Variation of the first Hilbert coefficients of parameters with a common   integral closure
  A problem posed by Wolmer V. Vasconcelos on the variation of the first Hilbert coefficients of parameter ideals with a common integral closure in a local ring is studied. Affirmative answers are given and counterexamples are explored as well. 
L. Ghezzi, S. Goto, J. Hong, K. Ozeki, T.T. Phuong, W.V. Vasconcelos
math.AC