Math 413/813 — Introduction to Algebraic Geometry

Minimum Requirements

• Each student is expected to learn a different theorem
• The written document must introduce, correctly state, and prove the theorem. It should also include at least one interesting example illustrating the theorem. The article should be as self-contained as possible. The new document must be typed, be at most eight pages in length (with 1 inch margins and 12 point font) and be available in PDF format.
• The verbal presentation must introduce and state the theorem. It should also include at least one interesting example illustrating the theorem.

Assessment and Deadlines: Project grades will be computed as follows:

Comments: By design this assignment is very open-ended. You are strongly encouraged to compute many examples. You are also encouraged to formulate, test, and prove your own conjectures.

Here is some advice on how to present mathematics.

• Paul Halmos on Writing.
• Steve Kleiman on Writing.
• Paul Halmos on Presenting.

Potential Topics

• Algebraic Statistics [St2] §5 (Theorem 5.3); [St3] §8 (Theorem 8.14)
• Alexander Duality [MS] §5 (Theorem 5.24); [HH] §8.1 (Theorem 8.1.6) — Nicole Pereira
• Automatic Theorem Proving [IVA] §6.4 (Proposition 6.5.8) — Ziqi Wang
• Belyi's theorem: [LZ] §2.1, 2.6 (Theorem 2.1.1) — Keenan McPhail
• Bernstein's theorem: [UAG] §7.5 (Theorem 7.5.4); [St3] §3 (Theorem 3.2) — Luke Steverango
• Bezout's Theorem — Landon McDougall
• Conics — A poor man's elliptic curve : [L]
• Combinatorial Nullstellensatz: [TV] §9.1 (Theorem 9.2) — Gage Thornewell
• Computation in local rings: [UAG] §4 (Theorem 4.2.2, Theorem 4.4.2); [GP] §6 (Theorem 6.2.6)
• Counting degree d rational curves in P²:
• Elliptic Curve Cryptography : [K] — Chelsea Crocker
• Fröberg's theorem: [HH] §9.2 (Theorem 9.2.3)
• Generic initial ideals: [E] §15.9 (Theorem 15.18); [HH] §4 (Theorem 4.1.2)
• Grassmannians: [H] §11 (Proposition 11.30) —Ilia Korchagin
• Hilbert syzygy theorem: [UAG] §6 (Theorem 6.2.1) [E] §15.5 (Theorem 15.10) — Didi Zhang
• Holonomy and metrics :
• Integer programming: [UAG] §8.1-2 (Theorem 8.1.11); [St2] §5 (Theorem 5.5) — Danny Oh
• Integral points on Conics
• Introduction to Gröbner bases: [IVA] §2; [AL] §1 — Paul Wilson
• Koszul complexes: [E] §17.1 (Theorem 17.1); [GP] §7 (Theorem 7.6.14) — Jason Kattan
• Linear partial differential equations: [St3] §10 (Theorem 10.3) — Ankai Liu
• Mordell-Weil Theorem:
• Noether normalization: [E] §13.1 (Theorem 13.3) — Nour Moustafa-Fahmy
• Puiseux series: [E] §13.3 (Corollary 13.15); [St3] §1.4 (Theorem 1.7)
• Quillen-Suslin theorem: [UAG] §5.1 (Theorem 5.1.8); [BG] §8 (Theorem 8.5) — Erin Crawley
• Quotients by finite groups: [IVA] §7 (Theorem 7.3.5); [St1] — Daniel Cloutier
• Resolutions of monomial ideals: [MS] §3.5 (Theorem 3.17); [HH] §7 (Theorem 7.1.1)
• Resultants: [UAG] §3 (Theorem 3.2.3); [St3] §4 (Theorem 4.6) — Samuel DeCoste
• SAGBI basis: [St2] §11 (Theorem 11.4); [MS] §14.3 (Theorem 14.11)
• Stickelberger's theorem: [UAG] §2 (Theorem 2.4.5); [St3] §2.3 (Theorem 2.6) — Katrina Parsche
• Straightening laws: [St1] §3 (Theorem 3.1.7, Theorem 3.2.1)
• Sums of squares: [St3] §7 (Theorem 7.3) — Sean Monahan
• Tropical hypersurfaces: [St3] § 9 (Theorem 9.17) — Sonja Ruzic
• Triangulations and toric ideals: [St2] §8 (Theorem 8.3); [BG] §7 (Theorem 7.18)
• Universal Gröbner bases: [St2] §1 (Theorem 1.4), §7 (Theorem 7.1)

References

[AL]  

William Adams and Philippe Loustaunau, An introduction to Gröbner bases

[BG]  

Winfried Bruns and Joseph Gubeladze, Polytopes, rings, and K-theory, Monographs in Mathematics. Springer, 2009 ISBN 978-0-387-76355-2

[IVA]  

David A. Cox, John B. Little, and Don O'Shea, Ideals, Varieties, and Algorithms, third edition, Springer, 2007 ISBN 978-0-387-35650-1

[UAG]

David A. Cox, John B. Little, and Don O'Shea, Using Algebraic Geometry, GTM 185, Springer, 2005 ISBN 978-0-387-20733-9

[E]    

David Eisenbud, Commutative algebra with a view towards algebraic geometry, GTM 150. Springer, 1995 ISBN 0-387-94268-8

[GP]  

Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, 2nd edition, Springer, 2008 ISBN 978-3-540-73541-0

[H]    

Brendan Hassett, Introduction to Algebraic Geometry, Cambridge University Press, 2007 ISBN 978-0-521-69141-3

[HH]  

Jürgen Herzog and Takayuki Hibi, Monomial ideals, GTM 260. Springer, 2011 ISBN 978-0-85729-105-9

[K]  

Neal Koblitz, Elliptic Curve Cryptosystems, Math. Comp. 48 (1987), no. 177, 203–209

[L]  

Franz Lemmermeyer, Conics — A poor man's elliptic curve

[LZ]  

Sergei Lando and Alexander Zvonkin, Graphs on surfaces and their applications, Springer 2004 ISBN 978-3-540-38361-1

[MS]  

Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, GTM 227. Springer, 2005 ISBN 0-387-22356-8

[St1]  

Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation. Springer, 1993 ISBN 3-211-82445-6

[St2]  

Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8. American Mathematical Society, 1996 ISBN 0-8218-0487-1

[St3]  

Bernd Sturmfels, Solving systems of polynomial equations, CBMS 97, American Mathematical Society, 2002 ISBN 0-8218-3251-4

[TV]  

Terence Tao and Van H. Vu, Additive combinatorics, Cambridge University Press, 2010 ISBN 978-0-521-13656-3