Local borcherds products

The local Picard group at a generic point of the one-dimensional Baily-Bore l boundary of a Hermitean symmetric quotient of type O(2,n) is computed. Th e main ingredient is a local version of Borcherds' automorphic products. Th e local obstructions for a Heegner divisor to be principal are given by cer tain theta series with harmonic coefficients. Sometimes they generate Borch erds' space of global obstructions. In these particular cases we obtain a s imple proof of a result due to the first author: Suppose that Gamma subset of O(2,n) is the orthogonal group attached to an even unimodular lattice. T hen every meromorphic modular form for Gamma, whose zeros and poles lie on Heegner divisors, is given by a Borcherds product.