′ → X be a blow-up at a torus-invariant point x in X. Then Pic(X′ ) ∼= Pic(X) ⊕ Z and the intersection theory on X′ is determined by • (a) if C, D ∈ PicX, then (π ∗C).(π ∗D) = C.D 27 • (b) if C ∈ PicX, then (π ∗C).E = 0 • (c) E2 = −1. By induction, one can prove the following: Corollary 3.3.1. Suppose X is obtained from finite successive blowups of P 2 . The Chow ring of X is given by Z[D, E1, · · · , Em]/(Ei · Ej , D · Ei , E2 i + D2 , D3 ), where each Ei is the exceptional divisor of the i-th blowup, and D is the generator of the Chow ring of P 2 . For convenience, let I = (Ei · Ej , D · Ei , E2 i + D2 , D3 ). Let H• (X,I • ) := L2 i=0 Hi (X,I i ). Recall: The reduction homomorphism ρ : H• (X,I • ) → Ch• (X) is given by θ 7→ D¯ 2 , ui 7→ D¯ − E¯ i . Now consider ∂ : CH• (Xm) → R, where ∂i : CHi (Xm) → Hi+1(Xm,I i+1). Theorem 3.3.1. Ker ∂1 ∼= Z m+1, generated by 2D, D − Ei , i = 1, · · · m. Proof. Consider the commutative diagram that comes from the lower right corner of the key diagram in (3.2) CHi (Xm) Hi+1(Xm,I i+1) Chi (Xm) Hi+1(Xm,I i+1) Chi+1(Xm) mod 2 ∂ ≃ β Sq2 (3.3) 28 Explicitly, ∂0 : Z → Lm i=1 Z ∂1 : ZD ⊕ ZE1 ⊕ · · · ⊕ ZEm → Z/2 ∂2 : ZD2 → 0. The interesting case is when i = 1, the composition Sq2 : CH1 (Xm) → Ch(Xm) → Ch2 (Xm) ∼= Z/2 sends D to D2 ̸= 0 ∈ Ch2 (Xm) ∼= Z/2. Similarly, Sq2 (Ei) = E2 i = −D2 = D2 ̸= 0 ∈ Ch2 (Xm). Since we also have H2 (Xm,I 2 ) ∼= H2 (Xm(R), Z) ∼= Z/2 ∼= Ch2 (Xm), ∂1 must send both D and Ei to the nontrivial element in H2 (Xm,I 2 ), i.e., Sq¯ 2 (D − Ei) = 0. This shows that D − Ei lies in Ker ∂. By now, we are ready to compute the Chow-Witt groups of the real surface Xm. Theorem 3.3.2. CHfi (Xm) =    Z ⊕ Z i = 0 Z 2m+1 i = 1 Z i = 2 Proof. First note that Xm is connected, CHf 0 (Xm) ∼= GW(R) ∼= Z⊕Z. When n = 1, the pullback diagram CHf n (X) Ker ∂n Hn (X,I n ) Chn (X) is explicitly CHf 1 (Xm) Z · (2D) Lm j=1 Z · (D − Ei) Z ⊕m (Z/2)⊕(m+1) . f g In other words, CHf 1 (Xm) is the kernel of Z m+1MZ m (f,−g) −−−−→ (Z/2)⊕(m+1) , 29 where F = (f, −g) has a matrix representation F[u0, · · · , um, v1, · · · , vm] = [D, ¯ E¯ 1, · · · , E¯m]                 0 1 1 · · · 1 1 1 · · · 1 0 −1 0 · · · 0 −1 0 · · · 0 0 0 −1 · · · 0 0 −1 · · · 0 · · · · · · · · · · · · · · · · · · · · · · · · · · · 0 0 0 · · · −1 0 0 · · · −1.                 It follows that Ker(F) = spanZ {u0, u1 + v1, u1 − v1, · · · , um + vm, um − vm}. Similarly, from the pullback diagram CHf 2 (Xm) Z Z/2 Z/2 we have CHf 2 (Xm) ∼= Z. Remark. One can identify the pair Z · (ui + vi) ⊕ Z · (ui − vi) as a copy of GW(R) as follows. The symbol m⟨1⟩ + n⟨−1⟩ ∈ GW(R) with m, n ∈ Z has rank m + n and is sent to (m − n)⟨1⟩ + n · h = (m − n)⟨1⟩ ∈ W(R). The pullback diagram for GW(R) as in diagram (2.1) is exactly Z ⊕ Z Z Z Z/2 + − mod 2 mod 2 Under this identification, u0 = 2D ∈ CH1 (Xm)lifts to x = h·D ∈ CHf 1 (Xm)since the GW(R) coefficient in x has rank 2 and is zero in W(R). We will also use this identification later on to describe the Chow-Witt ring CHf • (Xm) as a Z-graded GW(R)-algebra. 