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We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points., Comment: 22 pages, 1 figure