Bounds by multiplicity for generators and projective dimension of complete local domains

Determine whether there exist bounds depending only on the Hilbert–Samuel multiplicity e(R) for both (i) the minimal number of generators of the defining ideal I and (ii) the projective dimension of R over Q, where R=Q/I is a complete local domain with algebraically closed residue field, (Q,m_Q) is a regular local ring, and I is contained in m_Q^2; additionally, investigate the same questions when the Krull dimension of R is fixed in addition to the multiplicity.

Background

Theorem tGred establishes that for local rings of fixed multiplicity whose associated graded ring is reduced, there are uniform bounds (depending only on the multiplicity) for projective dimension and Betti numbers, and only finitely many possible h-polynomials. The authors then discuss examples showing that finiteness of h-polynomials fails without the reduced associated graded hypothesis.

However, the necessity of the reduced associated graded hypothesis for bounding projective dimension and Betti numbers remains unresolved. Motivated by this, the authors pose the question of whether one can bound the minimal number of generators of I and the projective dimension of R over Q solely in terms of the Hilbert–Samuel multiplicity, possibly also fixing the Krull dimension of R.