30 3.4 The Multiplicative Structure The Chow-Witt ring of P 1 × P 1 Note that (P 1 × P 1 )(R) ∼= S 1 × S 1 . Since Hi (S 1 , Z) is free in each degree, by Kunneth formula, H• (S 1 × S 1 , Z) ∼= H• (S 1 , Z) ⊗ H• (S 1 , Z) ∼= Z[α, β]/(α 2 , β2 ), where α, β denotes the generator in each H1 (S 1 , Z). Note that the Chow ring of P 1 × P 1 is also given by Z[H1, H2]/(H2 1 , H2 2 ). Note that ker ∂ = CH• (P 1 × P 1 ), i.e., ∂i : CHi (P 1 × P 1 ) → Hi+1(P 1 × P 1 ,I i+1) is zero for all i. In this case, the fundamental square is CHf 1 (P 1 × P 1 ) Z 2 Z 2 (Z/2)2 , where both maps Z 2 → (Z/2)2 are given by mod 2. It is not hard to see that CHf • (P 1 × P 1 ) ∼= (Z ⊕ Z)[u, v]/(u 2 , v2 ) ∼= GW(R)[u, v]/(u 2 , v2 ). In general, we have CHf • (P 1 × P 1 ) ∼= GW(k)[u, v]/(u 2 , v2 ) for a perfect field of characteristic not equal to 2. For the case X = Xm, the ring structure of Ker ∂ is as follows. Lemma 3.4.1. Ker ∂ ∼= Z[u0, u1, · · · , um, D2 ]/I′ , 31 where u0 = 2D, ui = D − Ei , and I ′ is generated by (i) u 2 0 − 4D2 (ii) For all 1 ≤ i < j ≤ m, uiuj − D2 (iii) For all i > 0, u 2 i , u0ui − 2D2 (iv) For all i ≥ 0, uiD2 . Proof. By Theorem 3.3.1, Ker ∂1 is additively generated by u0 = 2D, ui = D − Ei for 1 ≤ i ≤ m. Direct computation gives that for all 1 ≤ i < j ≤ m uiuj = (D − Ei) · (D − Ej ) = D2 − D · Ei − D · Ej + Ei · Ej = D2 (Since D · Ei = 0, Ei · Ej = 0 by Corollary 3.3.1) It follows that 4uiuj = 4D2 = u 2 0 . Similarly, 2u0uj = u 2 0 = 4D2 . One can also verify that for all 1 ≤ i ≤ m, u 2 i =(D − Ei) 2 =(D2 + E 2 i ) − 2D · Ei =0 ( by Corollary 3.3.1) 32 Remark. From the computation in Lemma 3.4.1, the multiplication table of Ker ∂ is given by   u0 u1 u2 · · · um u0 4D2 2D2 2D2 . . . 2D2 u1 2D2 0 D2 . . . D2 u2 2D2 D2 0 . . . D2 . . . . . . . . . . . . . . . . . . um 2D2 D2 D2 . . . 0   (3.4) Note that there is a bijection between the generators v1, · · · , vm of H1 (Xm(R), Z) and the m generators u1, · · · , um of Ker ∂1 as they both sent to D¯ − E¯ 1, · · · , D¯ − E¯m in (Z/2) L(m+1). In fact, there is a ring isomorphism Ker ∂ /(u0) ∼= H• (X(R), Z) by Corollary 3.2.1. We are now ready to describe the ring structure of CHf • (Xm). Theorem 3.4.1. The Chow-Witt ring of the real toric surface Xm is given by CHf• (Xm) ∼= GW(R)[x, y1, · · · , ym, θ]/J, 33 where J =< x2 − 4θ, y2 i , yiyj − θ, xyi − 2θ, xθ, x3 , θ2 ,(1 − ϵ)x, I(R)θ >, deg(x) = deg(yi) = 1, i = 1, · · · , m, and deg(θ) = 2. The ring homomorphism CHf• (Xm) → CH• (Xm) is given by x 7→ 2D yi 7→ D − Ei θ 7→ D2 and the ring homomorphism CHf• (Xm) → H• (Xm,I • ) is given by x 7→ 0 yi 7→ vi θ 7→ ¯θ Proof. By Lemma 3.4.1 and Corollary 3.2.1, the