The (Euclidean) hull of a linear code is defined to be the intersection of the code and its Euclidean dual. It is clear that the hulls are self-orthogonal codes, which are an important type of linear codes due to their wide applications in communication and cryptography. Let Fq\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb F_q$$\end{document} be the finite field of order q and n=qm-1q-1\documentclass[12pt]{minimal}
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\begin{document}$$n = \frac{q^m-1}{q-1}$$\end{document}, where q is a power of a prime and m≥2\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 2$$\end{document} is an integer. Let C(q,n,δ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_{(q,n,\delta )}$$\end{document} be a projective narrow-sense BCH code over Fq\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb F_q$$\end{document} with designed distance δ\documentclass[12pt]{minimal}
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\begin{document}$$\delta $$\end{document}. In this paper, we will investigate both the dimensions and the minimum distances of Hull(C(q,n,δ))\documentclass[12pt]{minimal}
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\begin{document}$$\text {Hull}({\mathcal {C}}_{(q,n,\delta )})$$\end{document}, where 2≤δ≤2(qm+12-1)q-1\documentclass[12pt]{minimal}
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\begin{document}$$2 \le \delta \le \frac{2(q^{\frac{m+1}{2}} -1)}{q-1}$$\end{document} if m≥5\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 5$$\end{document} is odd and 2≤δ≤qm2+1-1q-1-q+1\documentclass[12pt]{minimal}
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\begin{document}$$2 \le \delta \le \frac{q^{\frac{m}{2}+1}-1}{q-1}-q+1$$\end{document} if m≥6\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 6$$\end{document} is even. As a byproduct, a sufficient and necessary condition on the Euclidean dual-containing BCH code C(q,n,δ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_{(q,n,\delta )}$$\end{document} is documented. In addition, we present some characterizations of the hulls of ternary projective narrow-sense BCH codes when dim(Hull(C(3,n,δ)))=k-1,k-2\documentclass[12pt]{minimal}
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\begin{document}$$\dim \Big (\text {Hull} ({\mathcal {C}}_{(3,n,\delta )})\Big )=k-1, \ k-2$$\end{document} for even m≥2\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 2$$\end{document}; and dim(Hull(C(3,n,δ)))=k-1,k-2m-1\documentclass[12pt]{minimal}
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\begin{document}$$\dim \Big (\text {Hull} ({\mathcal {C}}_{(3,n,\delta )})\Big )=k-1, \ k-2m-1$$\end{document} for odd m≥3\documentclass[12pt]{minimal}
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\begin{document}$$m\ge 3$$\end{document}, where k is the dimension of C(3,n,δ)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {C}}_{(3,n,\delta )}$$\end{document}.
