Abstract：It is known by Hironaka's work in 1968 that on a smooth projective variety defined over an infinite field, any algebraic 1-cycle is rationally equivalent to a smooth one. In this talk, I will show that the result is also true when the variety is defined over a finite field. In this setting, several results on the density of smooth divisors satisfying certain conditions are needed to construct the rationally equivalent smooth 1-cycle. I will also sketch how Poonen's closed point sieve works to get such density results.
