Sharedwww / hartnotes / hartnotes.tocOpen in CoCalc
\contentsline {section}{\numberline {1}Preface}{4}
\contentsline {section}{\numberline {2}Ample Invertible Sheaves}{4}
\contentsline {section}{\numberline {3}Introduction to Cohomology}{5}
\contentsline {section}{\numberline {4}Cohomology in Algebraic Geometry}{6}
\contentsline {section}{\numberline {5}Review of Derived Functors}{6}
\contentsline {subsection}{\numberline {5.1}Examples of Abelian Categories}{7}
\contentsline {subsection}{\numberline {5.2}Exactness}{7}
\contentsline {subsection}{\numberline {5.3}Injective and Projective Objects}{8}
\contentsline {section}{\numberline {6}Derived Functors and Homological Algebra}{8}
\contentsline {subsection}{\numberline {6.1}Construction of $R^{i}F$}{9}
\contentsline {subsection}{\numberline {6.2}Properties of Derived Functors}{9}
\contentsline {section}{\numberline {7}Long Exact Sequence of Cohomology and Other Wonders}{10}
\contentsline {section}{\numberline {8}Basic Properties of Cohomology}{10}
\contentsline {subsection}{\numberline {8.1}Cohomology of Schemes}{10}
\contentsline {subsection}{\numberline {8.2}Objective}{10}
\contentsline {section}{\numberline {9}Flasque Sheaves}{11}
\contentsline {section}{\numberline {10}Examples}{12}
\contentsline {section}{\numberline {11}First Vanishing Theorem}{13}
\contentsline {section}{\numberline {12}\v {C}ech{} Cohomology}{14}
\contentsline {subsection}{\numberline {12.1}Construction}{14}
\contentsline {subsection}{\numberline {12.2}Sheafify}{14}
\contentsline {section}{\numberline {13}\v {C}ech Cohomology and Derived Functor Cohomology}{16}
\contentsline {subsection}{\numberline {13.1}History of this Module $E$}{17}
\contentsline {section}{\numberline {14}Cohomology of $\mathbf {P}_k^n$}{18}
\contentsline {section}{\numberline {15}Serre's Finite Generation Theorem}{18}
\contentsline {subsection}{\numberline {15.1}Application: The Arithmetic Genus}{19}
\contentsline {section}{\numberline {16}Euler Characteristic}{19}
\contentsline {section}{\numberline {17}Correspondence between Analytic and Algebraic Cohomology}{21}
\contentsline {section}{\numberline {18}Arithmetic Genus}{22}
\contentsline {subsection}{\numberline {18.1}The Genus of Plane Curve of Degree $d$}{22}
\contentsline {section}{\numberline {19}Not Enough Projectives}{23}
\contentsline {section}{\numberline {20}Some Special Cases of Serre Duality}{24}
\contentsline {subsection}{\numberline {20.1}Example: $\mathcal {O}_X$ on Projective Space}{24}
\contentsline {subsection}{\numberline {20.2}Example: Coherent sheaf on Projective Space}{24}
\contentsline {subsection}{\numberline {20.3}Example: Serre Duality on $\mathbf {P}_k^n$}{25}
\contentsline {section}{\numberline {21}The Functor $\ext$}{25}
\contentsline {subsection}{\numberline {21.1}Sheaf $\ext$}{25}
\contentsline {subsection}{\numberline {21.2}Locally Free Sheaves}{26}
\contentsline {section}{\numberline {22}More Technical Results on $\ext$}{27}
\contentsline {section}{\numberline {23}Serre Duality}{28}
\contentsline {section}{\numberline {24}Serre Duality for Arbitrary Projective Schemes}{30}
\contentsline {section}{\numberline {25}Existence of the Dualizing Sheaf on a Projective Scheme}{32}
\contentsline {subsection}{\numberline {25.1}Relative Gamma and Twiddle}{33}
\contentsline {section}{\numberline {26}Generalized Grothendieck Duality Theory}{35}
\contentsline {section}{\numberline {27}}{35}
\contentsline {section}{\numberline {28}Review of Differentials}{38}
\contentsline {subsection}{\numberline {28.1}The Sheaf of Differentials on a Scheme}{40}
\contentsline {section}{\numberline {29}Differentials on $\mathbf {P}^n$}{40}
\contentsline {section}{\numberline {30}Sheaf of Differentials and Canonical Divisor}{42}
\contentsline {section}{\numberline {31}Definitions}{45}
\contentsline {section}{\numberline {32}Genus}{45}
\contentsline {section}{\numberline {33}Riemann-Roch Theorem}{46}
\contentsline {section}{\numberline {34}Serre Duality}{47}
