User:Wundzer

Rando on the internet interested in derived categories, string theory, and algebraic geometry. Interested in helping me build out wikipedia in these sections and write content with examples? Put a message on my talk page! I can give instructions on stuff todo. Another option is to start a weekly reading group and write up the results on wikipedia. Thoughts?

Pages I've edited

• Pursuing Stacks – working on writing up a sketch of material covered in the manuscript, writing out articles on related homotopy theory constructions
• Postnikov tower – rewrote most of the article and gave relation to homotopy theory and computing homotopy groups
• Derived Noncommutative Algebraic Geometry – wrote entire article
• Chern classes – gave computational examples
• Pontryagin class – gave computation of K3 surfaces
• Todd genus – gave examples here too
• Homotopy groups – added homotopy groups of orthogonal groups and relation to sphere bundles
• Local systems – discussed bivariant topologicla theory with examples
• Coherent sheaf cohomology – Added Kunneth formula and related computation for genus g curves
• Algebraic curve – gave computations for genera of plane curves and curves in $\mathbb {P} ^{1}\times \mathbb {P} ^{1}$.
• Quot scheme – updated text and gave constructive examples
• Hilbert scheme – updated examples section and added some insightful examples
• Grothendieck riemann roch – added examples of vector bundles on curves, smooth proper morphisms and moduli of curves example, and closed embeddings
• Convexity (algebraic geometry) – Wrote first article with lots of examples.
• Proper morphism – Added examples and geometric intuition for valuative criterion of properness.
• Schubert calculus – Wrote up construction, examples, and lines on a cubic surface
• Field norm – added examples and reorganized page
• Linear system of divisors – added examples from curves, mentioned hyperelliptic curves, trigonal curves, g.r.d.'s and Brill noether, and improved other examples
• Dual number – added examples of tangent vectors on the scheme to show how this technology works.
• Fiber functor – gave definitions and additional references
• Galois group – refactored the examples section and added some additional examples and a computational proposition. Also added some much needed references
• Opposite ring – added examples of the oppositve algebra for the free algebra and quaternion algebra
• Formally smooth map – added examples and non-examples
• Locally compact field – rewrote most of article, added structure theorems, intuition, and examples from p-adic numbers
• Solvable group – added motivation for definition and reformatted examples section to be more readable. In addition, I added some more examples
• Noetherian scheme – added examples and non-examples, including one which motivates the study for schemes over a non-Noetherian base
• Deformation theory – added deformations of germs of analytic functions and mentioned tangent cohomology
• Derived algebraic geometry – added examples of derived schemes and spectral schemes
• Kodaira–Spencer map – rewrote article, added constructions of map and examples of it
• Exalcomm – updated page with definition of square-zero extension, construction, and structure theorems.
• Derivator – added sections on motivation, definition, etc.
• Jacobian ideal – added relations to hodge theory and deformation theory
• Solvable Lie algebra – updated examples section to be more illuminating, also made it more organized
• Nilpotent Lie algebra – reorganized and added examples
• Lie algebra – helped make this page more user friendly by adding examples and explanations
• Unipotent – fixed up definitions, added example section, added classification using nilpotent lie algebras
• Normal scheme – added cusp example of normalization and formatted examples section
• Minimal polynomial – explained tool to compute minimal polynomial and gave examples
• Integral element – reorganzied and added examples
• Quadratic integer – explained computation of ring of integers
• Annihilator (ring theory) – explained annihilators for commutative rings and gave techniques required for their calculation. In addition, added complete calculation for all finite modules over the integers
• Quintic threefold – rewriting the article...
• Length of a module – updated article, added references, examples, and related topics, esp to intersection theory
• Moduli of abelian varieties – created article
• Moduli of algebraic curves – updated article with construction, properties, and examples of low genera
• Moduli stack of elliptic curves – expanded upon article by constructing moduli space over characteristic 0 and gave detailed description of the points
• Gerbe – Added examples of root stacks
• Stack (mathematics) – reorganized examples section to it's more readable for beginners, added reference to local structure of algebraic stacks, also added examples
• Azumaya algebra – added examples
• Homotopy Lie algebra – updated with references and examples
• Homotopy associative algebra – created page
• Mirror Symmetry Conjecture – created page
• Mixed Hodge Structure – created page
• Intersection homology – added example of intersection cohomology sheaf
• Milnor map – improved definition with references, added some main theorems
• Mixed Hodge module – created page
• Kan complex – expanded and improved article with better organization, examples, application, and structure of Kan complexes
• Dold–Kan correspondence – stated functorial equivalence + sketched construction
• ∞-groupoid – adding material
• Symplectic matrix - reformatted and added theorem about generating set of matrices

