# 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.

Presentations Date Room
Schedule April 16/25 Jeff 422/319

Due Date Element Weight
2019-02-11 Outline 10%
2019-03-25 Rough Draft 10%
2019-04-01 Feedback on other's drafts 10%
2019-04-16/25 Presentation 30%
2019-04-16/25 Feedback on other's presentations 10%
2019-04-25 Final Paper 30%

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.

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