בתר מעיקרא או בתר תבר מנא אזלינן Bava Kama 17b

Rashi explains the din of Rabba where a person threw an item and then someone else broke it, that patur mean that the thrower is chayav and thus we see that Rabba holds בתר מעיקרא. The Ramban explains that the Rif understood Rabba differently, that Rabba actually holds בתר תבר מנא אזלינן and the meaning of patur is that therefore no one is chayav, since the thrower did not break it and the person who actually broke it broke a worthless object since it was about to be broken.

Rav Soloveitchik (רשימות שיעורים) explained this machlokes between Ramban and Rashi as follows:
According to the Ramban, each perpetrator should be looked at inidividually. For this reason, there is no chiyuv. The one who threw it would have been chayav if we held בתר מעיקרא and even had he died before the nezek happened, would have been chayav. But since we hold בתר תבר מנא, the only significant time is the breakage when the item was worthless. However, Rashi holds that we need to consider both times -- מעיקרא and תבר מנא, as both together caused the damage. The question of the sugya is who do we pin the chiyuv on (not who is responsible). Whichever we pick will create a chiyuv on the one individual and uproot the other's chiyuv, and creates an obligation on the entire act (thus having to pay even if we say בתר תבר מנא).

One other interesting note on this. The Ramban asked on Rashi from a case of murder: where a person throws a baby off a roof and someone else catches the baby on his sword (26b). In this case,the sugya says that the thrower is patur. What is the difference? Rav Soloveitchik points out the Ramban's question is very difficult. Unlike an inanimate object that we may consider worthless and broken as soon as it will break, this is not true in the halacha of people. As long as the baby is still alive it has a full status of being alive in the spheres of יבום and ערכין. Therefore, there is no possibility to consider בתר מעיקרא by murder.

Bava kama 9b - hiddur is an increase of 1/3 of what?

Tosfos suggests that the 1/3 increase is not in cost but in the size of your esrog that if you have one that is the size of a nut, then increase by 1/3. The Beis Yosef explains this idea based on others, that the meaning of Tosfos is that you are only being asked to do this hiddur of a 1/3 if you are in a state where you have the bare minimum of the mitzva. Then you should try to increase 1/3. But if you have a decent size esrog, then no 1/3 increase is necessary.

bava kama 9b - Hiddur Mitzva

There is a famous chakira about the din of hiddur mitzva, זה א-לי ואנוהו if it is a separate halacha we apply to mitzvos or if it is incorporated into the mitzva itself. Rav Soloveitchik explained that this chakira explains the two side of the question in our gemara of the 1/3 hiddur is מלבר or מלגיו. He explained that if we say that it is part of the mitzva, then it makes sense that when we look at the chefza you have at the end, that 1/3 of it be hiddur, thus it would be מלבר but if it is just a separate halacha then using what you have now, you tack on the hiddur of 1/3