Cartesian diagram is explicitly CHf • (Xm) Z[u0, u1, · · · , um, D2 ]/I′ Z[v1, · · · , vm, ¯θ]/J¯ Z/2[D, ¯ u¯1 · · · , u¯m]/ ¯I By the remark after Theorem 3.3.2, we have CHf • (Xm) ∼= GW(R)[h · D, D − E1, · · · , D − Em, D2 ]/J. Let x = h · D, yi = D − Em in CHf 1 (Xm) and let θ denote the generator in CHf 2 (Xm). The images of generators under the ring homomorphisms CHf • (Xm) → CH• (Xm) and CHf • (Xm) → H• (Xm,I • ) are 34 determined by Theorem 3.3.2. One can verify the relations via the multiplicative structures in CH• (Xm) and H• (Xm,I • ). 35 Chapter 4 Chow-Witt Rings of Toric Surfaces Over a More General Field 4.1 The Computation of Hi (X,W) Throughout this chapter, let k be a perfect field of characteristic not equal to 2 and let X be a smooth toric surface associated to a polyhedral fan Σ. Let x be a fixed point of the torus action. We want to proceed by considering the Mayer-Vietoris long exact sequence for the open covering X = (X\x) ∪ A 2 . First, we need to understand X\x. Theorem 4.1.1. X\x ≃A1 Wn i=1 P 1 . Proof. Let σ ∈ Σ be the fan that corresponds to the fixed point x ∈ X. Let vi , 0 ≤ i ≤ k denote all the arrows in Σ. Without loss of generality, we may assume σ = Cone{v0, v1}. Let σi = Cone{vi , vi+1}. Since X is smooth, all the vectors vi are primitive and Sk i=1 σi = Z 2 . It follows that Xσi ∼= A 2 and Σi = {σi , σi+1} defines a total space of a line bundle OP1 (n) for some n. Therefore, XΣi ≃A1 P 1 . Let Γi = Σ1 ∪ · · · ∪ Σi . Observe that Γi ∩ Σi+1 = {σi}, i.e., XΓi ∩ XΣi+1 = XΓi∩Σi+1 = Xσi ∼= A 2 . This gives a pushout diagram A 2 XΓi XΣi+1 XΓi+1 and thus XΓi+1 ≃A1 XΓi ∨ XΣi+1 ≃A1 XΓi ∨ P 1 . Since X\x = XΓk−1 , by induction, we are done. 36 Remark. Note that each P 1 in the wedge sum is obtained by the zero section of the line bundle OP1 (n) that is glued by any pair of adjacent cones in Σ. For the effective computation of Hi (X, KMW j ), one may expect similar techniques as in algebraic topology, for instance, the Mayer-Vietoris long exact sequence. Below are two important distinguished triangles in the stable motivic homotopy category due to Voevodsky’s work that lead to such exact sequences. Proposition 4.1.1. (See [25, Proposition 4.11].) For any elementary distinguished square in the stable motivic homotopy category, p −1 (U) V U X p j where p is étale and j is a closed immersion, there is a canonical distinguished triangle of the form p −1 (U+) → U+ ⊕ V+ → X+ → p −1 (U)+[1]. In particular, for the distinguished square associated to a Zariski covering X = U ∪ V , we have the MayerVietoris distinguished triangle (U ∩ V )+ → U+ ⊕ V+ → X+ → (U ∩ V )+[1]. 