MA 205 0 Algebraic Geometry homewrok
Fall 2018-2019

Homework.

For Monday September 17 (and discussion Friday September 14):
read Hartshorne Chapter 1 Sections 1.1-1.3 or equivalent in other texts.
Read Hartshorne Chapter 2 Section 1 or equivalent.
Discuss, write and submit: Hartshorne I.1 exercises 1.1, 1.2, 1.3, 1.4, 1.6, 1.8. Try your hand at 1.11.

For Monday September 24 (and discussion Friday September 21):
Re-read Hartshorne Chapter 1 Sections 1.1-1.3 or equivalent in other texts.
Read Hartshorne Chapter 2 Section 1-2 or equivalent.
Discuss, write and submit: Hartshorne I.2 exercises 2.1, 2.12, 2.15.
Hartshorne II.1 exercises 1.1, 1.2, 1.3, 1.15

For Monday October 15 (Friday October 5):
Read Hartshorne Chapter 1 Sections 1.3-1.4 or equivalent in other texts.
Read Hartshorne Chapter 2 Section 2-3 or Liu 2.1-2.3 or equivalent.
Discuss, write and submit:
Hartshorne I.3 exercises 3.1, 3.2,3.6, 3.21
Liu 2.1 exercises 1.1, 1.3, 1.8
Liu 2.3 exercise 2.3

Consider $A = \mathbb{Z}, B = \mathbb{Z}[x]/(x^2-5)$ and $C = \mathbb{Z}[t]/(t^2-t-1)$.

  • Show that $x \mapsto 2t-1$ gives rise to a homomnorphism $B \to C$.
  • Describe the schemes $\text{Spec}\ A$, $\text{Spec}\ B$ and $\text{Spec}\ C$.
  • Describe the morphisms $\text{Spec}\ C \to \text{Spec}\ A$, $\text{Spec}\ B \to \text{Spec}\ A $ and $\text{Spec}\ C \to \text{Spec}\ B$.

    • For Monday October 22 (discussion Friday October 19):
    Hartshorne II.2 exercises 2.3, 2.4, 2.5
    Hartshorne II.3 exercises 3.1, 3.2, 3.3
    Hartshorne I.4 exercises 4.1, 4.3, 4.9

    For Monday November 5 (discussion Friday November 2):
    Read Hartshorne Chapter 1 Sections 1.5-1.6 or equivalent in other texts.
    Read Hartshorne Chapter 2 Section 4-5.
    Discuss, write and submit:
    Hartshorne II.4 exercises 4.1, 4.2, 4.3, 4.6, 4.8
    Hartshorne I.5 exercises 5.1, 5.2, 5.4

    For Friday November 16 (discussion Friday November 9):
    Read Hartshorne Chapter 1 Sections 1.5-1.6 or equivalent in other texts.
    Read Hartshorne Chapter 2 Section 4-5.
    Read Liu Chapter 5 Section 5.1
    Discuss, write and submit:
    Liu Section 5.1 exercises 5.1.1-5.1.4
    Hartshorne I.5 exercise 5.6
    Hartshorne I.6 exercises 6.1, 6.2

    For Discussion Friday November 30 on:
    Read again Liu Chapter 5 Section 5.1
    Read Hartshorne Chapter II, Section 5-6, Chapter I section 7,
    Discuss and solve Hartshorne Chapter I, 7.1, 7.2 and 7.7
    also chapter I 1.9, 2.11, 3.5, 3.17

    For Discussion Friday December 7 on:
    Read Hartshorne Chapter II, Section 5-8,
    Discuss and solve problems to be announced.