General references

L-functions of motives

• (Conjectures in Arithmetic Algebraic Geometry) https://link.springer.com/chapter/10.1007%2F978-3-663-09505-7_4

Jacobians of hyperelliptic curves and other arithmetic

• A user's guide to the local arithmetic of hyperelliptic curves

Picard group + Moduli of vector bundles

• Adelic presentation (theorem of Weil) for a curve (On the Geometry of Higher Tate Spaces - Aron Heleodoro - Northwestern)
• For a reductive group $G$ and a curve $X$, $Bun_{G}(X)$ can be described as $G(k(X))\backslash G(\mathbb {A} _{X})/G(\mathbb {O} _{X})$
• There also exists a similar result for arbitrary varieties, but this uses a co-simplicial ring
• Adelic Descent Theory - https://arxiv.org/abs/1511.06271
• Residues and adeles - Beilinson (2 pages)

Coherent sheaves

• https://mathoverflow.net/questions/364678/is-mathbbq-p-otimes-mathbbq-mathbbq-p-coherent/364690

Modular forms

• https://mathoverflow.net/questions/24604/why-are-modular-forms-interesting
• https://math.stackexchange.com/questions/325364/intuition-for-the-importance-of-modular-forms
• The Web of Modularity: Arithmetic of the Coefficients of Modular Forms
• https://mathoverflow.net/questions/77278/introductory-text-on-galois-representations/77328#77328
• http://math.uchicago.edu/~emerton/research.html#students
• https://www.dpmms.cam.ac.uk/~ajs1005/preprints/mf.pdf -> motives of modular forms

Abelian varieties

General

• Debarre on Abelian varieties – (good discussion on Prym varieties) – https://www.math.ens.fr/~debarre/AV.pdf
• Products of Families of Abelian Varieties – https://www.jstor.org/stable/1970125?seq=1

Moduli of abelian varieties

• Geometry of A_g and compactifications – https://arxiv.org/abs/0711.0094
• Intersection cohomology of Satake compactification of A_g – http://arxiv.org/pdf/1603.02343.pdf

Schottky

• https://arxiv.org/pdf/1009.0369.pdf

Intermediate Jacobians

• https://arxiv.org/pdf/1510.08891.pdf

Pathologies in nature

This will contain references to pathological objects which occur in nature and not by manual hacking.

• Authormophism Groups of Topological Groups – https://www.jstor.org/stable/1990752?seq=1
• Moduli of vector bundles - write down covering
• Genus 2 curve with ample line bundle having no global sections - https://math.stackexchange.com/questions/764085/an-example-of-ample-sheaf-with-no-global-section
• Bestiary of topological objects - Strickland - http://neil-strickland.staff.shef.ac.uk/courses/bestiary/bestiary.pdf
• http://neil-strickland.staff.shef.ac.uk/courses/formalgroups/fg.pdf

Representation theory

• Lie algebras - https://web.archive.org/web/20191029165616/http://pi.math.cornell.edu/~dmehrle/notes/partiii/liealg_partiii_notes.pdf
• Finite groups - https://web.archive.org/web/20200110092941/http://pi.math.cornell.edu/~dmehrle/notes/partiii/reptheory_partiii_notes.pdf
• https://sbseminar.wordpress.com/2008/07/08/how-to-write-down-the-representations-of-gl_n/ Peter-Weyl theorem for algebraic groups
• http://www.math.lsa.umich.edu/~kesmith/rep.pdf pg 36 for additive group

Groupoids

• https://arxiv.org/abs/math/0503266 Loop groupoid (inertia groupoid of a constant quotient groupoid)
• https://arxiv.org/abs/math/0105039 Gerbes over Orbifolds and Twisted K-theory

GIT

• http://www.hanbommoon.net/wp-content/uploads/2013/08/Invariant_theory_and_moduli_spaces.pdf

Pages needing work

Projection valued measures

• Add examples, try using Rodriguez polynomials to form a basis
• http://dsp-book.narod.ru/FCWAVE/8274_PDF_C01.pdf
• https://www.math.upenn.edu/~deturck/m426/notes03.pdf
• http://math.gmu.edu/~dwalnut/teach/Math772/Fall07/772f07lec01.pdf
• https://math.stackexchange.com/questions/2791638/trying-to-construct-simple-example-of-a-spectral-measure-mu-psi-for-the-ex
• https://math.stackexchange.com/questions/1365019/how-exactly-does-one-define-the-spectral-measure-of-an-operator

Algebraic stacks

There really needs to be a separate algebraic stacks page which is focused entirely on that subset of stacks. This should include definitions, recent theorems (slice theorem), applications, and morphisms of different stacks.