37 Proposition 4.1.2. (See [25, Proposition 4.13].) Let i : Z → X be a closed embedding of smooth schemes over S and p : BlZX → X the blow-up of Z in X. Then there exists a canonical blow-up distinguished triangle in the stable motivic homotopy category p −1 (Z)+ → Z+ ⊕ (BlZX)+ → X+ → p −1 (Z)+[1]. Note that the W sheaf cohomology is representable in the stable motivic homotopy category, we have a long exact sequence · · · → Hi (X,W) → Hi (Z,W) ⊕ Hi (BlZX,W) → Hi (p −1 (Z),W) → Hi+1(X,W) → · · · We also need the localization sequence for the Witt sheaf cohomology. To be more precise, we have the following due to Ananyevskiy. Proposition 4.1.3. (See [1, Definition 12].) Let X be a smooth scheme and let Z ⊆ X be a subscheme of pure codimension c and let U = X\Z be the open complement. Let i : Z → X, j : U → X be inclusions and let N be the determinant of the normal bundle of Z in X. Then there is a localization sequence for W-cohomology · · · → Hi (U, W(L)) ∂ −→ Hi−c+1(Z, W(L ⊗ N )) i∗−→ Hi+1(X, W(L)) j ∗ −→ Hi+1(U, W(L)) → · · · We are now ready to compute the W cohomology of toric surfaces. Recall that toric surfaces are classified as successive blowups of P 2 or P 1 × P 1 up to A 1 -homotopy. More generally, if X is a smooth complete toric surface and let π : X′ → X denote the blow-up of X at a toric fixed point x ∈ X. We have a pushout diagram A 2\{0} A 2 X\x X. 38 Blowing up at x gives another diagram A 2\{0} Bl0(A 2 ) X\x X′ and one can patch these diagrams as follows Bl0A 2 A 2\{0} A 2 X′ X\x X. p η π Remark: Note that the back square is equivalent to P 1 X′ ∗ X (4.1) up to A 1 homotopy. Theorem 4.1.2. Let Xm = Blxm(· · · Blx1 (P 2 )) be a successive blow-up of P 2 . Then for m ≥ 0, Hi (Xm, W) =    W(k) i = 0 W(k) ⊕m i = 1 0 i = 2 . Proof. By (4.1), we have the Mayer-Vietoris long exact sequence 0 → H1 (X,W) → H1 (X′ ,W) → H1 (P 1 ,W) → H2 (X,W) → H2 (X′ ,W) → 0 (4.2) in W-cohomology. 39 By Fasel’s computation in [10], we have H2 (P 2 ,W) = 0, so the case m = 0 holds. Clearly, H2 (X,W) = 0 implies that H2 (X′ ,W) = 0. By induction, we have H2 (Xm,W) = 0 for all m ≥ 0. Now (4.2) is reduced to 0 → H1 (Xm,W) → H1 (Xm+1,W) → H1 (P 1 ,W) → 0 (4.3) Again, by [10] Hi (P 1 ,W) =    W(k) i = 0 W(k) i = 1. Since H1 (P 1 ,W) is a free W(k)-module, (4.3) is split. Therefore, H1 (Xm+1,W) ∼= H1 (Xm,W) ⊕ H1 (P 1 ,W) ∼= H1 (Xm,W) ⊕ W(k). By induction, we are done. 4.2 The Computation of Hi (X,I i ) From the computation in the previous section, Hi (Xm+1,W) is a free W(k)-module in each degree. In this case, we can apply Wendt’s splitting result to compute Hi (X,I i ) as in [26]. The result is as follows. Proposition 4.2.1. (See [26, Lemma 2.3].) Let X be a smooth scheme over k with characteristic ̸= 2. If Hn (X, W) is free as a W(k)-module, then we have a splitting Hn (X,I n ) ∼= Im(βn) ⊕ Hn (X, W), 40 where βn : Chn−1 (X) ∼= Hn−1 (X, KM n−1 /2) → Hn (X,I n ) is the morphism in the Bar sequence Hn−1 (X, K M n−1/2) βn −→ Hn (X,I n ) → Hn (X,I n−1 ) → Hn (X, K M n−1/2) = 0. In this case, the