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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!

Update--- For future readers, here are some more sources I found good.

https://www.birs.ca/events/2023/5-day-workshops/23w5101/videos (Isaebel's talk- seems to be a very clean modern proof via degeneration, simpler than harris).

Arend bayers k3 brill noether via wall crossing

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    $\begingroup$ Harris and Morrison, “Moduli of curves.” $\endgroup$
    – Jason Starr
    Commented Jun 14, 2023 at 17:46
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    $\begingroup$ There should be newer lecture notes available online by Brian Osserman, although I cannot recall how complete they are. $\endgroup$
    – Tabes Bridges
    Commented Jun 14, 2023 at 21:43
  • $\begingroup$ @TabesBridges Do you have a link? $\endgroup$
    – user135743
    Commented Jun 15, 2023 at 7:41
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    $\begingroup$ This is just a short survey, but "Brill-Noether theory" by Joe Harris (birs.ca/workshops/2014/14w5133/files/Osserman-reference.pdf) in Surveys in Differential Geometry XIV is a very nice 1st introduction to the subject. $\endgroup$
    – Sam Hopkins
    Commented Jun 15, 2023 at 19:34
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    $\begingroup$ @user135743 Hmm, it looks like Brian has left academia. His research papers are on the ArXiV, but it doesn't look like he ever uploaded the lecture notes. You could try looking for old versions of his UCDavid webpage, or contact him directly. $\endgroup$
    – Tabes Bridges
    Commented Jun 16, 2023 at 0:24

3 Answers 3

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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".

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  • $\begingroup$ For future readers I want to note that the book gives an account of BN (by the authors words) about 25 years in to the past, so it is also very out-dated compared to modern techniques sadly. $\endgroup$
    – user135743
    Commented Sep 8, 2023 at 1:37
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[Lazarsfeld, Robert. Brill-Noether-Petri without degenerations. J. Differential Geom. 23 (1986), no. 3, 299--307] is highly recommended.

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  • $\begingroup$ Thanks for the recommandation $\endgroup$
    – user135743
    Commented Jun 15, 2023 at 7:41
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This would certainly be

Casas-Alvero, Eduardo. Algebraic curves, the Brill and Noether way. Universitext. Springer, Cham, 2019.

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    $\begingroup$ That book is elementary, it doesn't even prove the main theorem of Brill-Noether, so it's not what I'm looking for. I think it's an introduction to curves $\endgroup$
    – user135743
    Commented Jun 15, 2023 at 7:44

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