galois-notes
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Notes on Galois theory in enumerative geometry
Notes from a minicourse on Galois groups of enumerative problems, following Joe Harris' 1979 paper of the same name. Notes are in rough shape but slowly improving, please suggest changes by emailing or submitting a pull request.
We also are slowly building up a repository of visualizations in enumerative geometry and Galois theory. Check it out!
Owner
- Name: Thomas Brazelton
- Login: tbrazel
- Kind: user
- Website: https://www2.math.upenn.edu/~tbraz/
- Repositories: 1
- Profile: https://github.com/tbrazel
Citation (citations_galois.bib)
@book{3264,
title = {3264 and {{All That}}},
author = {Eisenbud, David and Harris, Joe},
year = {2016},
publisher = {Cambridge University Press}
}
@book{alekseevAbelsTheoremProblems2004,
title = {Abel's Theorem in Problems and Solutions},
author = {Alekseev, V. B.},
year = {2004},
publisher = {Kluwer Academic Publishers, Dordrecht},
mrnumber = {2110624}
}
@article{Arnold-fields2,
title = {Symplectization, Complexification and Mathematical Trinities},
author = {Arnol'd, V. I.},
year = {1997},
journal = {Fields Institute Communications}
}
@misc{Beauville_delPezzo_lattices,
title = {Reduction Mod. 2 of Del {{Pezzo}} Lattices},
author = {Beauville, Arnaud},
year = {2022},
month = sep,
number = {arXiv:2208.12962},
eprint = {2208.12962},
publisher = {arXiv},
doi = {10.48550/arXiv.2208.12962},
urldate = {2025-07-08},
archiveprefix = {arXiv},
keywords = {Mathematics - Algebraic Geometry,Mathematics - Group Theory}
}
@article{BKT-conics,
title = {Enumerative {{Algebraic Geometry}} of {{Conics}}},
author = {Bashelor, Andrew and Ksir, Amy and Traves, Will},
year = {2008},
journal = {Amer. Math. Monthly},
volume = {115},
number = {8},
pages = {701--728},
urldate = {2025-04-20}
}
@book{bourbakiLieGroupsLie1998,
title = {Lie Groups and {{Lie}} Algebras. {{Chapters}} 1--3},
author = {Bourbaki, Nicolas},
year = {1998},
series = {Elements of {{Mathematics}} ({{Berlin}})},
publisher = {Springer-Verlag, Berlin},
mrnumber = {1728312}
}
@book{bourbakiLieGroupsLie2002,
title = {Lie Groups and {{Lie}} Algebras. {{Chapters}} 4--6},
author = {Bourbaki, Nicolas},
year = {2002},
series = {Elements of {{Mathematics}} ({{Berlin}})},
publisher = {Springer-Verlag, Berlin},
doi = {10.1007/978-3-540-89394-3},
mrnumber = {1890629}
}
@book{bourbakiLieGroupsLie2005,
title = {Lie Groups and {{Lie}} Algebras. {{Chapters}} 7--9},
author = {Bourbaki, Nicolas},
year = {2005},
series = {Elements of {{Mathematics}} ({{Berlin}})},
publisher = {Springer-Verlag, Berlin},
mrnumber = {2109105}
}
@article{brugalleEnumerationRealConics2013,
title = {Enumeration of Real Conics and Maximal Configurations},
author = {Brugall{\'e}, Erwan and Puignau, Nicolas},
year = {2013},
journal = {Journal of the European Mathematical Society (JEMS)},
volume = {15},
number = {6},
pages = {2139--2164},
issn = {1435-9855,1435-9863},
doi = {10.4171/JEMS/418},
mrnumber = {3120740}
}
@article{cartanReductionSaForme1896,
title = {{Sur la Reduction a sa Forme Canonique de la Structure d'un Groupe de Transformations Fini et Continu}},
author = {Cartan, E.},
year = {1896},
month = jan,
journal = {American Journal of Mathematics},
volume = {18},
number = {1},
eprint = {2369787},
