Knowledge and understanding. At the end of the course the student is required to prove that he/she knows the fundamental results of the polynomial ring in several variables with real and complex coefficients, and of plane affine and projective algebraic curves. Apply knowledge and understanding. At the end of the course the student must know how to apply the acquired knowledge to solve problems and exercises. The exercises can also be proposed as elementary theoretical results. In particular the student should be able to handle elementary properties of algebraic plane curves. Making judgment. At the end of the course the student will know how to recognize and how to apply the acquired techniques, and will also recognize the situations and problems in which these techniques can be advantageously used. Communication skills. At the end of the course the student will be able to express himself/herself appropriately on the above topics. Learning skills. At the end of the course the student will be able to consult standard books on algebraic curves and basic algebraic geometry.

Algebra 2, Geometria 2, Analisi 3

Polynomial rings in several variables. Plane algebraic curves, affine and projective. Resultant. Bézout theorem. Conics and rational curves. Cubics.

G. Fisher, Plane Algebraic Curves F. Kirwan, Complex algebraic curves, LMS (1992) R.J. Walker, Algebraic curves, Princeton (1950) D. Cox, J. Little, D. O'Shea, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Undergraduate Texts in Mathematics. Springer, Cham, 2015.

Basic properties of the polynomial rings in several variables with coefficients in a field. Hilbert's basis theorem. Basic facts of elimination theory and resultants. Factorization in polynomial rings. Plane affine algebraic curves: definition and first properties. Reviews on complex projective spaces. Complex projective plane algebraic curves. The theorem of Bézout. Points of inflection. Conics and rational curves. Classification of cubic curves.

Lessons and exercises sessions. To support the activities we will use the website Moodle.

There will be a moodle web-page of the course in which it will be reported: the diary of the lessons; the dates of the exams; additional material.

The final examination is designed to ascertain knowledge of the topics of the entire course programme. It consists of a written and an oral examination. In the written test, you will be asked to solve 4 exercises similar to those carried out in class and in the exercises, and assigned during the course. The mark for the written test will be expressed in thirtieths; the weight of each exercise carried out correctly is 7/8 points; the condition for admission to the oral test is to have obtained a mark greater than or equal to 18/30. A positive written test remains valid until the autumn exam session in September (included) of the same academic year. The written test can be repeated: handing in a written test substitutes the possible preceding written exam. The purpose of the oral test will be to test the students' theoretical knowledge of the discipline, their ability to express themselves and their language skills. It will focus on the understanding of the definitions and statements of the theorems discussed in class, and may include a discussion of the writing and demonstrations of some theorems. The oral examination will be marked in thirtieths. The oral test is passed if the mark is greater than or equal to 18/30. The final mark is awarded by means of a mark expressed in thirtieths calculated taking into account the written test and the oral test. The examination is passed with a mark of 18/30. To achieve the minimum mark (18/30), knowledge of the main definitions and results is required, together with the ability to solve elementary exercises with a fair command of the language of the subject. In order to obtain the maximum mark (30/30 with distinction), the student must demonstrate excellent knowledge of all the topics covered in the course, answer all the questions correctly and carry out all the exercises correctly. Students with disabilities, specific learning disabilities (DSA) or special educational needs (BES) are asked to contact the Disability Service or the University's DSA Service in advance in order to benefit from examination aids.

This course explores topics closely related to goal 4 of the United Nations 2030 Agenda for Sustainable Development (SDGs)