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As long as you don't have a UI, your ,net *framework* app should run on linux/mono. Keep in mind that mono (last time I checked) was SEVERELY behind the power curve as far as .net versions are concerned. If your app DOES have a UI, you may be able tio run it under Wine, but I see that as at least as much of a crap shoot as using mono.
As an alternative, you might want to try writing your code in C++.Net, get it working there, and then ciopying the source code (the cpp files) to a Linux C++ project, and try compiling it native on linux.
I've generally found that as we progress in dev technology, we get further away from being able to do cross-platform dev. This has been a problem ever since Windows was first introduced.
The 'cost saving' action of getting rid of the IT dept has crippled the Kaspersy updates coming over the network. Yes someone took their eye off the ball and didn't update the licence, they now have but can't push it from the server. Before we upgraded to proper IT support we never had this. (Three guys in a HOT room full of PC's and it rarely went wrong, and install codes and manuals could be found in minutes)