New York Journal of Mathematics

New York J. Math. 1 (1995) 178–183.

Division Algebras that Ramify Only Along a Singular Plane Cubic Curve

T. J. Ford

Abstract. Let K be the field of rational functions in 2 variables over an algebraically closed field k of characteristic 0. Let D be a finite dimensional K-central division algebra whose ramification divisor on the projective plane over k is a singular cubic curve. It is shown that D is cyclic and that the exponent of D is equal to the degree of D.

Let k be an algebraically closed field of characteristic 0. Let ℙ^{2} = Proj k[x, y, z] denote the projective plane over k and K the function field of ℙ^{2}. We view K as the set of all rational functions of the form f/g ∈ k(x, y, z) where f and g are homogeneous forms in k[x, y, z] of the same degree.

The Brauer group of the projective plane, B(ℙ^{2}), is trivial. Therefore a division algebra D that is central and finite dimensional over K necessarily ramifies at some prime divisor of ℙ^{2}. By [1, Theorem 1] there is a canonical exact sequence

0 ⟶ B(K) ⟶^{a} ⨁_{C} H^{1}(K(C), ℚ/ℤ) . (1)

The map a measures the ramification of a central K-division algebra D along a prime divisor C on ℙ^{2}. The group H^{1}(K(C), ℚ/ℤ) is the first étale cohomology group of the function field K(C) of C, with coefficients in the constant sheaf ℚ/ℤ. By Kummer theory [4, pp. 125–126] H^{1}(K(C), ℚ/ℤ) classifies the finite cyclic Galois extensions of K(C). The “ramification of D along C” is a cyclic extension L of K(C) obtained in the following way. Let A be a maximal order for D over the local discrete valuation ring 𝒪_{C}. Then L = A ⊗ K(C)/(radical) is a cyclic extension of K(C), which represents an element of H^{1}(K(C), ℚ/ℤ). Those C for which L is non-trivial make up the ramification divisor of D. A division algebra D is completely determined by its ramification data.

In this article we consider the case where D is a finite dimensional K-central division algebra whose ramification divisor is a reduced cubic curve C that is singular. Our main result is Theorem 1 below which states that every such algebra D is a cyclic algebra with exponent(D) = degree(D). By exponent(D) we mean the exponent of the class of D in the Brauer group B(K). By degree(D) we mean the square root of the dimension of the vector space D over K.

\begin{matrix}c \\ ◯ \end{matrix}1995 State University of New York

ISSN 1076-9803/95

Received May 15, 1995.

Mathematics Subject Classification. Primary 13A20; Secondary 12E15, 14F20, 11R52.

Key words and phrases. Brauer group, division algebra, central simple algebra, symbol algebra, cyclic algebra.

If D has ramification divisor C, a nonsingular cubic curve on ℙ^{2}, then it is known that exponent(D) = degree(D). The reader is referred to [3] and its bibliography for a discussion of this case. M. Van den Bergh has recently announced a proof that if D has odd exponent, then D is cyclic.

In our context, each irreducible component of C is a rational curve whose normalization is isomorphic to ℙ^{1}. Let C be a reduced curve on ℙ^{2} each of whose irreducible components is a rational curve. Write C = C_{1} ∪ ⋯ ∪ C_{m} as a union of irreducible curves. Let C̃_{i} denote the normalization of C_{i}. By our assumption C̃_{i} ≅ ℙ^{1}. Let C̃ be the disjoint union C̃_{1} ∐ ⋯ ∐ C̃_{m}. Let Z denote the singular locus of C, which is a finite set of points, hence Z = {Z_{1}, …, Z_{s}}. Let π : C̃ → C be the natural projection and W = π^{−1}(Z). Then W is a finite set of points, hence W = {W_{1}, …, W_{e}}. The square

\begin{matrix}W & ⟶ & C̃ \\ ↓^{π} & & ↓^{π} \\ Z & ⟶ & C \end{matrix} (2)

is commutative. Define a graph Γ = Γ(C). The vertex set of Γ is {Z_{1}, …, Z_{s}, C̃_{1}, …, C̃_{m}} and the edge set is {W_{1}, …, W_{e}}. The edge W_{i} has positive end the C̃_{j} containing W_{i} and negative end the Z_{t} defined by Z_{t} = π(W_{i}). Let M be the incidence matrix of Γ. Then M induces a boundary map, also denoted M,

M : (ℤ/n)^{(e)} → (ℤ/n)^{(m)} ⊕ (ℤ/n)^{(s)} (3)

for any positive integer n. The kernel of M is the combinatorial cycle space H_{1}(Γ, ℤ/n) of Γ. Since we are assuming each C̃_{i} ≅ ℙ^{1} is simply connected, it follows that H^{1}(C, ℤ/n) = 0. Since ℙ^{2} is simply connected, H^{1}(ℙ^{2}, ℤ/n) ≅ H^{3}(ℙ^{2}, ℤ/n) = 0. Combining Lemma 0.1 and Corollary 1.3 of [2], there is an isomorphism _{n} B(ℙ^{2} − C) ≅ H_{1}(Γ, ℤ/n). Therefore the K-division algebras D with exponent dividing n and that ramify only along C make up a subgroup of B(K) that is isomorphic to H_{1}(Γ, ℤ/n).

