A cross-linkable copolymer comprising a copolymerization product of a monomer mixture consisting substantially of a vinyl lactam (a) and at least one further vinyl monomer (b) of a different type selected from the group consisting of hydrophobic, hydrophilic and functional vinyl monomers, wherein the monomers are present in the copolymer in the form of statistically distributed building blocks in accordance with the Lewis-Mayo equation, and, if a functional vinyl monomer is present as a building block in the copolymer, that building block is, where appropriate, modified with a reactive vinyl monomer (c), the reactive vinyl monomer (c) being linked to a building block of a functional vinyl monomer with retention of its vinylic group and with the formation of a covalent bond, wherein the cross-linkable copolymer is water-soluble, and wherein the hydrophilic vinyl monomer (b) is selected from hydroxy-substituted lower alkyl acrylates and methacrylates, acrylamide, methacrylamide, lower alkyl-acrylamides and -methacrylamides, ethoxylated acrylates and methacrylates, hydroxy-substituted lower alkyl-acrylamides and -methacrylamides, hydroxy-substituted lower alkyl vinyl ethers, sodium ethylenesulfonate, sodium styrenesulfonate, 2-acrylamido-2-methylpropanesulfonic acid, N-vinylpyrrole, 2- and 4-vinylpyridine, vinylically unsaturated carboxylic acids having a total of from 3 to 5 carbon atoms, amino-lower alkyl, mono-lower alkyl and di-lower alkylamino-lower alkyl acrylates and methacrylates and allyl alcohol.