CONTENTS -- Lecture LI: Quadratic Equations -- 1: Word Problems -- 2: Sets and Maps -- 3: Groups and Fields -- 4: Rings and Ideals -- 5: Modules and Vector Spaces -- 6: Polynomials and Rational Functions -- 7: Euclidean Domains and Principal Ideal Domains -- 8: Root Fields and Splitting Fields -- 9: Advice to the Reader -- 10: Definitions and Remarks -- 11: Examples and Exercises -- 12: Notes -- 13: Concluding Note -- Lecture L2: Curves and Surfaces -- 1: Multivariable Word Problems -- 2: Power Series and Meromorphic Series -- 3: Valuations -- 4: Advice to the Reader -- 5: Zorn's Lemma and Well Ordering -- 6: Utilitarian Summary -- 7: Definitions and Exercises -- 8: Notes -- 9: Concluding Note -- Lecture L3: Tangents and Polars -- 1: Simple Groups -- 2: Quadrics -- 3: Hypersurfaces -- 4: Homogeneous Coordinates -- 5: Singularities -- 6: Hensel's Lemma and Newton's Theorem -- 7: Integral Dependence -- 8: Unique Factorization Domains -- 9: Remarks -- 10: Advice to the Reader -- 11: Hensel and Weierstrass -- 12: Definitions and Exercises -- 13: Notes -- 14: Concluding Note -- Lecture L4: Varieties and Models -- 1: Resultants and Discriminants -- 2: Varieties -- 3: Noetherian Rings -- 4: Advice to the Reader -- 5: Ideals and Modules -- 6: Primary Decomposition -- 6.1: Primary Decomposition for Modules -- 7: Localization -- 7.1: Localization at a Prime Ideal -- 8: Affine Varieties -- 8.1: Spectral Affine Space -- 8.2: Modelic Spec and Modelic Affine Space -- 8.3: Simple Points and Regular Local Rings -- 9: Models -- 9.1: Modelic Proj and Modelic Projective Space -- 9.2: Modelic Blowup -- 9.3: Blowup of Singularities -- 10: Examples and Exercises -- 11: Problems -- 12: Remarks -- 13: Definitions and Exercises -- 14: Notes -- 15: Concluding Note -- Lecture L5: Projective Varieties -- 1: Direct Sums of Modules.

2: Graded Rings and Homogeneous Ideals -- 3: Ideal Theory in Graded Rings -- 4: Advice to the Reader -- 5: More about Ideals and Modules -- (Ql) Nilpotents and Zerodivisors in Noetherian Rings -- (Q2) Faithful Modules and Noetherian Conditions -- (Q3) Jacobson Radical Zariski Ring and Nakayama Lemma -- (Q4) Krull Intersection Theorem and Artin-Rees Lemma -- (Q5) Nagata's Principle of Idealization -- (Q6) Cohen's and Eakin's Noetherian Theorems -- (Q7) Principal Ideal Theorems -- (Q8) Relative Independence and Analytic Independence -- (Q9) Going Up and Going Down Theorems -- (Q10) Normalization Theorem and Regular Polynomials -- (Qll) Nilradical Jacobson Spectrum and Jacobson Ring -- (Q12) Catenarian Rings and Dimension Formula -- (Q13) Associated Graded Rings and Leading Ideals -- (Q14) Completely Normal Domains -- (Q15) Regular Sequences and Cohen-Macaulay Rings -- (Q16) Complete Intersections and Gorenstein Rings -- (Q17) Projective Resolutions of Finite Modules -- (Q18) Direct Sums of Algebras Reduced Rings and PIRs -- (Q18.1) Orthogonal Idempotents and Ideals in a Direct Sum -- (Q18.2) Localizations of Direct Sums -- (Q18.3) Comaximal Ideals and Ideal Theoretic Direct Sums -- (Q18.4) SPIRs = Special Principal Ideal Rings -- (Q19) Invertible Ideals Conditions for Normality and DVRs -- (Q20) Dedekind Domains and Chinese Remainder Theorem -- (Q21) Real Ranks of Valuations and Segment Completions -- (Q22) Specializations and Compositions of Valuations -- (Q23) UFD Property of Regular Local Domains -- (Q24) Graded Modules and Hilbert Polynomials -- (Q25) Hilbert Polynomial of a Hypersurfaces -- (Q26) Homogeneous Submodules of Graded Modules -- (Q27) Homogeneous Normalization -- (Q28) Alternating Sum of Lengths -- (Q29) Linear Disjointness and Intersection of Varieties.

(Q30) Syzygies and Homogeneous Resolutions -- (Q31) Projective Modules Over Polynomial Rings -- (Q32) Separable Extensions and Primitive Elements -- (Q33) Restricted Domains and Projective Normalization -- (Q34) Basic Projective Algebraic Geometry -- (Q34.1) Projective Spectrum -- (Q34.2) Homogeneous Localization -- (Q34.3) Varieties in Projective Space -- (Q34.4) Projective Decomposition of Ideals and Varieties -- (Q34.5) Modelic and Spectral Projective Spaces -- (Q34.6) Relation between AfRne and Projective Varieties -- (Q35) Simplifying Singularities by Blowups -- (Q35.1) Hypersurface Singularities -- (Q35.2) Blowing-up Primary Ideals -- (Q35.3) Residual Properties and Coefficient Sets -- (Q35.4) Geometrically Blowing-up Simple Centers -- (Q35.5) Algebraically Blowing-up Simple Centers -- (Q35.6) Dominating Modelic Blowup -- (Q35.7) Normal Crossings Equimultiple Locus and Resolved Ideals -- (Q35.8) Quadratic and Monoidal Transformations -- (Q35.9) Regular Local Rings -- 6: Definitions and Exercises -- 7: Notes -- 8: Concluding Note -- Lecture L6: Pause and Refresh -- 1: Summary of Lecture LI on Quadratic Equations -- 2: Summary of Lecture L2 on Curves and Surfaces -- 3: Summary of Lecture L3 on Tangents and Polars -- 4: Summary of Lecture L4 on Varieties and Models -- 5: Summary of Lecture L5 on Projective Varieties -- 6: Definitions and Exercises -- BIBLIOGRAPHY -- DETAILED CONTENT -- NOTATION-SYMBOLS -- NOTATION-WORDS -- INDEX.