Let α, β be elements of K, n ≥ 2 an integer, and ζ a fixed nth root of unity in K. The symbol algebra (α, β)_{n} is the associative K-algebra generated by u, v subject to the relations u^{n} = α, v^{n} = β, uv = ζvu. The ramification divisor of the algebra (α, β)_{n} is contained in the union of the sets of zeros and poles of the functions α and β on ℙ^{2}.

The main tool used in proving Theorem 1 is [2, Theorem 2.1] which tells us how to map a symbol algebra (α, β)_{n} over K to a sum of weighted edges in the graph Γ. This sum of weighted edges is an element in the edge space, ℤ/n^{(e)}, that is in ker M = H_{1}(Γ, ℤ/n). According to [2, Theorem 2.1], the weights on the edges of the graph can be computed in terms of the local intersection multiplicities of the various components of α and β. Suppose the zeros and poles of α and β are contained in C. Let P ∈ Z be a singular point on C. Let A_{1}, …, A_{t} be the components of C corresponding to vertices in Γ that are adjacent to P, as shown in Figure 1. Assume first that the curve Ã_{1} has only one point W_{1} lying over P. Then the weight (as

[figure: A_{1}; W_{1}; W_{2}; P; W_{3}; W_{t}; A_{2}; A_{3}; A_{t}] Figure 1

an element of ℤ/n) assigned to the edge W_{1} connecting P to A_{1} is

∑_{i=2}^{t} [v_{1}(β)v_{i}(α) − v_{1}(α)v_{i}(β)] (A_{1}.A_{i})_{P} , (4)

where (A_{1}.A_{i})_{P} is the local intersection multiplicity and v_{i} is the discrete valuation on K given by the local ring 𝒪_{A_{i}}. If A_{1} has multiple tangents at P, then there will be several edges connecting A_{1} to P in Γ. In this case (4) gives the weight for any one branch W_{1} of A_{1} at P where instead of (A_{1}.A_{i})_{P} the local intersection multiplicity for the branch that is associated with W_{1} is used.

Theorem 1. Let C be a reduced cubic curve in ℙ^{2} and assume C is singular. Let D be a finite dimensional central K-division algebra whose ramification divisor on ℙ^{2} is C. Then D is a cyclic algebra and exponent(D) = degree(D).

Proof. Let n be the exponent of the class of D in the Brauer group of K. We use the techniques of [2, Sec. 2] that were mentioned above. Upon desingularization, the singular cubic C consists of one, two or three components each of which is isomorphic to ℙ^{1}. Therefore the subgroup of B(K) consisting of classes of division algebras annihilated by n that ramify only along C is isomorphic to H_{1}(Γ, ℤ/n). Here Γ is the graph associated to C and H_{1} is simply the combinatorial cycle space of the graph. In each example below, Γ is a planar graph hence the ℤ/n-rank of H_{1}(Γ, ℤ/n) simply counts the number of regions of Γ.

There are only 6 cases to consider. In each case we show that D is a symbol algebra (α, β)_{n} hence is cyclic.

Case 1: C is irreducible and has a cuspidal singularity. In this case C is simply connected, H_{1}(Γ(C), ℤ/n) = 0, hence no non-trivial division algebra can have ramification divisor equal to C.

Case 2: C is irreducible and has a nodal singularity. Let l_{1} = 0 and l_{2} = 0 be the equations of the tangent lines to C at the node. The line l_{1} = 0 intersects the first branch of C with multiplicity 2 and the second branch with multiplicity 1. Similarly, l_{2} intersects the first branch of C with multiplicity 1 and the second branch with multiplicity 2. Consider the symbol algebra

Λ = (\frac{l_{1}}{l_{2}} , \frac{c}{l_{2}^{3}} )_{n}

over K. The ramification divisor of Λ must be contained in the curve l_{1}l_{2}c = 0. The graph of l_{1}l_{2}c = 0 is shown in Figure 2. Let W_{1} denote the edge of Γ corresponding

[figure: P; l_{1}; l_{2}; W_{2}; W_{1}; W_{3}; W_{4}; C]

Figure 2. The graph for the symbol Λ in Case 2.

[figure: P; q; Q; l_{1}; l_{2}; W_{4}; W_{5}; W_{1}; W_{2}; W_{3}]

Figure 3. The graph for the symbol Λ in Case 3.

to the first branch of C. We apply (4) to determine the weights w_{i} for the edges W_{i} of the element in the cycle space corresponding to Λ. In the notation above, we have α = l_{1}/l_{2}, β = c/l_{2}^{3}, A_{1} is the first branch of C, A_{2} is the curve l_{1} = 0, A_{3} is the curve l_{2} = 0, (A_{1}.A_{2})_{P} = 2, (A_{1}.A_{3})_{P} = 1, v_{1}(β) = 1, v_{2}(β) = 0, v_{3}(β) = −3, v_{1}(α) = 0, v_{2}(α) = 1, and v_{3}(α) = −1. From (4) we have

w_{1} = [(1)(1) − (0)(0)](2) + [(1)(−1) − (0)(−3)](1) = +1 .

