Let \(K\) be a number field, \(X/K\) be a smooth projective geometrically irreducible curve of genus \(> 1\), and let \(\mathcal X\) be a proper regular model of \(X\) over the ring of integers of \(K\). Given a \(\mathbb Q\)-divisor \(D\) of degree one on \(X\), the authors locally construct vertical \(\mathbb Q\)-divisors \(V_D\) and \(U_D\) associated to \(D\), from which they construct an associated hermitian line bundle \(\overline{\mathcal L_D}\). Theorem 1.1 compares the height with respect to \(\overline{\mathcal L_D}\) with the Néron-Tate height on the Jacobian after embedding via \(D\) (Theorem 1.1). This implies a positivity result for \(\overline{\mathcal L_D}^2\) when \(\overline{\mathcal L_D}\) is relatively semipositive (Proposition 1.2). Proposition 1.2 then implies an explicit lower bound for the self-intersection number of the relative dualizing sheaf \(\overline \omega\) on \(\mathcal X\) (Theorem 1.3). This enables them to recover a result from the admissible intersection theory that when \(\mathcal X\) is semistable and minimal and has at least one reducible fiber, then there is an effectively computable positive lower bound for \(\overline \omega^2\). The authors’ approach also applies to non-semistable cases, and as examples, they compute lower bounds for \(\overline \omega^2\) for the minimal regular models of (1) modular curves \(X_1(N)\) for certain \(N\) (Proposition 6.1) and (2) the Fermat curve of prime exponent \(>3\) (Theorem 6.6).