reduction homomorphism ρ : Hn (X, W) → Chn (X) is injective on the image of βn. Remark: Note that ρ ◦ β = Sq2 : Chn−1 (X) → Chn (X). It suffices to understand the image of Sq2 to compute Im(β). Since Chi (Xm) =    Z/2 i = 0, 2 (Z/2)⊕(m+1) i = 1, and the Steenrod operation Sq2 : Chn−1 (Xm) → Chn (Xm) is surjective when n = 2 and is equal to 0 otherwise, we have Im(βi) ∼=    Z/2 i = 2 0 otherwise. (4.4) Corollary 4.2.1. Hi (Xm,I i ) ∼=    W(k) i = 0 W(k) ⊕m i = 1 Z/2 i = 2. Proof. This follows directly from Theorem 4.1.2 and Proposition 4.2.1. To recover the ring structure of H• (Xm,I • ) = L2 i=0 Hi (Xm,I i ), one needs to understand the reduction map ρ : H• (Xm,I • ) → Ch• (Xm). As in Chapter 3.2, let v1, · · · , vm denote the generators of H1 (Xm,I 1 ) and let ¯θ denote the generator of H2 (Xm,I 1 ). 41 Lemma 4.2.1. The reduction map ρ : H• (Xm,I • ) → Ch• (Xm) is given by ¯θ 7→ D¯ 2 vi 7→ D¯ − E¯ i Proof. The first assignment comes from the fact that Sq2 : Ch1 (Xm) → Ch2 (Xm) is surjective. We want to show the second one. Let x be a torus-invariant point in Xm. Consider the closed immersion x ,→ Xm with the open complement U = Xm\x. By Proposition 4.1.3, there exists a localization sequence 0 → H1 (Xm,W) → H1 (Xm\x,W) → H0 (pt,W(N )) → H2 (Xm,W) → · · · where N is the determinant of the normal bundle of x in Xm. Clearly, N is trivial and one can drop the twist and obtain H0 (pt,W(N )) ∼= W(k). By Theorem 4.1.2, we have H2 (Xm,W) = 0. By Theorem 4.1.1, Xm\x ≃A1 Wm+1 i=1 P 1 and this gives H1 (Xm\x,W) ∼= Lm+1 i=1 H1 (P 1 ,W) ∼= W(k) m+1 by Fasel’s computation in [10]. The reduction map ρ ′ : H1 (Xm,I 1 ) → Ch1 (X\x) is given by (rank,rank, · · · ,rank) : W(k) m+1 → (Z/2)m+1 Now consider the commutative diagram 0 H1 (Xm,I 1 ) H1 (Xm\x,I 1 ) 0 Ch1 (Xm) Ch1 (Xm\x), ρ ρ ′ (4.5) where each horizontal arrow comes from a corresponding localization sequence for the closed embedding x ,→ X. By (4.4) and Proposition 4.2.1, we have Im(β1) = 0 and thus H1 (Xm,I 1 ) ∼= H1 (Xm,W). Similarly, since the Steenrod operation Sq2 = ρ ◦ β acts trivially on P 1 , the same argument shows that 42 H1 (P 1 ,I 1 ) ∼= H1 (P 1 ,W) and thus H1 (Xm,I 1 ) ∼= H1 (Xm,W). Therefore, the diagram (4.5) comes from the comparison map 0 H1 (Xm,W) H1 (Xm\x,W) H0 (x,W) 0 Ch1 (Xm) Ch1 (Xm\x) 0, ρ ρ ′ between the localization sequence in Witt cohomology and the localization sequence in Chow ring modulo 2. Further contemplations indicate that the map H1 (Xm\x,W) → H0 (x,W) restricts to the identity map on each copy H1 (P 1 ,W) → H0 (x,W). Therefore, the generators v1, · · · , vm are sent to D − E1, · · · , D − Em as the case of real toric surfaces. Corollary 4.2.2. The ring structure of R = L2 i=0 Hi (Xm,I i ) is W(k)[v1, · · · , vm, ¯θ]/IR, where IR =< v2 i − ¯θ, vivj , vi ¯θ, ¯θ 2 , 2 ¯θ >. Here deg(vi) = 1 and deg( ¯θ) = 2. Proof. This