To compute w_{2} using (4), we have A_{1} is the second branch of C, A_{2} is the curve l_{1} = 0, A_{3} is the curve l_{2} = 0, (A_{1}.A_{2})_{P} = 1, (A_{1}.A_{3})_{P} = 2, and the v_{i} values are the same as for w_{1}. From (4) we have

w_{2} = [(1)(1) − (0)(0)](1) + [(1)(−1) − (0)(−3)](2) = −1 .

To compute w_{3} using (4), we have A_{1} is the curve l_{1} = 0, A_{2} = C, A_{3} is the curve l_{2} = 0, (A_{1}.A_{2})_{P} = 3, (A_{1}.A_{3})_{P} = 1, v_{1}(β) = 0, v_{2}(β) = 1, v_{3}(β) = −3, v_{1}(α) = 1, v_{2}(α) = 0, and v_{3}(α) = −1. From (4) we have

w_{3} = [(0)(0) − (1)(1)](3) + [(0)(−1) − (1)(−3)](1) = 0 .

Similarly, using (4) we find w_{4} = 0. Therefore Λ has ramification divisor C and exponent n. Since _{n} B(ℙ^{2} − C) ≅ H_{1}(Γ, ℤ/n) ≅ ℤ/n we see that every algebra class of exponent n is some power of the class of Λ, and therefore has degree n.

Case 3: C factors into a line and an irreducible conic and has 2 nodes. Let q = 0 be the equation of the conic and l_{1} = 0 the equation of the line. Let P and

[figure: P_{1}; l_{3}; P_{3}; l_{1}; P_{2}; l_{2}; W_{1}; W_{2}; W_{3}; W_{4}; W_{5}; W_{6}]

Figure 4. The graph for the symbol Λ in Case 5.

Q denote the 2 nodes of C. Let l_{2} = 0 be the equation of the tangent to q = 0 at P. Consider the symbol algebra

Λ = (\frac{l_{1}}{l_{2}} , \frac{q}{l_{2}^{2}} )_{n}

over K. The ramification divisor of Λ is contained in the curve l_{1}l_{2}q = 0. The graph for Λ is shown in Figure 3. We apply (4) to compute the weight w_{1} of edge W_{1} for the algebra Λ. In the notation above, we have α = l_{1}/l_{2}, β = q/l_{2}^{2}, A_{1} is the curve l_{1} = 0, A_{2} is the curve l_{2} = 0, A_{3} is the curve q = 0, (A_{1}.A_{2})_{P} = 1, (A_{1}.A_{3})_{P} = 1, v_{1}(α) = 1, v_{2}(α) = −1, v_{3}(α) = 0, v_{1}(β) = 0, v_{2}(β) = −2, and v_{3}(β) = 1. From (4) we have

w_{1} = [(0)(−1) − (1)(−2)](1) + [(0)(0) − (1)(1)](1) = +1 .

Similarly we compute w_{2} = −1, w_{3} = +1, w_{4} = −1, and w_{5} = 0. Therefore Λ has ramification divisor C and exponent n. Since _{n} B(ℙ^{2} − C) ≅ H_{1}(Γ, ℤ/n) ≅ ℤ/n we see that every algebra class of exponent n is some power of the one given, and therefore has degree n.

Case 4: C factors into a line and an irreducible conic and has a cuspidal singularity. In this case C is simply connected, H_{1}(Γ, ℤ/n) = 0, hence no division algebra can have ramification divisor equal to C.

Case 5: C factors into 3 lines and has 3 nodes. Let the equation of C be written l_{1}l_{2}l_{3} = 0 where each l_{i} is a linear form. Consider the symbol algebra

Λ = (\frac{l_{1}}{l_{3}} , \frac{l_{2}}{l_{3}} )_{n}

over K. The graph for Λ in this case is the hexagon shown in Figure 4. Using (4) and the same ideas as in the earlier cases, we find that the ramification divisor of Λ is C and exponent(Λ) = n. Since _{n} B(ℙ^{2} − C) ≅ H_{1}(Γ, ℤ/n) ≅ ℤ/n we see that every algebra class of exponent n is some power of the one given, and therefore has degree n.

Case 6: C factors into 3 lines and has 1 singular point. In this case C is simply connected, H_{1}(Γ, ℤ/n) = 0, hence no division algebra can have ramification divisor equal to C.

Division Algebras 183

References

[1] M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3) 25 (1972), 75–95.

[2] T. J. Ford, On the Brauer group of a localization, J. Algebra 147 (1992), 365–378.

[3] , Products of symbol algebras that ramify only on a nonsingular plane elliptic curve, The Ulam Quarterly 1 (1992), 12–16.

[4] J. Milne, Etale Cohomology, Princeton Mathematical Series, no. 33, Princeton University Press, Princeton, N.J., 1980.

Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431

Ford@acc.fau.edu