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Additional info for Belyi's proof of a conjecture of Grothendieck (2001)(en)(1s)

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140]). We consider X := {(x, y, z, w) ∈ C4 | xy + zw + z 3 + w3 = 0}. 5. The variety X is Q-factorial. More precisely, X is factorial, that is, R = C[x, y, z, w]/(xy + zw + z 3 + w3 ) is a UFD. Proof. 196]), it is sufficient to check that x · R is a prime ideal of R and R[1/x] is a UFD. This is an easy exercise. Note that Q-factoriality is not a local condition in the analytic topology. 6. Let X an be the underlying analytic space of X. Then X an is not analytically Q-factorial at (0, 0, 0, 0).

We define (simple) normal crossing divisors, which will play an important role in the following sections. 10 (Normal crossing and simple normal crossing). Let X be a smooth variety. A reduced effective divisor D is said to be a simple normal crossing divisor (resp. normal crossing divisor ) if for each closed point p of X, a local defining equation of D at p can be written as f = z1 · · · zjp in OX,p (resp. OX,p ), where {z1 , · · · , zjp } is a part of a regular system of parameters. 11. The notion of normal crossing divisor is local for the ´etale topology (cf.

WHAT IS LOG TERMINAL? 2. The set of integers (resp. rational numbers, real numbers) is denoted by Z (resp. Q, R). We will work over an algebraically closed field k of characteristic zero; my favorite is k = C. 2. Preliminaries on Q-divisors Before we introduce singularities of pairs, let us recall the basic definitions about Q-divisors. 1 (Q-Cartier divisor). Let D = di Di be a Q-divisor on a normal variety X, that is, di ∈ Q and Di is a prime divisor on X for every i. Then D is Q-Cartier if there exists a positive integer m such that mD is a Cartier divisor.

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