proof goes parallel with that in Corollary 3.2.1. Essentially, all the relations follows from the commutative diagram H1 (X,I 1 ) × H1 (Xm,I 1 ) H2 (Xm,I 2 ) ∼= Z/2 Ch1 (Xm) × Ch1 (Xm) Ch2 (Xm) ∼= Z/2. ∪ ρ×ρ ≃ ∪ Recall that the Chow ring of Xm is given by CH• (Xm) = Z[D, E1, · · · , Em]/(EiEj (i ̸= j), D · Ei , E2 i + D2 i , D3 ), 43 where D denotes the strict transform of the hyperplane class in CH• (P 2 ). Next, consider ∂i : Hi (Xm, KM i ) → Hi+1(Xm,I j+1). Explicitly, we have ∂0 : Z → Mm i=1 W(k) ∂1 : Z · D ⊕ Z · E1 ⊕ · · · ⊕ Z · Em → Z/2 ∂2 : Z · D2 → 0 Note that the computation of Ker ∂ for real toric varieties also works here. Indeed, we can apply the same diagram as in (3.3) and use the fact that Sq2 is nontrivial on each generator of H1 (Xm,I 1 ). Therefore, Ker ∂ remains the same as for real toric varieties. This gives Theorem 4.2.1. Ker ∂i =    Z i = 0 Z · (2D) Lm i=1 Z · (D − Ei) i = 1 Z · D2 i = 2 Now we are ready to compute the Chow-Witt rings of the toric surfaces Xm. Theorem 4.2.2. (Main Theorem) Let Xm denote the toric surface obtained by blowing up m points from the surface P 2 . Then the Chow-Witt ring of Xm is given by CHf• (Xm) ∼= GW(k)[x, y1, · · · , ym, θ]/J, where J =< x2 − 4θ, y2 i , yiyj − θ, xyi − 2θ, xθ, x3 , θ2 ,(1 − ϵ)x, I(k)θ >, deg(x) = deg(yi) = 1, i = 1, · · · , m, and deg(θ) = 2, deg(θ) = 2, and deg(x) = deg(yi) = 1, 1 ≤ i < j ≤ m. 44 Proof. We proceed by applying the diagram (3.1). The diagram computing CHf 1 (Xm) is reduced to a pullback diagram CHf 1 (Xm) Z m+1 W(k) Lm (Z/2) L(m+1) . fk gk i.e., a short exact sequence 0 → CHf 1 (Xm) → Z m+1 ⊕ W(k) ⊕m (fk,−gk) −−−−−→ (Z/2)⊕(m+1) → 0. • (a) fk is the mod 2 map, i.e., sending u0 = 2D to 0 and sending ui = D − Ei to D¯ − E¯ i . • (b) gk sends the i-th copy of W(k) to Z/2 ·(D¯ −E¯ i). On each copy of W(k), the map W(k) → Z/2 is exactly the rank map mod 2. By (a) and (b), the i-th component of W(k) together with the i-th component of Z get pulled back to a copy of GW(k), since restricting fk and gk to the i-th component pulls back to a copy of GW(k) as in the diagram GW(k) Z W(k) Z/2. rank mod 2 rank Let yi , 1 ≤ i ≤ m denote the generators in CHf 1 (Xm)such that GW(k)·yi maps to W(k)·vi in Hi (Xm,I i ) and maps to Z · (D − Ei) ∈ CH1 (Xm). Let x denote the generator in CHf 1 (Xm) such that it correspond to 2D in CHf 1 (Xm) and sent to 0 in W(k). This shows that x represents the class of the hyperbolic form in 45 GW(k). Therefore, we have m + 1 generators in CHf 1 (Xm) and CHf 1 (Xm) ∼= 2Z ⊕ GW(k) ⊕m. Similarly, CHf 2 (Xm) can be computed by the pullback diagram CHf 2 (Xm) Z Z/2 Z/2. mod 2 Id This gives CHf 2 (Xm) ∼= Z. The unique generator θ is the lift of the 2-torsion component ¯θ in H2 (Xm,I 2 ). Remark: Note that W(R) ∼= Z, GW(R) ∼= Z ⊕ Z, and I(R) ∼= Z. One can recover the same result for