Resolution of singularities and the positivity conjecture of SerreKazuhiko Kurano (Meiji University)
Let p and q be prime ideals of a regular local ring (A, m) satisfying (1) p + q is anm-primary ideal, (2) dim A/p + dim A/q = dim A. In [3], Serre conjectured that theintersection multiplicity between Spec(A/p) and Spec(A/q) is positive, i.e.,X
χA (A/p, A/q) : =(−1)i `A (TorAi (A/p, A/q)) > 0.
i≥0Let p∗ be the kernel of the natural ring homomorphism GA (m) → GA/p (m·A/p). Sometimes p∗ is called the initial ideal of p. Recently, Dutta [1] proved the inequality χA (A/p, A/q) ≥em (A/p) × em (A/q) if Proj(GA (m)/p∗ + q∗ ) is a finite set. Here, em (A/p) denotes the multiplicity of A/p at m. There are two key points in his proof. The first one is to use theformulaXH
χA (A/p, A/q) = em (A/p) × em (A/q) + η∗ ( (−1)i [TorOi (OV , OW )])ias in Example 20.4.3 in [2]. Here, π : H → Spec(A) is the blow-up at m, and V (resp.
W ) is the strict transform of Spec(A/p) (resp. Spec(A/q)). Set η = π|π−1 (m) : π −1 (m) →Spec(A/m). We denote η∗ the induced push-forward map G0 (π −1 (m)) → G0 (A/m). The
second key point is to use Gabber’s non-negativity theorem.
The formula as above (Example 20.4.3 in [2]) is proved by blowing up the affine scheme
Spec(A[T ]) at (m, T ).
The aim of my talk is to show that, if there exists a birational map good enough (otherthan the blowing-up of Spec(A[T ]) at (m, T )), then Serre’s positivity conjecture is true. Itis proved in the same way as Dutta’s method.
The following is a problem on the existence of a birational map good enough that implies
the positivity conjecture.
Problem 1 Let P and Q be prime√ideals of a regular local ring (B, n) satisfying (1)ht(P ) + ht(Q) = dim(B) − 1, (2) B/ P + Q is a discrete valuation ring. Do there exist
an ideal I of B that satisfies the following four conditions?
1. X̃ : = Proj(B[It]) is regular.
√
2. I 6⊂ P + Q.
3. Let Ỹ = Proj(B/P [(I·B/P )t]), Z̃ = Proj(B/Q[(I·B/Q)t]), and ϕ : X̃ → Spec(B) be
the blow-up along I. Then, Ỹ ∩ Z̃ ∩ ϕ−1 (n) is a finite set.

4. There exists an irreducible component E of ϕ−1 (n) such that codimX̃ E = 1 andỸ ∩ Z̃ ∩ reg(E) 6= ∅.
I do not know if Problem 1 is true or not even in the case where B = C[s, t, u](s,t,u) andP and Q are principal ideals.
Putting B = A[T ](m,T ) , P = pB and Q = pB, we have the following theorem.
Theorem 2 If Problem 1 is true, then Serre’s positivity conjecture is true.
Problem 1 seemes to be very difficult since existence of resolutions of singularity is still
a open problem.
Next we weaken the assumption of birationality, i.e., we replace birationality with generically finiteness.
Problem 3 Let P and Q be prime√ideals of a regular local ring (B, n) satisfying (1)ht(P ) + ht(Q) = dim(B) − 1, (2) B/ P + Q is a discrete valuation ring. Do there exist
proper surjective and generically finite map ϕ : W → Spec(B) that satisfies the following
four conditions?
1. W is regular.
2. There exists an openset U of Spec(B) such that U 3finite.

√P + Q and ϕ−1 (U ) → U is

3. There exist closed subschemes Ỹ and Z̃ such that (1) ϕ(Ỹ ) = V (P ), (2) ϕ(Z̃) = V (Q),(3) ϕ(Ỹ ∩ Z̃) = V (P + Q), and (4) Ỹ ∩ Z̃ ∩ ϕ−1 (n) is a finite set.
4. There exists an irreducible component E of ϕ−1 (n) such that codimW E = 1 andỸ ∩ Z̃ ∩ reg(E) 6= ∅.
Theorem 4 If Problem 3 is true, then Serre’s positivity conjecture is true.
We give some examples in my talk.

References[1] S. P. Dutta, A special case of positivity II, preprint.
[2] W. Fulton, Intersection Theory, Springer-Verlag, Berlin, New York, 1984.
[3] J-P. Serre, Algèbre locale. Multiplicités, Lect. Note in Math., vol. 11, SpringerVerlag, Berlin, New York, 1965.