The default value is false. When defined to be true, the symbolic power is calculated as defined in Villarreal. In particular, symbolicPower has the option to restrict to minimal primes versus use all associated primes with UseMinimalPrimes. With this option the symbolic power of an ideal $I$ is defined as $$I^{(n)} = \cap_{p \in Min(R/I)}(I^nR_p \cap R ),$$ where $Min(R/I)$, is the set of minimal primes in $I$.
R. Villarreal. "Monomial algebras" Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. xviii+686 pp. ISBN: 978-1-4822-3469-5.
The object UseMinimalPrimes is a symbol.