real toric varieties when we set k = R. 4.3 Geometric Interpretation Recall that in the singular cohomology theory, the attaching map of a connected cellular topological space X is given by the map i : S n−1 ≃ ∂Dn → X(n)\x ≃ Wk i=1 S n−1 , where Xn denotes the n-th skeleton of X. Let pj denote the quotient map Wk i=1 S n−1 → Wk i=1 S n−1/ W i̸=j S n−1 ∼= S n−1 . Let nj be the degree of the map φj = pj ◦ i. On the level of singular cohomology, the attaching map Hn−1 (X\x) ∼= Ln i=1 Hn−1 (S n−1 ) → Hn−1 (S n−1 ) is given by Z k (n1,··· ,nk) −−−−−−→ Z. We wish to establish a similar result of cellular spaces in terms of A 1 -homotopy. Let pj denote the quotient map X\x ≃ Wk i=1 P 1 → Wk i=1 P 1/ W i̸=j P 1 ≃ P 1 and let i : A 2\{0} → X\x be the open embedding. 46 Claim: φj : H1 (P 1 , KMW 1 ) → H1 (A 2\{0}, KMW 1 ), induced by pj ◦ i : A 2\{0} → P 1 , is multiplication by 0 or η. When j = 2, we have H1 (P 1 , KMW 2 ) → H2 (X, KMW 2 ) → H2 (X′ , KMW 2 ) → 0. Since H1 (P 1 , KMW 2 ) ∼= KMW 1 (k) ∼= k ×, which is divisible, and note that there is no nontrivial group homomorphism from k × to Z, one can prove by induction to show that H2 (X, KMW 2 ) ∼= Z for all toric surfaces. When j = 1, we have H1 (P 1 , KMW 1 ) ∼= KMW 0 (k) ∼= GW(k). By (1), we have 0 → H1 (X, K MW 1 ) → H1 (X′ , K MW 1 ) → GW(k) → H2 (X, K MW 1 ) . To understand the map GW(k) → H2 (X, KMW 1 ), one can compare Mayer-Vietoris long exact sequences (the right-hand-side maps to 0 since H2 (X\x, KMW 1 ) ∼= 0.) 0 H1(X′ , KMW 1 ) H1(X\x, KMW 1 ) ⊕ H1(Bl0(A 2), KMW 1 ) H1(A 2\{0}, KMW 1 ) H2(X′ , KMW 1 ) 0 0 H1(X, KMW 1 ) H1(X\x, KMW 1 ) H1(A 2\{0}, KMW 1 ) H2(X, KMW 1 ) 0. ∼= 47 Note that H1 (X\x, KMW 1 ) ∼= H1 ( Wl i=1 P 1 , KMW 1 ) ∼= H1 (P 1 , KMW 1 ) ⊕l ∼= KMW 0 (k) ⊕l ∼= GW(k) ⊕l . Also, Hi (A n − 0, K MW j ) ∼= Hi (S n−1 ∧ G ∧n m , K MW j ) ∼= Hi−n+1(G ∧n m , K MW j ) ∼= Hi−n+1(Spec(k), K MW j−n ) ∼=    0 i ̸= n − 1 KMW j−n (k) i = n − 1 gives H1 (A 2\{0}, KMW 1 ) ∼= KMW −1 (k) ∼= W(k), one may expect Claim: The morphism ϕ : H1 (X\x, KMW 1 ) → H1 (A 2\{0}, KMW 1 ) is given by (ϕ1, · · · , ϕl) : GW(k) ⊕l → W(k), where each ϕj can only be η or 0. 48 Bibliography [1] Alexey Ananyevskiy. “The special linear version of the projective bundle theorem”. In: Compositio Mathematica 151.3 (2015), pp. 461–501. [2] Aravind Asok and Jean Fasel. “Comparing Euler classes”. In: The Quarterly Journal of Mathematics (2016), pp. 1–33. [3] Aravind Asok and Christian Haesemeyer. “The 0-th stable Aˆ 1-homotopy sheaf and quadratic zero cycles”. In: arXiv preprint arXiv:1108.3854 (2011). [4] Aravind Asok and Fabien Morel. “Smooth varieties up to A1-homotopy and algebraic h-cobordisms”. In: Advances in Mathematics 227.5 (2011), pp. 1990–2058. 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Algebraic geometry. Vol. 52. Springer Science & Business Media, 2013. [14] Allen Hatcher. Algebraic topology. Cambridge University Press, 2002. 