• Lecture notes: https://folk.uio.no/fredrme/algstacks.pdf
• Could include some material from https://sites.math.northwestern.edu/~pgoerss/papers/modfg.pdf
• fppf vs fpqc and https://mathoverflow.net/questions/200990/do-algebraic-stacks-satisfy-fpqc-descent
• Mention smooth groupoids result: https://stacks.math.columbia.edu/tag/04T5
• Mention lifting a topology from a category fibered in groupoids over a site (useful fundamental result, should put on topos or grothendieck topology page)

Quasi-coherent sheaves on algebraic stacks

• https://stacks.math.columbia.edu/tag/0440 qcoh on algebraic groupoids
• https://stacks.math.columbia.edu/tag/06WS : can this result be glued using the slice theorem?

• https://stacks.math.columbia.edu/tag/06WU : smooth groupoid equivalence

Deformation theory

• Obstruction theories are explained really well in "Introduction to Singularities and Deformations"
• https://arxiv.org/abs/1302.1149 -> applications to deformations of log calabi-yau manifolds
• http://www.numdam.org/item/RSMUP_2003__110__221_0/ -> MHS on log deformations
• https://arxiv.org/abs/1708.00097

Deligne cohomology

• https://arxiv.org/abs/alg-geom/9601025

Picard groups of moduli spaces

• mumford picard groups of moduli problems
• https://mathoverflow.net/questions/25323/picard-groups-of-moduli-problems for general case

Level structures

Mention how $GL_{2}(Z)$ gives infinitely many bases for $Z^{\oplus 2}$, hence we need to consider level structures to get finite etale coverings of moduli spaces

• https://jeffyelton.weebly.com/uploads/4/4/5/6/44566759/moduli_spaces-1.pdf has good motivation for level structures
• Silverman Arithmetic – C.13 pg 439 gives excellent overview of classical theory
• Also describe Hodge theory giving relation between elliptic curve and its hodge structure, exploit this as a tool for level-structure intuition https://arxiv.org/pdf/1412.8499.pdf
• https://www.math.purdue.edu/~arapura/preprints/shimura2.pdf last 2 pages has excellent examples
• https://warwick.ac.uk/fac/sci/maths/people/staff/fbouyer/modular_curves.pdf

Drinfeld modules

• http://www-personal.umich.edu/~asnowden/seminar/2017/drinfeld/
• http://math.bu.edu/people/jsweinst/Teaching/MA841Fall17/DrinfeldModules.pdf
• https://www.mit.edu/~fengt/6Drinfeld.pdf
• https://www.math.ias.edu/~lurie/papers/LevelStructures1.pdf
• http://math.stanford.edu/~conrad/papers/kmpaper.pdf

Moduli of Abelian varieties

• check out page 9 of https://arxiv.org/abs/alg-geom/9609001 for a whirlwind tour of constructing moduli of abelian varieties over characteristic 0
• also look at https://www.google.com/books/edition/Moduli_Spaces_of_Abelian_Surfaces/F2vzHJXwq5gC?hl=en&gbpv=0 which is referenced in the article
• discuss semi-stable reduction, look at intro of irreduciblity of moduli of curves DM
• http://www-personal.umich.edu/~asnowden/teaching/2013/679/L08.html has a great discussion for the case of elliptic curves
• General construction using Mumford relations and Hilbert schemes – https://www.math.purdue.edu/~arapura/preprints/abelian.pdf
• Also, Mumford has a more in-depth overview http://www.dam.brown.edu/people/mumford/alg_geom/papers/1970b--ICMNice-IMU.pdf
• Serre-Tate – http://math.mit.edu/~zzyzhang/Lubin-Tate.pdf and https://www.math.upenn.edu/~chai/papers_pdf/CMIPchaioort.pdf
• Example 3.4 in https://www.math.wustl.edu/~matkerr/SV.pdf

