I live in Manchester, UK, which is known for our rainy weather, and Sukkot often requires a lot of covering and uncovering the Sukkah with a waterproof cover. I have been wondering whether it is possible to find a plant with leaves that could be overlapped on a sloping frame in a similar layout to roof tiles so that the water would run off the lower end of the frame (possibly even into a gutter). Clearly there are a number of practical problems, including finding a suitable plant, finding a halachically acceptable way to retain the structure without it blowing apart and possibly finding a way to stop it drying out too fast (especially with heating being needed), but are there any halachic reasons why this wouldn't be allowed if the practicalities could be overcome?
Tosafos, as DonielF wrote, states that nailing down the schach to stop the rain invalidates the succah. This is because a sukka needs to be temporary and by stopping the rain it is considered to be permanent. Some say the problem is the nails - even if they do not stop the rain and some say that stopping the rain alone is a problem.
The Rash in Siman 11 and Mordechai 733 brings Rabbeinu Tam who says clearly that a sukka must allow rain in. He proves this from the fact that one may leave the sukka if it rains. Why not be obligated to put up schach and there be no reason to leave? Must be that a kosher sukka doesn’t block rain.
The Tur brings this opinion in siman 731 but does not agree with it. The Bach however says one should be stringent and keep the opinion of Rabbeinu Tam. The shulchan Aruch does not mention it. However the Levush like the Bach says to be stringent as does the Mishna Berura who says it is good bedieved.
However, it is not clear if this is all only when the schach is nailed in. The Piskei Teshuva brings down that someone actually created schach made two layers of gutters and Rav Eliyashav and Rav Chaim Kanievsky both said it is good but the Minchas Yitzchak and Lehoros Nassan said it is not kosher.
There you have it in time for next Sukkos ;)