Let C be the conic in A³ defined by X₀²+X₁²+X₂². Then:

• The conic C is not smooth at the origin O.
• The local ring of C at O is not a unique factorization. In fact, X₀, X₁ and X₂ are all irreducible, but we have the factorizations X₀²+X₁² = (X₀ + i X₁) (X₀ - i X₁) = -X₂²