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} $?