Module MATH3321: ALGEBRAIC GEOMETRY III

Department: Mathematical Sciences

MATH3321: ALGEBRAIC GEOMETRY III

Prerequisites

• Complex Analysis II (MATH 2011) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Actuarial Mathematics II (MATH2171) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Symmetries II (MATH2111) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Hyperbolic Geometry II (MATH2121) ) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061); OR Contours and Probability II (MATH2561) ) AND Linear Algebra II (MATH2021) AND Algebra and Number Theory II (MATH2061).

Corequisites

• None.

Excluded Combination of Modules

• Algebraic Geometry IV (MATH4011).

Aims

• To introduce the basic theory of algebraic varieties and birational geometry, with particular emphasis on plane curves.

Content

• Affine and projective plane curves over a field k.
• Conics, Pappus' Theorem.
• Unique factorisation in polynomial rings.
• Study's lemma, irreducibility.
• Singular points, tangents, points of inflection.
• Dual plane, linear systems and curves.
• Bezout's theorem: Resultants, weak form of Bezout, applications of Pascal's theorem, Cayley-Bacharach theorem, group law on a cubic. Intersection multiplicity, strong form of Bezout.
• Bezout's theorem: applications, flexes, Hessian, configuration of flexes of a cubic.
• Elliptic curves, Weierstrass normal form.
• Complex curves as real surfaces.
• Basic topology and manifolds.
• Degree-genus formula.
• Resolution of singularities and non-singular models.

Learning Outcomes

• By the end of the module students will: be able to solve complex, unpredictable and specialised problems in Algebraic Geometry.
• have an understanding of specialised and complex theoretical mathematics in the field of Algebraic Geometry.
• have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas:
• Affine and projective plane curves.
• Unique factorisation in polynomial rings.
• Linear systems.
• Bezout's theorem and applications.
• Topology of curves over the complex numbers.

• Students will have highly specialised and advanced mathematical skills which will be used with minimal guidance in the following areas: spatial awareness, abstract reasoning.

• Students will have enhanced problem solving skills.

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

• Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
• Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
• In addition, formatively assessed assignments provide feedback for students and the lecturer on student progress and opportunities for the lecturer to test and enhance development of modelling and computation skills.
• Summative examination assesses acquired knowledge, problem-solving skills and arrange of modelling and computational skills.

Teaching Methods and Learning Hours

Summative Assessment

Formative Assessment:

Four written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University