Reider's proof of the Fujita conjecture that on a smooth surface, K+4A is very ample whenever A is ample, relies fundamentally on the Bogomolov instability theorem. I.e. It is precisely the Bogomolov instability of the vector bundle produced by Serre's construction associated to a possible base point of a linear series K+L, under certain conditions on the chern classes, that allows one to conclude the result. This technique seems to have played a major role in the study of linear series since that time. See Igor Reider's paper in the Annals of Math, (1988), or the article of Lazarsfeld in volume 3 of the IAS Park City volume, Complex algebraic geometry.

If you prefer to phrase it as bad behavior related to instability, I guess you could, since this shows that the presence of something bad, namely a base point, implies instability, which forces something else (the presence of certain curves though the base point on the surface) which if not bad, is at least rather special. And this lets you classify all cases where the original bad behavior occurs. So certain specific bundles can only be unstable in rather unusual ways.

Reider's proof of the Fujita conjecture that on a smooth surface, K+4A is very ample whenever A is ample, relies fundamentally on the Bogomolov instability theorem. I.e. It is precisely the Bogomolov instability of the vector bundle produced by Serre's construction associated to a possible base point of a linear series K+L, under certain conditions on the chern classes, that allows one to conclude the result. This technique seems to have played a major role in the study of linear series since that time. See Igor Reider's paper in the Annals of Math, (1988), or the article of Lazarsfeld in volume 3 of the IAS Park City series, Complex algebraic geometry.

If you prefer to phrase it as bad behavior related to instability, I guess you could, since this shows that the presence of something bad, namely a base point, implies instability, which forces something else (the presence of certain curves though the base point on the surface) which if not bad, is at least rather special. And this lets you classify all cases where the original bad behavior occurs. So certain specific bundles can only be unstable in rather unusual ways.

This may not interest you, since it is also in the direction of very good, rather than bad, but Reider's proof of the Fujita conjecture that on a smooth surface, K+4A is very ample whenever A is ample, relies fundamentally on the Bogomolov instability theorem. I.e. It is precisely the Bogomolov instability of the vector bundle produced by Serre's construction associated to a possible base point of a linear series K+L, under certain conditions on the chern classes, that allows one to conclude the result. This technique seems to have played a major role in the study of linear series since that time. See Igor Reider's paper in the Annals of Math, (1988), or the article of Lazarsfeld in volume 3 of the IAS Park City volume, Complex algebraic geometry.

Reider's proof of the Fujita conjecture that on a smooth surface, K+4A is very ample whenever A is ample, relies fundamentally on the Bogomolov instability theorem. I.e. It is precisely the Bogomolov instability of the vector bundle produced by Serre's construction associated to a possible base point of a linear series K+L, under certain conditions on the chern classes, that allows one to conclude the result. This technique seems to have played a major role in the study of linear series since that time. See Igor Reider's paper in the Annals of Math, (1988), or the article of Lazarsfeld in volume 3 of the IAS Park City volume, Complex algebraic geometry.

If you prefer to phrase it as bad behavior related to instability, I guess you could, since this shows that the presence of something bad, namely a base point, implies instability, which forces something else (the presence of certain curves though the base point on the surface) which if not bad, is at least rather special. And this lets you classify all cases where the original bad behavior occurs. So certain specific bundles can only be unstable in rather unusual ways.