Are there modern good lecture notes/book about Brill-Noether theory of curves.


Most interesting theorems here are proved via limit linear series, which I found no lecture notes on (instead there is the original paper from the 80's which references several older papers and while well written is hard to read due to this).


The other class of interesting theorems are proved via topicalization methods on which I also haven't found good sources
Thanks!



Modern references to Brill-Noether theory on curves?
I recommend also Geometry of Algebraic Curves, Volume I, (1984), chs. V and VII, by Arbarello, Cornalba, Griffiths and Harris, as well as Volume II, (2011), ch. XXI, of the same title, by Arbarello, Cornalba, and Griffiths. (The authors present in Volume II, a "simplified" version, due to Pareschi, of Lazarsfeld's proof.) The history of the problem and proofs are exposited very thoroughly in the bibliographical notes to the cited chapters.
I would add to this history that the "Brill Noether matrix" appears even earlier in Roch's paper proving his half of the Riemann-Roch theorem, and that the "Brill-Noether number" is a generalization, for r > 1, of the formula d ≥ (g/2) + 1, in the section numbered 5. of the Erste Abtheilung, of Riemann's paper on Abelian Functions, which Riemann asserts to hold whenever a general curve of genus g has a non constant meromorphic function with only d poles.


Here is another modern source, in which a masters student from Holland generalizes the results of ACGH to arbitrary alg closed fields. <http://abarbon.com/assets/Andrea%20Barbon%20-%20Algebraic%20Brill-Noether%20Theory.pdf>


but the webpage is "not secure".