eprinttype = {jstor},
pages = {1},
issn = {00029327},
doi = {10.2307/2369787},
urldate = {2025-05-19},
langid = {french}
}
@article{Cayley-octad,
title = {A {{Memoir}} on {{Quartic Surfaces}}},
author = {Cayley, Prof},
year = {1869},
journal = {Proceedings of the London Mathematical Society},
volume = {3},
pages = {19--69},
issn = {0024-6115},
doi = {10.1112/plms/s1-3.1.19},
mrnumber = {1577187}
}
@article{cayleyTripleTangentPlanes1849,
title = {On the Triple Tangent Planes of Surfaces of the Third Order},
author = {Cayley, Arthur},
year = {1849},
journal = {Cambridge and Dublin Mathematical Journal},
volume = {4},
pages = {118--132}
}
@article{Chasles-conics,
title = {Construction Des Coniques Qui Satisfont {\`a} Cinq Conditions},
author = {Chasles, Michel},
year = {1864},
journal = {Compte rendu des s{\'e}ances de l'acad{\'e}mie des sciences},
volume = {58},
pages = {297--315}
}
@book{Coble-book,
title = {Algebraic Geometry and Theta Functions},
author = {Coble, Arthur B.},
year = {1961},
series = {American {{Mathematical Society Colloquium Publications}}},
volume = {Vol. X},
publisher = {American Mathematical Society, Providence, RI},
mrnumber = {123958}
}
@book{Coolidge,
title = {A Treatise on Algebraic Plane Curves},
author = {Coolidge, Julian Lowell},
year = {1959},
publisher = {Dover Publications, Inc., New York},
mrnumber = {120551}
}
@book{Dolgachev,
title = {Classical {{Algebraic Geometry}}: A Modern View},
author = {Dolgachev, Igor V},
year = {2012},
publisher = {Cambridge University Press},
langid = {english}
}
@article{DolgachevOrtland,
title = {{Point sets in projective spaces and theta functions}},
author = {Ortland, David and Dolgachev, Igor V},
year = {1988},
journal = {Ast{\'e}risque},
volume = {165},
langid = {french}
}
@book{EKM,
title = {The {{Algebraic}} and {{Geometric Theory}} of {{Quadratic Forms}}},
author = {Elman, Richard and Karpenko, Nikita and Merkurjev, Alexander},
year = {2008},
month = jul,
series = {Colloquium {{Publications}}},
volume = {56},
publisher = {American Mathematical Society},
address = {Providence, Rhode Island},
doi = {10.1090/coll/056},
urldate = {2022-07-08},
langid = {english}
}
@book{Farquhar-deception,
title = {A {{Treasury}} of {{Deception}}\_ {{Liars}}, {{Misleaders}}, {{Hoodwinkers}}, and the {{Extraordinary True Stories}} of {{History}}'s {{Greatest Hoaxes}}, {{Fakes}} and {{Frauds}}},
author = {Farquhar, Michael},
year = {2005},
publisher = {Penguin Books}
}
@book{Forster,
title = {Lectures on {{Riemann Surfaces}}},
author = {Forster, Otto},
year = {1981},
series = {Graduate {{Texts}} in {{Mathematics}}},
volume = {81},
publisher = {Springer New York},
address = {New York, NY},
doi = {10.1007/978-1-4612-5961-9},
urldate = {2025-03-09},
copyright = {http://www.springer.com/tdm},
langid = {english}
}
@book{Gray-worlds,
title = {Worlds {{Out}} of {{Nothing}}: {{A Course}} in the {{History}} of {{Geometry}} in the 19th {{Century}}},
shorttitle = {Worlds {{Out}} of {{Nothing}}},
author = {Gray, Jeremy},
year = {2010},
series = {Springer {{Undergraduate Mathematics Series}}},
publisher = {Springer London},
address = {London},
doi = {10.1007/978-0-85729-060-1},
urldate = {2023-03-26},
langid = {english}
}
@incollection{GrossHarris-theta,
title = {On Some Geometric Constructions Related to Theta Characteristics},
