• A. W.~Apter and J. D. Hamkins, “Indestructibility and the level-by-level agreement between strong compactness and supercompactness,” J.~Symbolic Logic, vol. 67, iss. 2, pp. 820-840, 2002.  
  Journal MR Citation arχiv

  @ARTICLE{ApterHamkins2002:LevelByLevel,
  AUTHOR = {Arthur W.~Apter and Joel David Hamkins},
  TITLE = {Indestructibility and the level-by-level agreement between strong compactness and supercompactness},
  JOURNAL = {J.~Symbolic Logic},
  FJOURNAL = {The Journal of Symbolic Logic},
  VOLUME = {67},
  YEAR = {2002},
  NUMBER = {2},
  PAGES = {820--840},
  ISSN = {0022-4812},
  CODEN = {JSYLA6},
  MRCLASS = {03E35 (03E55)},
  MRNUMBER = {1905168 (2003e:03095)},
  MRREVIEWER = {Carlos A.~Di Prisco},
  DOI = {10.2178/jsl/1190150111},
  URL = {http://wp.me/p5M0LV-2i},
  eprint = {math/0102086},
  archivePrefix = {arXiv},
  primaryClass = {math.LO},
  }

Can a supercompact cardinal $\kappa$ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above $\kappa$, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.