\contentsline {section}{\numberline {35}A Bird's Eye View of Curves}{48}
\contentsline {section}{\numberline {36}Moduli Space}{48}
\contentsline {section}{\numberline {37}Embeddings in Projective Space}{48}
\contentsline {section}{\numberline {38}Elementary Curve Theory}{48}
\contentsline {subsection}{\numberline {38.1}Definitions}{48}
\contentsline {subsection}{\numberline {38.2}Maps to Projective Space}{49}
\contentsline {section}{\numberline {39}Low Genus Projective Embeddings}{49}
\contentsline {subsection}{\numberline {39.1}Genus $0$ curves}{49}
\contentsline {subsection}{\numberline {39.2}Genus $1$ curves}{49}
\contentsline {subsection}{\numberline {39.3}Moduli Space}{50}
\contentsline {section}{\numberline {40}Curves of Genus 3}{51}
\contentsline {section}{\numberline {41}Curves of Genus 4}{53}
\contentsline {subsection}{\numberline {41.1}Aside: existence of $g^1_d$'s in general}{54}
\contentsline {subsection}{\numberline {41.2}Classifying curves of genus $4$}{54}
\contentsline {section}{\numberline {42}Curves of Genus 5}{55}
\contentsline {subsection}{\numberline {42.1}The space of quadrics containing an embedded genus 5 curve.}{56}
\contentsline {subsection}{\numberline {42.2}Genus 5 curves with very ample canonical divisor}{56}
\contentsline {section}{\numberline {43}Homework Assignment}{58}
\contentsline {section}{\numberline {44}Curves of genus 6}{58}
\contentsline {section}{\numberline {45}Oral Report Topics}{60}
\contentsline {section}{\numberline {46}Curves of general genus}{61}
\contentsline {section}{\numberline {47}Halphen's Theorem}{63}
\contentsline {section}{\numberline {48}Hurwitz's Theorem}{64}
\contentsline {section}{\numberline {49}Elliptic Curves}{66}
\contentsline {section}{\numberline {50}Automorphisms of Elliptic Curves}{68}
\contentsline {section}{\numberline {51}Moduli Spaces}{71}
\contentsline {section}{\numberline {52}The Jacobian Variety}{72}
\contentsline {subsection}{\numberline {52.1}Consequence of existence of the Jacobian}{74}
\contentsline {section}{\numberline {53}The Jacobian}{74}
\contentsline {subsection}{\numberline {53.1}Consequences of existence}{75}
\contentsline {subsubsection}{\numberline {53.1.1}Group structure}{75}
\contentsline {subsubsection}{\numberline {53.1.2}Natural fibration, dimension}{75}
\contentsline {subsubsection}{\numberline {53.1.3}The Zariski tangent space}{76}
\contentsline {section}{\numberline {54}Flatness}{77}
\contentsline {subsection}{\numberline {54.1}Technical definitions}{77}
\contentsline {subsubsection}{\numberline {54.1.1}General nonsense}{77}
\contentsline {subsection}{\numberline {54.2}Examples}{77}
\contentsline {subsection}{\numberline {54.3}Algebraic geometry definitions}{79}
\contentsline {subsection}{\numberline {54.4}Families}{79}
\contentsline {section}{\numberline {55}Theorem about flat families}{80}
\contentsline {section}{\numberline {56}Examples of Flat Families}{80}
\contentsline {section}{\numberline {57}Homework problems}{82}
\contentsline {subsection}{\numberline {57.1}Exercise on basic cohomology and abstract nonsense}{82}
\contentsline {subsection}{\numberline {57.2}Chapter III, 4.8, 4.9, 5.6}{83}
\contentsline {subsubsection}{\numberline {57.2.1}Exercise III.4.8: Cohomological Dimension}{83}
\contentsline {subsubsection}{\numberline {57.2.2}Exercise III.4.9}{86}
\contentsline {subsubsection}{\numberline {57.2.3}Exercise III.5.6: Curves on a nonsingular quadric surface}{86}
\contentsline {subsection}{\numberline {57.3}IV, 3.6, 3.13, 5.4, Extra Problems}{91}
\contentsline {subsubsection}{\numberline {57.3.1}Exercise IV.3.6: Curves of Degree $4$}{91}
\contentsline {subsubsection}{\numberline {57.3.2}Exercise IV.3.12}{93}
\contentsline {subsubsection}{\numberline {57.3.3}Exercise IV.5.4}{95}
\contentsline {subsubsection}{\numberline {57.3.4}Extra Problem 3, by William Stein}{96}
\contentsline {subsubsection}{\numberline {57.3.5}Extra Problem 4, by Nghi Nguyen}{96}