## commutative algebra: in a special type of local Gorenstein ring of \$ 2 dimension

Leave $$(R, mathfrak m, k)$$ be a ring of local dimension of Gorenstein $$2$$ such that $$mu ( mathfrak m ^ 2) (= dim_k mathfrak m ^ 2 / mathfrak m ^ 3) le 4$$ .

So it is true that $$R$$ it is regular? Or at least it's true that $$R$$ It has minimal multiplicity, that is $$e (R) = mu ( mathfrak m) -1$$ ?

here $$e (R) = d! lim_ {n to infty} dfrac {l (R / mathfrak m ^ n)} {n ^ d}$$ , where $$d = dim R$$ so in our case $$e (R) = 2 lim_ {n to infty} dfrac {l (R / mathfrak m ^ n)} {n ^ 2}$$

## Geometry ag.algebraica – On resolution of singularities of Gorenstein

Leave $$(X, o)$$ Being a singularity of Gorenstein isolated, normal, at least. $$3$$. Leave $$pi: widetilde {X} to X$$ be the singularity resolution of $$X$$. Is there a hypersurface? $$H subset X$$ (defined by an equation) such that strict transformation $$widetilde {H}$$ of $$H$$ It is regular and has the property: there is a $$1-1$$-correspondence between the irreducible components of the exceptional divisor of $$widetilde {X}$$ and that of $$widetilde {H}$$?