Hodge

• https://arxiv.org/abs/1310.1900
• https://arxiv.org/abs/1203.2776
• https://arxiv.org/abs/1310.1773

Gromov–Witten invaritants

• sagemath computations of GW invariants and schubert calculus – https://web.archive.org/web/20200518215736/https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/3750/file/Hiep_Dang_thesis.pdf
• https://arxiv.org/abs/math/9807109 -> computes 2875 using GW calculations
• https://www.maths.ed.ac.uk/~abayer/Bmur-quantum-coh-and-rth-root.pdf
• https://arxiv.org/abs/math/0702234
• GW on orbifolds lecture – https://arxiv.org/abs/math/0512372v2
• https://www.ihes.fr/~maxim/TEXTS/WithManinCohFT.pdf
• Intro – https://arxiv.org/abs/1407.1260v1
• https://www.dailymotion.com/video/x8juco

Motivic

• https://arxiv.org/abs/alg-geom/9506023
• Motives of DM-stacks – https://hal.archives-ouvertes.fr/file/index/docid/773027/filename/motdm.pdf

Picard–Fuchs

• Mirror Symmetry Cox, Katz – page 83
• https://www.ams.org/journals/bull/2007-44-04/S0273-0979-07-01178-0/S0273-0979-07-01178-0.pdf -> pg 5 has Picard-Fuchs for Dwork family
• Remark 2.3.8.3 on page 37 of https://eudml.org/doc/142177
• https://www.jstor.org/stable/1970727?seq=1
• references/guide to Dworks study – https://web.math.princeton.edu/~nmk/dworkfam64.pdf

Ring of integers

• https://www.jmilne.org/math/CourseNotes/ANT.pdf page 40 for algorithm using the discriminant
• Also look at the pages before for more info and relations to discriminant

Jacobian ideal

• Page 84 of Mirror Symmetry book by Cox Katz has a *much* better explanation for reductions of pole orders
• Also, this has a lot of results/explanations on computing Gauss-Manin connections

Jacobians and Periods

• http://rin.io/calculating-the-period-matrix-1/
• https://mathoverflow.net/questions/277410/how-can-i-calculate-the-period-matrix-of-this-riemann-surface
• Hyperelliptic shiga curves -> https://arxiv.org/abs/1211.6910
• Period matrix of Bring's curve -> https://projecteuclid.org/euclid.pjm/1102635736
• The group law on the Jacobian of a curve of genus 2 – Flynn – https://people.maths.ox.ac.uk/flynn/arts/art5.pdf
• https://mathoverflow.net/questions/214619/realizing-a-jacobian-as-a-projective-variety
• Mumford's theta relations
• https://mathoverflow.net/questions/115440/equation-for-simple-jacobian-of-a-genus-two-curve -> hyperelliptic curves
• https://arxiv.org/abs/math/0112321
• https://arxiv.org/abs/0802.3014
• Equations for the Jacobian of a hyperelliptic curve – https://www.ams.org/journals/tran/1998-350-08/S0002-9947-98-02056-X/S0002-9947-98-02056-X.pdf
• https://mathoverflow.net/questions/102440/deformations-of-smooth-projective-hypersurfaces-and-the-jacobian-ring -> could also be part of deformation theory page
• https://projecteuclid.org/euclid.pjm/1102634742 -> formal group of jacobian
• Non-Isomorphic curves with isomorphic Jacobians (Note PPAV hypothesis required for torelli theorem) -> https://www.sciencedirect.com/science/article/pii/S0022314X96900268
• Algorithm to calculate the period matrix of the curve x^m + y^m = 1 – https://www.jstor.org/stable/43686230?seq=1
• Period Matrices of hyperelliptic curves – https://eudml.org/doc/155814
• https://projecteuclid.org/euclid.nmj/1118775097
• https://arxiv.org/abs/1810.03818 – feynman integrals and period matrices
• https://arxiv.org/abs/1810.03818 – feynman integrals and twisted intersection theory

Computing period matrices and Hodge theory

• http://w3.impa.br/~hossein/books.html

Numerical aspects

• Computing period matrices and the Abel-Jacobi map of superelliptic curves – Molin, Neurohr https://webusers.imj-prg.fr/~pascal.molin/pdf/periods.pdf (poster https://www.mathematik.uni-kl.de/~thofmann/ants/postersession/poster_neurohr.pdf)
• https://www.sciencedirect.com/science/article/pii/S0747717198902401
• Jacobian ideal gives basis of holomorphic differentials for a stable curve

