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?
This will contain references to pathological objects which occur in nature and not by manual hacking.
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.
Mention how gives infinitely many bases for , hence we need to consider level structures to get finite etale coverings of moduli spaces
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
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.
Maybe add this to the Gauss-Manin page...
There really should be a page on Riemann's existence theorem. Here are some references
https://math.rice.edu/~av15/Files/AWS2015Notes.pdf
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 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
This is closely related to the moduli of curves. Here are some resources
https://deopurkar.github.io/research/papers/thesis.pdf
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.
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!
There should be a page discussing embedded points and cohen-macaulay schemes. Reference: https://stacks.math.columbia.edu/tag/05AJ
Consider the scheme
which is the axis with an embedded point at the origin. Then, this gives a non-example of a Cohen-Macaulay scheme.
Add examples an stuff from
Checkout the exercises in https://amor.cms.hu-berlin.de/~soldatea/alggeom_V4A2_SS16.html https://amor.cms.hu-berlin.de/~soldatea/V4A2/
https://link.springer.com/book/10.1007/978-3-540-69392-5 (stable reduction exercises are awesome!)
There should be a page on log schemes and log geometry. Checkout Log structure for links to pages not yet created.
Add in relative Euler sequence for projective bundles
https://amor.cms.hu-berlin.de/~soldatea/V4A2/AGUebungII3.pdf
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.
Checkout this link
and construct examples of azumaya algebras. As a corollary, the quot scheme will give some moduli space of modules of this azumaya algebra.
There should be page discussing the Kontsevich moduli spaces of curves. Some references are
Let be a quintic threefold defined by a degree 5 homogeneous polynomial , a section of . Using the map
fiber of at a point is the rational curve .
we can pullback and the push-forward is . This glues to a vector bundle of rank on . There is an associated section whose vanishing locus is the orbifold .
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
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.
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 , discussion of bundles on classified by . 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 have a nice decomposition into a few matrix subgroups which are multiplied together. Checkout the books
There should also be also be a discussion about what the standard symplectic structure "does" on using inner products. Again "Introduction to Symplectic Dirac Operators" has a nice discussion :)
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.
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.
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
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.
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
This page should be edited to include the case of flat vector bundles over whose monodromy is determined by a map , this could also include flat principal bundles, so . This article has a good description https://arxiv.org/pdf/1501.00730.pdf
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.
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?
This will contain references to pathological objects which occur in nature and not by manual hacking.
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.
Mention how gives infinitely many bases for , hence we need to consider level structures to get finite etale coverings of moduli spaces
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
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.
Maybe add this to the Gauss-Manin page...
There really should be a page on Riemann's existence theorem. Here are some references
https://math.rice.edu/~av15/Files/AWS2015Notes.pdf
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 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
This is closely related to the moduli of curves. Here are some resources
https://deopurkar.github.io/research/papers/thesis.pdf
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.
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!
There should be a page discussing embedded points and cohen-macaulay schemes. Reference: https://stacks.math.columbia.edu/tag/05AJ
Consider the scheme
which is the axis with an embedded point at the origin. Then, this gives a non-example of a Cohen-Macaulay scheme.
Add examples an stuff from
Checkout the exercises in https://amor.cms.hu-berlin.de/~soldatea/alggeom_V4A2_SS16.html https://amor.cms.hu-berlin.de/~soldatea/V4A2/
https://link.springer.com/book/10.1007/978-3-540-69392-5 (stable reduction exercises are awesome!)
There should be a page on log schemes and log geometry. Checkout Log structure for links to pages not yet created.
Add in relative Euler sequence for projective bundles
https://amor.cms.hu-berlin.de/~soldatea/V4A2/AGUebungII3.pdf
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.
Checkout this link
and construct examples of azumaya algebras. As a corollary, the quot scheme will give some moduli space of modules of this azumaya algebra.
There should be page discussing the Kontsevich moduli spaces of curves. Some references are
Let be a quintic threefold defined by a degree 5 homogeneous polynomial , a section of . Using the map
fiber of at a point is the rational curve .
we can pullback and the push-forward is . This glues to a vector bundle of rank on . There is an associated section whose vanishing locus is the orbifold .
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
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.
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 , discussion of bundles on classified by . 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 have a nice decomposition into a few matrix subgroups which are multiplied together. Checkout the books
There should also be also be a discussion about what the standard symplectic structure "does" on using inner products. Again "Introduction to Symplectic Dirac Operators" has a nice discussion :)
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.
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.
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
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.
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
This page should be edited to include the case of flat vector bundles over whose monodromy is determined by a map , this could also include flat principal bundles, so . This article has a good description https://arxiv.org/pdf/1501.00730.pdf
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.