49 [15] Jens Hornbostel and Matthias Wendt. “Chow–Witt rings of classifying spaces for symplectic and special linear groups”. In: Journal of Topology 12.3 (2019), pp. 916–966. [16] Jens Hornbostel, Matthias Wendt, Heng Xie, and Marcus Zibrowius. “The real cycle class map”. In: Annals of K-Theory 6.2 (2021), pp. 239–317. [17] Jesse Leo Kass and Kirsten Wickelgren. “An arithmetic count of the lines on a smooth cubic surface”. In: Compositio Mathematica 157.4 (2021), pp. 677–709. [18] John Milnor. “Algebraic K-theory and quadratic forms”. In: Inventiones mathematicae 9.4 (1970), pp. 318–344. [19] John Willard Milnor and Dale Husemoller. Symmetric bilinear forms. Vol. 73. Springer, 1973. [20] Fabien Morel. A1-algebraic topology over a field. Vol. 2052. Springer, 2012. [21] Fabien Morel. “Sur les puissances de l’idéal fondamental de l’anneau de Witt”. In: Commentarii mathematici helvetici 79.4 (2004), pp. 689–703. [22] Markus Rost. “Chow groups with coefficients”. In: Documenta Mathematica 1 (1996), pp. 319–393. [23] Burt Totaro. “Non-injectivity of the map from the Witt group of a variety to the Witt group of its function field”. In: Journal of the Institute of Mathematics of Jussieu 2.3 (2003), pp. 483–493. [24] D Orlov1 A Vishik2 V Voevodsky. “An exact sequence for Km/2 with applications to quadratic forms”. In: Annals of Mathematics 165 (2007), pp. 1–13. [25] Vladimir Voevodsky. “A1-homotopy theory”. In: Proceedings of the international congress of mathematicians. Vol. 1. Berlin. 1998, pp. 579–604. [26] Matthias Wendt. “Chow-Witt rings of grassmannians”. In: arXiv preprint arXiv:1805.06142 (2018). 50 
Abstract (if available)
Abstract We compute the Chow-Witt rings of smooth projective toric surfaces based on joint work by Hornbostel-Wendt-Xie-Zibrowius. By considering the real cycle class map, one can turn the real case of the problem into a problem in algebraic topology and interpret the $\I^j$-cohomology as an analogue of the singular cohomology with the integer coefficients. 
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Creator Jiang, Wenhan (author) 
Core Title On the Chow-Witt rings of toric surfaces 
School College of Letters, Arts and Sciences 
Degree Doctor of Philosophy 
Degree Program Mathematics 
Degree Conferral Date 2024-05 
Publication Date 05/17/2024 
Defense Date 03/21/2024 
Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital) 
Tag Chow-Witt rings,OAI-PMH Harvest,toric varieties 
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Advisor Asok, Aravind (committee chair), Friedlander, Eric (committee member), Pilch, Krzysztof (committee member), Williams, Harold (committee member) 
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Tags
Chow-Witt rings
toric varieties