Introduction to Algebraic Geometry (Projects)

Cubic plane curve
projective variety
Clebsch surface
David Hilbert
Task
Learn a new theorem related to the course material and communicate it as a written document and via a video presentation.
Minimum
Requirements
  • Each student will focus on a different result.
  • The written document will introduce/motivate, correctly state, and prove a theorem. It will also include at least one interesting example, construction, or special case illustrating the theorem. The article will be as self-contained as possible. The new document must be typed, be at most eight pages in length (with one inch margins and a 12pt font), and be available in the PDF format.
  • The video presentation must introduce and state the theorem. It should also include at least one example, construction, or special case illustrating the theorem. This new video must be at most 20 minutes in length and available in a common format such as the MP4 file type.
Assessment
Project grades will be computed as follows:
Due Date Element Weight
2021–01–29 research 10%
2021–02–12 outline 10%
2021–03–05 rough draft 15%
2021–03–19 feedback 15%
2021–04–06 paper 25%
2021–04–16 video 25%
Advice
Paul R. Halmos and Steven L. Kleiman each provide some suggestions on how to write mathematics. Paul R. Halmos also offers some suggestions about how to talk mathematics.
Comments
By design, this project is very open-ended. Students are strongly encouraged to create their own examples. Consider what was the original motivation or historical context for your theorem. Does your theorem have any interesting specializations or important applications?
Potential
Topics
The following are natural candidates:
  • Alexander duality; see [5, Theorem 5.24] or [4, Subsection 1.5.3]
  • Automatic theorem proving; see [1, Proposition 6.4.8]
  • Computations in local rings; see [2, Proposition 2.11]
  • Conditional independence models; see [8, Proposition 8.1] — Harley Easton
  • Descartes rule of signs; see [6, Theorem 2.1] or [8, Theorem 1.5] — Emily Cullingworth
  • Fröberg theorem; see [4, Theorem 9.3.3]
  • Generic initial ideals; see [4, Theorem 4.1.2] or [3, Theorem 15.18]
  • Going-up theorem; see [3, Proposition 4.15] — Ben Syms-Wilson
  • Gröbner fan, see [2, Theorem 4.1]
  • Hilbert syzygy theorem; see [2, Theorem 6.2.1] or [3, Theorem 15.10] — Gabby Wolfe
  • Integer programming; see [2, Theorem 8.1.11] or [7, Theorem 5.5] — Junhee Park
  • Invariant theory; see [1, Theorem 7.3.5] — Curtis Wilson
  • Kushnirenko theorem; see [6, Theorem 3.2] — Konstantin Uvarov
  • Lexsegment ideals; see [4, Theorem 6.3.1] or [5, Theorem 2.22]
  • Linear partial differential equations; see [8, Theorem 10.3]
  • Multivariate polynomial splines; see [2, Proposition 3.7] — Hayden Pfeiffer
  • Newton polytopes; see [7, Lemma 2.2]
  • Noether normalization; see [3, Theorem 13.3] — Neil Utom
  • Puiseux series; see [3, Corollary 13.15] or [8, Theorem 1.7]
  • Quillen–Suslin theorem; see [2, Theorem 5.1.8]
  • Real subvarieties; see [8, Theorem 7.2]
  • Sagbi basis; see [7, Theorem 11.4]
  • Strickelberger theorem; see [2, Theorem 2.4.5] or [8, Theorem 4.6]
  • Symmetric polynomials; see [1, Theorem 7.1.3] — Sara Stephens
  • Triangulations and toric ideals; see [7, Theorem 8.3]
  • Universal Gröbner bases; see [7, Theorem 7.1] — Victor Bubis
References
  1. David A. Cox, John B. Little, and Donal O’Shea, Ideals, Varieties, and Algorithms, An Introduction to Computational Algebraic Geometry and Commutative Algebra, Fourth Edition, Springer, 2015.
  2. David A. Cox, John B. Little, and Donal O’Shea, Using Algebraic Geometry, Second edition, Graduate Texts in Mathematics 185, Springer, 2005
  3. David Eisenbud, Commutative algebra with a view towards algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995
  4. Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer, 2011
  5. Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics 227, Springer, 2005
  6. Frank Sottile, Real solutions to equations from geometry, University Lecture Series 57, American Mathematical Society, Providence, RI, 2011
  7. Bernd Sturmfels, Gröbner bases and convex polytopes, University Lecture Series 8. American Mathematical Society, Providence, RI, 1996
  8. Bernd Sturmfels, Solving systems of polynomial equations, CBMS Regional Conference Series in Mathematics 97, American Mathematical Society, Providence, RI, 2002