Period Matrices and jacobians of higher dim varieties

• Computing Periods of Hypersurfaces – https://arxiv.org/abs/1803.08068
• Singularities of Differentiable Maps, Volume 2 Chapter (Integrals and differential equations) (MHS) (Period Map and Intersection Form)

• PERIODS OF ALGEBRAIC VARIETIES -> Oliver Debarre

• SHIMURA CURVES WITHIN THE LOCUS OF HYPERELLIPTIC JACOBIANS IN GENUS THREE -> example of family of period matrices giving family of curves https://www.uni-frankfurt.de/50566282/Generic_50566282.pdf

Coherent sheaf cohomology

Look at theorem's 4.4 and 4.5 in Altman-Kleiman's book on Grothendieck Duality for useful results of computations for sheaf cohomology

Dualizing complexes

• Create an article on Dualizing complex
• Use the appendix in https://arxiv.org/abs/0807.3283 as a reference
• Also, https://mathoverflow.net/questions/335211/definition-of-dualizing-complex is useful

Atyiah class

• Excellent overview here: https://arxiv.org/abs/1107.5076 : gives geometric motivation/understanding for Atiyah class: it comes from studying simultaneous deformations $V\to X$ for a complex vector bundle on a complex manifold $X$.
• https://www.ams.org/journals/tran/1957-085-01/S0002-9947-1957-0086359-5/S0002-9947-1957-0086359-5.pdf
• http://www.cs.umsl.edu/~rao/papersdir/chern.pdf
• Cotangent Complexe
• Simple vector bundles have trivial endomorphism sheaves, hence there is a nice comparison between the Atiyah class and chern class – https://mathoverflow.net/questions/47885/trace-of-the-atiyah-class-equals-chern-class
• https://projecteuclid.org/euclid.nmj/1118798682 (simple bundle examples with criterion)
• https://mathoverflow.net/questions/47885/trace-of-the-atiyah-class-equals-chern-class/115613#115613 (Apparently there is a relation between atiyah and chern classes using hodge to de rham spectral sequence)
• Relation between chern classes and atiyah classes given here: https://www.mat.uniroma2.it/~fbracci/download/Atiyahclass.pdf

Intersection forms

There should be a page discussing the intersection forms of manifolds and varieties. In addition, it should reference the Todd index theorem as a tool for computing the intersection forms using the decomposition of integral binary forms.

• K3 Surfaces – Hirzebruch discusses this computation
• Milnor Lattice (Dimca's book on singularities) https://link.springer.com/chapter/10.1007/BFb0090648
• Non-smoothable manifolds – mention how the intersection forms give examples of manifolds which are not smoothable
• Classification of manifolds – mention relation between the classification of manifolds and their cohomology with the cup product
• https://wstein.org/edu/Fall2003/252/lectures/ -> Modular symbols

• On the intersection matrix of a hypersurface: http://www.numdam.org/item/ASENS_1969_4_2_4_583_0/

Brieskorn lattice

Maybe add this to the Gauss-Manin page...

• THE DIFFERENTIAL STRUCTURE OF THE BRIESKORN LATTICE MATHIAS SCHULZE – https://www.mathematik.uni-kl.de/~mschulze/download/dsbl

Algebraic Number theory

• Look at chapter 2 in http://people.math.gatech.edu/~mbaker/pdf/ANTBook.pdf for computational tools
• Also, in the p-adic section, look at the field extensions and Hensel's lemma. This is material for the locally compact field page and Galois group pages

Locally compact field

• Unramified field extensions
• Totally ramified field extensions
• https://www.math.arizona.edu/~cais/Prelim/LocalFieldExt.pdf contains examples of both
• Algebraic number fields in this context
• pages 64,66,67 in Koblitz
• should be included with Galois groups page too https://web.archive.org/web/20200415182659/https://facstaff.elon.edu/cawtrey/ae-dihedral.pdf

Dessins d'enfants

• https://arxiv.org/abs/1910.12651 – elementary intro, also discusses Riemann's existence theorem
• https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/bachelor/2017-2018/vandersteen-scriptieherzien.pdf is an elementary discussion

• https://arxiv.org/pdf/1309.1968.pdf for a more complete discussion
• Also should use this page as a reference for a fiber functor

Riemann's existence theorem

There really should be a page on Riemann's existence theorem. Here are some references

