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Given an ample line bundle  L  on a  K 3 surface  S , we study the slope stability with respect to  L  of rank‐3 Lazarsfeld–Mukai bundles associated with complete, base‐point‐free nets of typeg  d  2on curves  C  in the linear system |  L |. When  d  is large enough and  C  is general, we obtain a dimensional statement for the varietyW  d  2   (  C  )   . If the Brill–Noether number is negative, then we prove that anyg  d  2on any smooth, irreducible curve in |  L | is contained in ag  e  rwhich is induced from a line bundle on  S , thus answering a conjecture of Donagi and Morrison. Applications towards transversality of Brill–Noether loci and higher‐rank Brill–Noether theory are then discussed.

Stability of rank‐3 Lazarsfeld–Mukai bundles on K 3 surfaces