booktitle = {Contributions to Automorphic Forms, Geometry, and Number Theory},
author = {Gross, Benedict H. and Harris, Joe},
year = {2004},
pages = {279--311},
publisher = {Johns Hopkins Univ. Press, Baltimore, MD},
mrnumber = {2058611}
}
@article{Harris-Galois,
title = {Galois Groups of Enumerative Problems},
author = {Harris, Joe},
year = {1979},
month = dec,
journal = {Duke Mathematical Journal},
volume = {46},
number = {4},
issn = {0012-7094},
doi = {10.1215/S0012-7094-79-04635-0},
urldate = {2024-08-07},
langid = {english}
}
@article{Harris-theta,
title = {Theta-Characteristics on Algebraic Curves},
author = {Harris, Joe},
year = {1982},
journal = {Transactions of the American Mathematical Society},
volume = {271},
number = {2},
langid = {english}
}
@book{harrisAlgebraicGeometry1992,
title = {Algebraic {{Geometry}}},
author = {Harris, Joe},
year = {1992},
series = {Graduate {{Texts}} in {{Mathematics}}},
volume = {133},
publisher = {Springer New York},
address = {New York, NY},
doi = {10.1007/978-1-4757-2189-8},
urldate = {2025-04-04},
copyright = {http://www.springer.com/tdm},
langid = {english}
}
@book{Hartshorne,
title = {Algebraic {{Geometry}}},
author = {Hartshorne, Robin},
year = {1977},
series = {Graduate {{Texts}} in {{Mathematics}}},
volume = {52},
publisher = {Springer New York},
address = {New York, NY},
doi = {10.1007/978-1-4757-3849-0},
urldate = {2020-11-12},
langid = {english}
}
@article{Hermite51,
title = {Sur Les Fonctions Alg{\'e}briques},
author = {Hermite, Charles},
year = {1851},
journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences},
volume = {XXXII}
}
@article{Hesse1855,
title = {{{\"U}ber Determinanten und ihre Anwendung in der Geometrie, insbesondere auf Curven vierter Ordnung.}},
author = {Hesse, Otto},
year = {1855},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {49},
pages = {243--264},
issn = {0075-4102; 1435-5345/e},
urldate = {2025-03-25},
langid = {deu}
}
@book{Hunt,
title = {The Geometry of Some Special Arithmetic Quotients},
author = {Hunt, Bruce},
year = {1996},
series = {Lecture Notes in Mathematics},
number = {1637},
publisher = {Springer},
address = {Berlin ; New York},
langid = {english},
lccn = {QA3 QA573 .L28 no. 1637},
keywords = {Moduli theory,Surfaces Algebraic,Threefolds (Algebraic geometry)}
}
@article{Jacobi1850,
title = {{Beweis des Satzes da{\ss} eine Curve nten Grades im Allgemeinen ... (n-2)(n2-9) Doppeltangenten hat.}},
author = {Jacobi, C. G. J.},
year = {1850},
journal = {Journal f{\"u}r die reine und angewandte Mathematik},
volume = {40},
pages = {237--260},
issn = {0075-4102; 1435-5345/e},
urldate = {2025-03-25},
langid = {deu}
}
@book{Jordan,
title = {Trait{\'e} Des Substitutions},
author = {Jordan, M. Camille},
year = {1870},
publisher = {Gauthier-Villars},
address = {Paris}
}
@book{Katz-book,
title = {Enumerative Geometry and String Theory},
author = {Katz, Sheldon},
year = {2006},
series = {Student {{Mathematical Library IAS}}/{{Park City Mathematical Subseries}}}
}
@book{Khovanskii,
title = {Topological {{Galois Theory}}: {{Solvability}} and {{Unsolvability}} of {{Equations}} in {{Finite Terms}}},
shorttitle = {Topological {{Galois Theory}}},
author = {Khovanskii, Askold},
year = {2014},
series = {Springer {{Monographs}} in {{Mathematics}}},
publisher = {Springer Berlin Heidelberg},