• https://www.math.upenn.edu/~harbater/RETppr.pdf
• https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/bachelor/2017-2018/vandersteen-scriptieherzien.pdf
• https://www.math.uci.edu/~mfried/booklist-ret/chpret4-firsthalf.pdf

Galois group

• include splitting fields as a definition
• Give examples from the splitting fields yielding symmetric group galois extensions, including theorem 32
• Dihedral groups
• Quaternion group as a galois group
• https://core.ac.uk/download/pdf/81182361.pdf has many techniques, these are also discussed in Dummit Foote
• Galois groups of local fields. Check out section 4.3 of http://diposit.ub.edu/dspace/bitstream/2445/122676/2/memoria.pdf and Chapter IV §1 and §2 in Serre's book on local fields
• Include examples from geometry, such as covering spaces, and use their function fields. Check out theorem 9.8 and corollary 8.10 inside http://faculty.ycp.edu/~fbutler/MastersThesis.pdf and the references therein.
• https://math.stackexchange.com/questions/257880/field-extension-with-dihedral-galois-group

Galois groups of polynomials

• Milne's section on galois groups of polynomials
• Dummit + Foote

• https://math.stackexchange.com/questions/609869/how-to-determine-the-galois-group-of-irreducible-polynomials-of-degree-3-4-5
• https://math.stackexchange.com/questions/45893/how-to-find-the-galois-group-of-a-polynomial
• https://math.stackexchange.com/questions/1766701/computing-the-degree-of-the-splitting-field-of-x318x3-over-bbb-q?noredirect=1&lq=1

Weil conjectures

• https://amslaurea.unibo.it/7385/1/tamiozzo_matteo_tesi.pdf has computations of the zeta function for elliptic curves. This could be done for other curves as well, such as genus 2 curves.
• https://mathoverflow.net/questions/70605/from-zeta-functions-to-curves/70606#70606

Arithmetic of K3 surfaces

https://math.rice.edu/~av15/Files/AWS2015Notes.pdf

Stable vector bundles

• This discusses classifications of vector bundles and stable vector bundles https://www.maths.tcd.ie/pub/ims/bull60/R6005.pdf
• Le Potier's book gives a theorem that the category of vector bundles with a fixed slope forms an abelian category

• https://userpage.fu-berlin.de/hoskins/Stratifications_course.html
• https://arxiv.org/abs/math/0310185 , http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.535.5191&rep=rep1&type=pdf

Voevodsky Motives

• https://userpage.fu-berlin.de/hoskins/motive_bun.pdf

Tautological Ring of Kontsevich Spaces

• http://math.ucsd.edu/~doprea/flags.pdf

Moduli of curves

Notes: https://deopurkar.github.io/teaching/moduli/

https://mathoverflow.net/questions/76585/moduli-space-of-genus-2-curves

There should be examples of the moduli of curves page. This could include the stacks ${\overline {\mathcal {M}}}_{1,1}$ genus 2 from Mumford's paper, and genus up-to 6. This paper has a great summary:

https://arxiv.org/abs/1307.6614

https://arxiv.org/abs/1904.08081

Mumford's paper: http://www.dam.brown.edu/people/mumford/alg_geom/papers/1983b--EnumGeomModuli-NC.pdf

https://www.math.brown.edu/~bhassett/papers/genus2/logmodel3.pdf

Riemann-Hurwitz theory

This is closely related to the moduli of curves. Here are some resources

https://deopurkar.github.io/research/papers/thesis.pdf

Hilbert polynomial

Add examples of hilbert polynomial for hypersurfaces. Reference is Kollar Rational curves on algebraic varieties. In addition, mention RR and HRR as tools for computing the hilbert polynomial.

Etale topology

There should be discussions about the local rings, strict henselization, and unramified extensions. Also, there should be discussions about geometric interpretations of Etale topology, Henselian traits, and what the points in the topology sees. The example given here https://math.stackexchange.com/questions/2321214/grothendiecks-vanishing-cycles is excellent!

Embedded points

There should be a page discussing embedded points and cohen-macaulay schemes. Reference: https://stacks.math.columbia.edu/tag/05AJ

Examples

Consider the scheme

${\text{Spec}}\left({\frac {\mathbb {Z} [x,y]}{(x^{2},xy)}}\right)$

which is the $y$ axis with an embedded point at the origin. Then, this gives a non-example of a Cohen-Macaulay scheme.

