The Cauvery river flows into Tamil Nadu and then back into Karnataka and then back again into Tamil Nadu in the stretch below.
Karnataka's point seems to be that somehow the lines of the Cauvery river are the borders of Karnataka, especially at the points along the river here where they will end up with more land as a result. These lines have been drawn since independence, so the argument is absurd. Plus they would have to give back Belgaum to Maharashtra with such an argument. So the Kannadiga "nationalist parties" are definitely trying to have their cake and eat it too. They have gotten used to Tamil Nadu water sufferings and feel infuriated that Tamil Nadu can do something to solve the problem without "the annual begging" that goes on during droughts.
Plus they keep attacking Tamil movie theaters at any available opportunity, prompting a token hunger strike by the Tamil film stars. Upcoming elections in Karnataka compound the issue; Tamil bashing is one of the best ways to get votes in Karnataka for nationalist Kannadiga parties.