In mathematics, particularly in the theory of abelian varieties and algebraic geometry, a *Theta divisor* is a specific kind of divisor associated with a principally polarized abelian variety (PPAV). More formally, if \( A \) is an abelian variety and \( \Theta \) is a quasi-projective variety corresponding to a certain polarization, then the theta divisor \( \theta \) is defined as the zero locus of a section of a line bundle on \( A \).
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The theta divisor consists of degree- line bundles with a nonzero global section. It is the irreducible image of the symmetric product of a curve , with its reduced Cartier divisor structure. The determinant description of the theta divisor supplies equations, multiplicities and pullbacks in families.