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The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials on a smooth real projective curve X. We show that there is a duality between its faces and totally real effective divisors on X. This allows us to fully describe the face lattice in case X has genus 1. The dual cone ^ is known as the moment cone, and it plays an important role in real analysis. We compute the Carath\'eodory number of ^ for an elliptic normal curve X, which measures the complexity of quadrature rules of measures supported on X. This number can also be interpreted as a maximal typical Waring rank with nonnegative coefficients. Interestingly, the topology of the real locus of X influences the Carath\'eodory number of ^. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.
Baldi et al. (Mon,) studied this question.