address = {Berlin, Heidelberg},
doi = {10.1007/978-3-642-38871-2},
urldate = {2025-03-09},
copyright = {https://www.springernature.com/gp/researchers/text-and-data-mining},
langid = {english}
}
@incollection{Kleiman-Chasles,
title = {Chasles's Enumerative Theory of Conics: A Historical Introduction},
shorttitle = {Chasles's Enumerative Theory of Conics},
booktitle = {Studies in Algebraic Geometry},
author = {Kleiman, Steven L.},
year = {1980},
series = {{{MAA Stud}}. {{Math}}.},
volume = {20},
pages = {117--138},
publisher = {Math. Assoc. America, Washington, DC},
mrnumber = {589410}
}
@incollection{landsbergRepresentationTheoryProjective2004,
title = {Representation Theory and Projective Geometry},
booktitle = {Algebraic Transformation Groups and Algebraic Varieties},
author = {Landsberg, Joseph M. and Manivel, Laurent},
year = {2004},
series = {Encyclopaedia {{Math}}. {{Sci}}.},
volume = {132},
pages = {71--122},
publisher = {Springer, Berlin},
doi = {10.1007/978-3-662-05652-3_6},
mrnumber = {2090671}
}
@misc{Looijenga-monster,
title = {Cubic Threefolds Moduli and the {{Monster}} Group},
author = {Looijenga, Eduard},
year = {2024},
month = nov,
number = {arXiv:2310.20124},
eprint = {2310.20124},
publisher = {arXiv},
doi = {10.48550/arXiv.2310.20124},
urldate = {2025-07-07},
archiveprefix = {arXiv},
keywords = {Mathematics - Algebraic Geometry,Mathematics - Group Theory}
}
@article{Manivel2006,
title = {Configurations of Lines and Models of {{Lie}} Algebras},
author = {Manivel, L.},
year = {2006},
month = oct,
journal = {Journal of Algebra},
volume = {304},
number = {1},
pages = {457--486},
issn = {00218693},
doi = {10.1016/j.jalgebra.2006.04.029},
urldate = {2024-07-12},
copyright = {https://www.elsevier.com/tdm/userlicense/1.0/},
langid = {english}
}
@book{MillerBlichfeldtDickson,
title = {Theory and Applications of Finite Groups},
author = {Miller, G. A. and Blichfeldt, H. F. and Dickson, L. E.},
year = {1916},
publisher = {Dover Publications, Inc., New York},
mrnumber = {123600}
}
@article{plaumannQuarticCurvesTheir2011a,
title = {Quartic Curves and Their Bitangents},
author = {Plaumann, Daniel and Sturmfels, Bernd and Vinzant, Cynthia},
year = {2011},
month = jun,
journal = {Journal of Symbolic Computation},
volume = {46},
number = {6},
pages = {712--733},
issn = {07477171},
doi = {10.1016/j.jsc.2011.01.007},
urldate = {2024-07-14},
copyright = {https://www.elsevier.com/tdm/userlicense/1.0/},
langid = {english}
}
@article{Ronga-Klein,
title = {Felix {{Klein}}'s Paper on Real Flexes Vindicated},
author = {Ronga, Felice},
year = {1998},
journal = {Banach Center Publications},
volume = {44},
number = {1},
pages = {195--210},
issn = {0137-6934, 1730-6299},
doi = {10.4064/-44-1-195-210},
urldate = {2025-03-10},
langid = {english}
}
@article{RongaTognoliVust,
title = {The Number of Conics Tangent to Five given Conics: The Real Case},
shorttitle = {The Number of Conics Tangent to Five given Conics},
author = {Ronga, Felice and Tognoli, Alberto and Vust, Thierry},
year = {1997},
journal = {Revista Matem{\'a}tica de la Universidad Complutense de Madrid},
volume = {10},
number = {2},
pages = {391--421},
issn = {0214-3577},
mrnumber = {1605670}
}
@article{Sameera-theta,
title = {On Intersections of Symmetric Determinantal Varieties and Theta Characteristics of Canonical Curves},
