Irakasgai has ingelesez irakatsiko da soilik. In this course we study (geometric) varieties that are described as the zero sets of polynomial equations. We start by understading the relation between varieties and polynomial rings using techniques in commutative algebra. Additionally, we set background in affine and projective geometry over algebraically closed fields (most of the time, over the field of complex numbers) and then, we study the Zariski topology, irreducible varieties, Bezout theorem for intersection of curves in the projective plane and the additive structure of irreducible cubic curves.



We recommend to have taken the following courses: Algebraic Structures (2nd year), Commutative Algebra (3rd year) and Algebraic Equations (3rd year), which are devoted to developing the fundamentals of abstract algebra and its main applications. Also, we recommned to have at least a B2 level in English to follow the course.

SPECIFIC COMPETENCES:



M11CM09 - To know the ring of fractions of Z and of ring of polynomials over a field (unity, prime ideals, maximal ideals, etc). To understand the relation between the ring of fractions and its quotients.

M11CM10 - To be able to apply the structure theorem of Artinian rings to the quotients of polynomial rings with two variables (with coefficients in a field) by ideals generated by two coprime polynomials.

M11CM11 - To be able to apply Hilbert's Nullstellensatz theorem to study the existence of solutions of a system of equations over an algebraically closed field.

M11CM12 - To be able to compute the index of intersection point of two planar curves.

M11CM13 - To be able to apply Bezout's theorem to study the planar curves: inflection points, parametrization of curves, etc. To be able to sum points in an irreducible cubic curve.



LEARNING RESULTS:



- Compute the unities, prime and maximal ideals, etc. of certain fractions of Z and of polynomial rings with coefficients in a field.

- To be able to apply the zeroes theorem of Hilbert to study the existence of solutions of a system of equations with coefficients in an algebraically closed field.

- To be able to compute the index of intersection point of two planar curves.

- To be able to apply Bezout's theorem to study the planar curves: inflection points, parametrization of curves, etc. To be able to sum points in an irreducible cubic curve.



Further key words in learning results: Noetherian rings, Zariski topology, algebraic varieties, coordinate rings, tangent space, multiple points, singularities.

1. RINGS OF FRACTIONS: Definition and main properties. Localization of a ring in a prime ideal. Ideals in rings of fractions.

2. NOETHERIAN RINGS: Definition properties and examples.

3. HILBERT NULLSTELLENSATZ: Integral extensions of rings, Zariski theorem. Maximal ideals of a polynomial ring over an algebraically closed field. Hilbert nullstellensatz.

4. PLANE CURVES: Tangents. