  We investigate the prospects of future supernova searches to get meaningful
constraints about the cosmic star formation history (CSFH) and the delay time
of type Ia supernovae from star formation (tau_{Ia}), based only on supernova
data. Here we parameterize the CSFH by two parameters, alpha and beta that are
the evolutionary indices (proportional to (1+z)^{alpha, beta}) at z <~ 1 and >~
1, respectively, and quantitatively examined how well the three parameters
(alpha, beta, and tau_{Ia}) can be constrained in ongoing and future supernova
surveys. We found that the type classification of detected supernovae down to
the magnitude of I_{AB} ~ 27 is essential, to get useful constraint on beta.
The parameter tau_{Ia} can also be constrained within an accuracy of ~ 1--2
Gyr, without knowing alpha that is somewhat degenerate with tau_{Ia}. This
might be potentially achieved by ground-based surveys but depending on the
still highly uncertain type-classification by imaging data. More reliable
classification will be achieved by the SNAP mission. The supernova counts at a
magnitude level of I_{AB} or K_{AB} ~ 30 will allow us to break degeneracies
between alpha and tau_{Ia} and independently constrain all the three
parameters, even without knowing supernova types. This can be achieved by the
SNAP and JWST missions, having different strength of larger statistics and
reach to higher redshifts, respectively. The dependence of observable
quantities on survey time intervals is also quantitatively calculated and
discussed.