author = {Kulkarni, Avinash and Vemulapalli, Sameera},
year = {2024},
month = may,
journal = {Journal of Pure and Applied Algebra},
volume = {228},
number = {5},
pages = {107538},
issn = {00224049},
doi = {10.1016/j.jpaa.2023.107538},
urldate = {2025-03-26},
langid = {english}
}
@misc{SottileYahl,
title = {Galois {{Groups}} in {{Enumerative Geometry}} and {{Applications}}},
author = {Sottile, Frank and Yahl, Thomas},
year = {2021},
month = aug,
number = {arXiv:2108.07905},
eprint = {2108.07905},
primaryclass = {math},
publisher = {arXiv},
doi = {10.48550/arXiv.2108.07905},
urldate = {2025-03-08},
archiveprefix = {arXiv},
langid = {english},
keywords = {Mathematics - Algebraic Geometry}
}
@misc{Stacks,
title = {Stacks {{Project}}},
shorttitle = {Stacks},
author = {Stacks Project Authors},
howpublished = {https://stacks.math.columbia.edu/}
}
@article{Steiner-conics,
title = {Elementare {{L{\"o}sung}} Einer Geometrischen {{Aufgabe}}, Und {\"U}ber Einige Damit in {{Beziehung}} Stehende {{Eigenschaften}} Der {{Kegelschnitte}}},
author = {Steiner, J.},
year = {1848},
month = jul,
journal = {Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
volume = {1848},
number = {37},
pages = {161-192b},
publisher = {De Gruyter},
issn = {1435-5345},
doi = {10.1515/crelle-1848-18483710},
urldate = {2025-04-19},
chapter = {Journal f{\"u}r die reine und angewandte Mathematik},
copyright = {De Gruyter expressly reserves the right to use all content for commercial text and data mining within the meaning of Section 44b of the German Copyright Act.},
langid = {english}
}
@article{Steiner-cubic-surfaces,
title = {{\"U}ber Die {{Fl{\"a}chen}} Dritten {{Grades}}},
author = {Steiner, Jakob},
year = {1857},
journal = {Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
volume = {53},
pages = {133--141},
doi = {10.1515/crll.1857.53.133}
}
@article{TBS-conics,
title = {3264 {{Conics}} in a {{Second}}},
author = {Timme, Sascha and Breiding, Paul and Sturmfels, Bernd},
year = {2020},
month = jan,
journal = {Notices of the American Mathematical Society},
volume = {67},
number = {01},
pages = {1},
issn = {0002-9920, 1088-9477},
doi = {10.1090/noti2010},
urldate = {2025-04-19},
langid = {english}
}
@article{vainsencherEnumerationNfoldTangent1995,
title = {Enumeration of N-Fold Tangent Hyperplanes to a Surface},
author = {Vainsencher, Israel},
year = {1995},
journal = {Journal of Algebraic Geometry},
volume = {4},
number = {3},
pages = {503--526},
issn = {1056-3911,1534-7486},
mrnumber = {1325790}
}
@book{Weber1896,
title = {Lehrbuch Der {{Algebra}}},
author = {Weber, H},
year = {1896},
publisher = {Braunschweig}
}
@article{Welschinger-GAFA,
title = {Towards Relative Invariants of Real Symplectic Four-Manifolds},
author = {Welschinger, Jean-Yves},
year = {2006},
month = dec,
journal = {GAFA Geometric And Functional Analysis},
volume = {16},
number = {5},
pages = {1157--1182},
issn = {1016-443X, 1420-8970},
doi = {10.1007/s00039-006-0576-5},
urldate = {2025-04-20},
copyright = {http://www.springer.com/tdm},
langid = {english}
}
@phdthesis{Zhang-thesis,
title = {Galois Groups of Enumerative Problems},
author = {Zhang, Yongquan},
year = {2019},
urldate = {2025-03-24},
langid = {english},
school = {Harvard University}
}
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