The author introduces the notion of curves with singularities of coordinate axes type (TAC). A basic example of TAC singularity is given by the union of the coordinate axes in an affine space. Over an algebraically closed field, any TAC singularity is étale locally isomorphic to this example. Let \(\mathcal{X}\) be a semi-stable curve over a discrete valuation ring. A rational point on the generic fiber \(\mathcal{X}_K\) (assumed to be smooth) defines an immersion \(\mathcal{X}_K \to J_K\), where \(J_K\) is the Jacobian variety of \(\mathcal{X}_K\). However, the induced rational map \(a: \mathcal{X} \cdots \to \mathcal{N}\) (where \(\mathcal{N}\) is the Néron model of \(J_K\)) may contract an irreducible component of the special fiber of \(\mathcal{X}\). Contracting such fibers, one obtains \(a_{\min}: \mathcal{X}_{\min} \cdots \to \mathcal{N}\) through which \(a\) factors. It is proved that \(\mathcal{X}_{\min}\) has only TAC singularities. Moreover, \(a_{\min}\) is defined on the smooth locus, and the restriction of \(a_{\min}\) to the smooth locus of any irreducible component of the special fiber of \(\mathcal{X}_{\min}\) is an immersion. As an application, a result of C. Deninger and A. Werner [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 4, 553–597 (2005; Zbl 1087.14026)] is proved under a milder assumption.