The core variety of a multisequence in the truncated moment problem

Let β≡β(m)={βi}i∈Z+n,|i|≤m, β0>0, denote a real n-dimensional multisequence of finite degree m. The Truncated Moment Problem concerns the existence of a positive Borel measure μ, supported in Rn, such that(0.1)βi=∫Rnxidμ(i∈Z+n,|i|≤m). We associate to β≡β(2d) an algebraic variety in Rn called the core variety, V≡V(β). The core variety contains the support of each representing measure μ. We show that if V is nonempty, then β(2d−1) has a representing measure. Moreover, if V is a nonempty compact or determining set, then β(2d) has a representing measure. We also use the core variety to exhibit a sequence β, with positive definite moment matrix and positive Riesz functional, which fails to have a representing measure.