Stability Conditions

Add examples an stuff from

• https://web.archive.org/web/20200303013703/http://www.math.utah.edu/~goller/S16_SAGS/Franco_StabilityConditions.pdf
• https://web.archive.org/web/20200303013834/https://web.northeastern.edu/iloseu/Xiaolei_S16.pdf
• https://faculty.math.illinois.edu/~dowdall/talk_resources--GEAR2012/Franco/Franco--Higgs_bundle_elliptic_curves_slides--GEAR2012.pdf

Algebraic geometry pages

Checkout the exercises in https://amor.cms.hu-berlin.de/~soldatea/alggeom_V4A2_SS16.html https://amor.cms.hu-berlin.de/~soldatea/V4A2/

Algebraic curves stuff...

https://link.springer.com/book/10.1007/978-3-540-69392-5 (stable reduction exercises are awesome!)

Formal schemes

• Add examples of deformations of curves
• Add perfection example, https://arxiv.org/pdf/1810.11853.pdf

Log geometry

There should be a page on log schemes and log geometry. Checkout Log structure for links to pages not yet created.

Euler sequence

Add in relative Euler sequence for projective bundles

https://amor.cms.hu-berlin.de/~soldatea/V4A2/AGUebungII3.pdf

Deformations of curves

There should be a page dedicated to the deformations of curves. This could include discussions of Kodaira-Spencer theory and applications, pointed curves, maps of pointed curves in Kontsevich moduli spaces.

Hilbert Schemes

• https://arxiv.org/abs/1512.07363 is a great reference with discussions about the virtual tangent sheaf. Material from this paper and its references could be used in other pages as well, such as enumerative geometry, equivariant k-theory, virtual fundamental classes, and others.

• Also, https://web.archive.org/web/20200304195241/http://www.mat.uc.cl/archivos/geometry-of-hilbert-and-quot-schemes-of-points-on-smooth-curves-and-smooth-surfaces.pdf has more examples of VFC technology for hilbert schemes of points on a surface
• https://escholarship.org/uc/item/70r7943w

Azumaya algebras

Checkout this link

• https://mathoverflow.net/questions/319993/explicit-examples-of-azumaya-algebras

and construct examples of azumaya algebras. As a corollary, the quot scheme will give some moduli space of modules of this azumaya algebra.

Kontsevich moduli spaces

There should be page discussing the Kontsevich moduli spaces of curves. Some references are

• https://arxiv.org/abs/alg-geom/9608011
• http://www.math.utah.edu/~yplee/teaching/gw/Koch.pdf
• http://www.math.snu.ac.kr/~kiem/KontMod-IJM.pdf

Quintic threefold

Let $X$ be a quintic threefold defined by a degree 5 homogeneous polynomial $Q(x_{0},\ldots ,x_{4})$, a section of ${\mathcal {O}}_{\mathbb {P} ^{4}}(5)$. Using the map

$\phi :{\overline {\mathcal {M}}}_{0,1}(\mathbb {P} ^{4},d)\to \mathbb {P} ^{4}$

$\phi (C;x_{1};f)=f(x_{1})$

$\pi :{\overline {\mathcal {M}}}_{0,1}(\mathbb {P} ^{4},d)\to {\overline {\mathcal {M}}}_{0,0}(\mathbb {P} ^{4},d)$

fiber of $\pi$ at a point $[C]\in {\overline {\mathcal {M}}}_{0,0}(\mathbb {P} ^{4},d)$ is the rational curve $C$.

$\phi |_{C}:C\to \mathbb {P} ^{4}$

we can pullback $\phi ^{*}{\mathcal {O}}_{\mathbb {P} ^{4}}(5)\cong {\mathcal {O}}_{C}(5d)$ and the push-forward is $\Gamma (C,\phi ^{*}{\mathcal {O}}_{\mathbb {P} ^{4}}(5))$. This glues to a vector bundle ${\mathcal {E}}_{d}$ of rank $5d+1$ on ${\overline {\mathcal {M}}}_{0,0}(\mathbb {P} ^{4},d)$. There is an associated section ${\tilde {Q}}\in \Gamma ({\mathcal {E}}_{d})$ whose vanishing locus is the orbifold ${\overline {\mathcal {M}}}_{0,0}(X,d)$.

