Do we really know Mup (i.e. the transition mass between Type Ia and core-collapse supernova progenitors)?

Mup is the minimum stellar mass that, after the core-helium burning, develops temperature and density conditions for the occurrence of a hydrostatic carbon burning. Stars whose mass is lower than this limit are the progenitors of C-O white dwarfs and, when belong to a close binary system, may give rise to explosive phenomena, such as novae or type Ia supernovae. Stars whose mass is only slightly larger than Mup ignite C in a degenerate core and, in turn, experience a thermonuclear runaway. Their final fate may be a massive O-Ne WDs or, if the core mass approaches the Chandrasekhar limit, an e-capture SNe. More massive objects ignite C in non-degenerate conditions. These "massive " stars are the progenitors of various kind of core-collapse supernovae (type IIp. IIL, IIN, Ib, Ic). It goes without saying that Mup is a fundamental astrophysical parameter. From its knowledge depends our understanding of the SNe progenitors, of their rates, of the chemical evolution, of the WD luminosity functions and much more. A precise evaluation of Mup relies on our knowledge of various input physics used in stellar modeling, such as the plasma neutrino rate, responsible of the cooling of the core, the equation of state of high density plasma, which affects the heating of the contracting core and its compressibility, and some key nuclear reaction rates, such as, in particular, the 12C+12C and the 12C+α. In this paper we review the efforts made to determine this important parameter and we provide an up-to-date evaluation of the uncertainties due to the relevant nuclear physics inputs.