Symbolic Powers of Monomial Ideals

الأصل · EN

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I in k[x₀,, xₙ] we show Iᵗ⁽ᵐ⁺ᵉ⁻¹⁾⁻ᵉ⁺ʳ⁾ is a subset of M⁽ᵗ⁻¹⁾⁽ᵉ⁻¹⁾⁺ʳ⁻¹(I⁽ᵐ⁾)ᵗ for all positive integers m, t and r, where e is the big-height of I and M = (x₀,, xₙ). This captures two conjectures (r=1 and r=e): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.