A infinity algebras

• http://www.claymath.org/library/monographs/cmim04.pdf has a great discussion about the abstract structure
• https://lada.math.ncsu.edu/FinDimAInfEx-final.pdf has a good example
• https://arxiv.org/abs/math/9910179
• Fukaya structure?
• https://arxiv.org/abs/math/0604379
• Massey triple product gives $A_{\infty }$-structure on $H_{\bullet }(A)$ for a dga $A$
• https://arxiv.org/abs/math/0606144 – $A_{\infty }$-structure on ext algebras – look at McCleary ~ pg 300 and nLab for more references

Fourier–Mukai transforms

• https://arxiv.org/abs/1109.3083
• Fourier-Mukai transforms for surfaces and moduli spaces of stable sheaves – https://era.ed.ac.uk/handle/1842/12070

Perverse sheaves

This page is in need of an upgrade. It should include results such as the decomposition theorem and examples of perverse sheaves. http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf has a ton of useful info for this, also https://web.math.princeton.edu/~smorel/faisceaux_pervers.pdf

Grothendieck–Riemann–Roch

There should be some example computations here. This could include some basic examples, like computations related to HRR, GRR on curves https://math.stanford.edu/~vakil/245/245class18.pdf, and mumford's results about tautological classes. Also, the extension to equivariant theories would be nice https://arxiv.org/abs/1205.4742.

Homotopy groups

There should be ample discussion about applications of the long exact sequence in homotopy theory. This should include the simply connectedness of lie groups, such as $SU(n)$, discussion of bundles on $S^{4}$ classified by $\pi _{3}(SU(2))$. Husemollers fiber bundles book contains useful info about this too. It would be nice if the exotic spheres milnor constructed were accessible through wikipedia articles.

Symplectic groups

Symplectic groups have a nice decomposition into a few matrix subgroups which are multiplied together. Checkout the books

• Introduction to Symplectic Dirac Operators
• Folland: Harmonic analysis in phase space (which gives the proofs)

Symplectic structures

There should also be also be a discussion about what the standard symplectic structure "does" on $\mathbb {R} ^{n}$ using inner products. Again "Introduction to Symplectic Dirac Operators" has a nice discussion :)

Algebraic curves

Mumford gives a complete list of ways to find algebraic curves. This should be included somewhere to give beginners a look at how to construct any genus of algebraic curve and where to look for more advanced examples.

Postnikov tower

The whitehead tower should be constructed, explained, and applications with spectral sequences given. This should have similar applications to computing homotopy groups.

Also, there should be these constructions for spectra as well.

Eilenberg–Maclane spaces

This page should have computations of the rational cohomology ring and the partial computations of the integral cohomology ring. The book "Homotopical Topology" has an *excellent* overview of how to accomplish this feat

Atyiah–Singer index theorem

This page should have some computations on it! There is an excellent reference giving some easily accessible formulas – https://www.maths.ed.ac.uk/~v1ranick/papers/gilkey3.pdf in particular, it could be computed for smooth complex projective hypersurfaces.

• Spectral triples: examples and index theory – Carie Phillips Reney – https://pdfs.semanticscholar.org/c22b/95d337533c5e276329ed068520fcfad581f5.pdf
• http://www.its.caltech.edu/~matilde/FiniteSp3.pdf – finite spectral triples
• Differential operators on NC torus – http://pub.math.leidenuniv.nl/~aricif2/Documents/AriciMasterThesis.pdf – elementary and starts from scratch

Chern–Weil theory

Lectures on Chern–Weil Theory and Witten Deformations by Weiping Zhang has a lot of great results for Chern-Weil theory. He gives an overview of Bott localization formula as an application

Also, he discusses 3-manifolds which apparently all have trivial tangent bundle and the Chern-Simons functional

Dupont fiber bundles – contains calculation for CP^n – https://data.math.au.dk/publications/ln/2003/imf-ln-2003-69.pdf

Flat vector bundle

This page should be edited to include the case of flat vector bundles over $S^{1}$ whose monodromy is determined by a map $\mathbb {Z} \to GL_{n}(\mathbb {F} )$, this could also include flat principal bundles, so $\mathbb {Z} \to G$. This article has a good description https://arxiv.org/pdf/1501.00730.pdf

Holomorphic vector bundles

The constructions of vector bundles in https://arxiv.org/pdf/1501.00730.pdf should be discussed, including discussions about theta functions as sections of line bundles on elliptic curves.

• https://arxiv.org/pdf/1009.3230.pdf

• https://math.berkeley.edu/~nadler/atiyah.classification.pdf
• http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf

Mirror Symmetry

https://math.berkeley.edu/~auroux/papers/cp2mirror.pdf