Relative effective Cartier divisor on a curve ($D\subset C\times T$)

A relative effective divisor of degree $m$ on a smooth projective curve is an effective Cartier divisor in $C\times T$ that is finite and flat of degree $m$ over $T$. Its associated line bundle has degree $m$ on every fibre. The symmetric product of a curve $C^{(m)}$ represents these families and carries their universal divisor. Locally on a smooth coordinate chart, the divisor is described by a monic polynomial, whose coefficients are elementary symmetric functions of its points; this description includes collisions and works in positive characteristic.