return to Fall 2024 MATH 5330 webpage
Required textbook:
Other possible books:
Ideals, Varieties, and Algorithms: An Introduction to
Computational Algebraic Geometry and Commutative
Algebra, Undergraduate Texts in Mathematics, by
David A. Cox, John Little, Donal O’Shea, Springer, 4th ed.
(denoted [CLO] in this course).
A list of known errata for this book is posted here.
A list of other books that might be helpful for learning this material.
- You should make sure you are viewing the most current version of this page and not the version in your browser’s cache; reload the page from source or clear the cache and reload.
- Homework will be collected every couple of weeks.
- There might be more than one correct answer for any given question.
- Dates indicate homework assigned in lecture on that date. Dates for future assignments are tentative and subject to change.
- If you notice some questions are in a nonincreasing order from the book, then it means that the order is recommended by your instructor and is usually chosen to match the order in which the material was presented in class.
- Any optional exercises are for those students who wish to explore some of the ideas in more depth, and are usually beyond the scope of the material covered in class.
- Skimming through the main ideas in a section shortly before that section is covered in class should help you understand the lecture – try it!
- The homework from Fall 2020 can be viewed here; in particular, it will give you a rough outline of what material will be presented and how many lectures will be spent on each item. (Note: in that semester, the class met for two lectures on one day per week.)
- LAST REVISION: 11/29/24.
| Aug 20 | Check your Canvas notifications to check you can receive Canvas announcements. Attendance will be noted, starting today. Read course syllabus carefully. Make a note of the test dates in your calendar. Review course website and repeat frequently during the semester. Review the course’s Canvas portal. Read this study tip and read https://www.jeffreybennett.com/pdf/How_to_Succeed_general.pdf for ideas on how to study most effectively. Read this file on writing proofs. Read your lecture notes (meaning the notes you should have taken during lecture) and read Section 1.1 & do Sec 1.1: 2, 3(d), 5. Due Sept 5 at the start of lecture. |
| Aug 22 | Read your lecture notes & Sec 1.2 & do Sec 1.2: 1, 3, 4(a)-(d), 5, H1, 6, 8, 15(a)-(c). Be careful to explain your answer to the parts of the questions that ask if the dimension matches what your intuition says it should be — this is the part I often have to return to students for them to redo and it hurts their grade!!! Due Sept 5 at the start of lecture. |
| Aug 27 | Read your lecture notes and & pgs 14-20 & do Sec 1.3: 1-4. Due Sept 5 at the start of lecture. |
| Aug 29 | Read your lecture notes and & Sec 1.4 & do Sec 1.4: 1, 2, 3 (with ℚ replaced by 𝕜 & assume ℕ⊂ 𝕜 ), 5, 7, 8, 15(a)(b), H2. Due Sept 5 at the start of lecture. |
| Sep 03 | Read your lecture notes and do H3; skim Sec 1.5 and do Sec 1.5: 1, 5. Read the statement of Theorem 4 on page 77. Do Sec 2.5: 17, 18, 11, H4, H5, 16. Due Sept 19 at the start of lecture (extended to Sept 24). |
| Sep 05 | Read your lecture notes and do H6-H9. Due Sept 19 at the start of lecture (extended to Sept 24). |
| Sep 10 | Read your lecture notes and pages 80-81 and do Sec 2.5:13, 14. Due Sept 19 at the start of lecture (extended to Sept 24). |
| Sep 12 | Read your lecture notes & pg 209.7(Defn 7)-210 & pgs 189-190 and do H10 & Sec 4.3: 3 (hint: read Definition 12 through Proposition 13 of Sec 4.3). Due Sept 19 at the start of lecture (extended to Sept 24). |
