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The goal of this paper is the fine structure of the ideals in the title, withemphasis on the properties of the associated Rees algebra and the specialfiber. The watershed between the present approach and some of the previous workin the literature is that here one does not assume that the ideals in questionsatisfy the common generic properties. One exception is a recent work of N. P.H. Lan which inspired the present work. Here we recover and extend his work. Westrongly focus on the behavior of the ideals of minors of the correspondingso-called Hilbert--Burch matrix and on conjugation features of the latter. Weapply the results to three important models: linearly presented ideals of planefat points, reciprocal ideals of hyperplane arrangements and linearly presentedmonomial ideals.

Linearly presented perfect ideals of codimension $2$ in three variables