Symbolic analytic spread: upper bounds and applications

The symbolic analytic spread of an ideal $I$ is defined in terms of the rate of growth of the minimal number of generators of its symbolic powers. In this article we find upper bounds for the symbolic analytic spread under certain conditions in terms of other invariants of $I$. Our methods also work for more general systems of ideals. As applications we provide bounds for the (local) Kodaira dimension of divisors, the arithmetic rank, and the Frobenius complexity. We also show sufficient conditions for an ideal to be a set-theoretic complete intersection.