| Sep 17 | Read your lecture notes & do H11. Due Oct 3 at the start of lecture. |
| Sep 19 | Read your lecture notes & pgs 190-192 and do H12-H14. Due Oct 3 at the start of lecture. |
| Sep 24 | Read your lecture notes and do H15 and read Definition 3, Theorem 4 & Theorem 6 of Sec 4.6 and do Sec 4.6: 9. Due Oct 3 at the start of lecture. |
| Sep 26 | Read your lecture notes and do H16-H19 and Sec 4.8: 4. Note that this latter exercise is on pg 231 and appears to have an erroneous section number at the start of those exercises. Due Oct 17 at the start of lecture. |
| Oct 01 | Read your lecture notes and do Sec 4.4: 1 (this question can be done in an informal manner). Due Oct 17 at the start of lecture. |
| Oct 03 | Read your lecture notes and Sec 5.1 & Sec 5.2 (but note that Definition 8(iii) is not correct on pg 243 – what is wrong with it?). Do Sec 5.1: 1, 2, 5, 6 & Sec 5.2: 6, 9(use FIT), 10(a), 18. Due Oct 17 at the start of lecture. |
| Oct 08 | Read your lecture notes. |
| Oct 10 | Read your lecture notes and do H20(a)-(f) and read Sec 5.4 and do Sec 5.4: 4 (assume V ⊆ 𝕜³), 5 (assume f, g ∊ 𝕜[x, y, z]) & 14 (14 is optional for any undergraduate students). For 14(a), ignore the instruction to use material developed earlier; instead use the fact that the field is ℝ so that 5th roots of real numbers exist. For 14(c): use y5 = x2 in ℝ[V] to show that g5 = f2 in ℝ[t]; then deduce f = h5, g = h2 for some h ∊ ℝ[t]; & then use (b) and the uniqueness part of (a) to prove that y = Φ(h)2 = (a + bx)2, where a, b ∊ ℝ[y], yields a contradiction. Due Nov 7 at the start of lecture. |
| Oct 15 | Read your lecture notes and Sec 5.5 and do H21-H22. Due Nov 7 at the start of lecture. |
| Oct 17 | Read your lecture notes. |
| Oct 22 | Read your lecture notes and Sec 8.1 and do H23. Due Nov 7 at the start of lecture. |
| Oct 24 | Read your lecture notes and Sec 8.2 and do H24-H25 and Sec 8.2: 11, 12, 21(a). Due Nov 7 at the start of lecture. |
| Oct 29 | Read your lecture notes and do Sec 8.2: 7 (note that Proposition 5 in Sec 1.1 should be useful in (b)). The last exercise completes the part of the proof of Proposition 4 in Section 8.3 that was not proved completely in lecture. Due Nov 21 at the start of lecture. |
| Oct 31 | Read your lecture notes and do H26. Due Nov 21 at the start of lecture. |
| Nov 5 | Read your lecture notes and do Sec 8.4: 9 (hint: use the fact that the projective closure of W is the smallest projective variety that contains W) Due Nov 21 at the start of lecture. |
| Nov 7 | Read your lecture notes and do H27-H28. Due Nov 21 at the start of lecture. |
| Nov 12 | Read your lecture notes. |
| Nov 14 | Read your lecture notes and do H29-H30. Due Nov 21 at the start of lecture. |
| Nov 19 | Read your lecture notes and pages 451-454. |
| Nov 21 | Read your lecture notes. |
| Nov 26 | Read your lecture notes. |
| Nov 28 | THANKSGIVING HOLIDAY — see the academic schedule at https://www.uta.edu/academics/academic-calendar |
| Dec 03 | Read your lecture notes. Please remember to complete online the student feedback survey by 11:59 pm on Dec 3 — check your mymav e-mail for the link. I appreciate the feedback (e.g., use of website, the use of Canvas, the choice of homework questions from the book and/or the questions not from the book, the examples provided in class, feedback from grading, corniness of jokes …??). Thank you! I will have my usual office hours through today inclusive, and I will have an additional office hour as noted below. |
| Dec 04 | 5:00-6:30 PM: office hour in PKH 462. |
The assignments from Fall 2020 can be viewed in their entirety here .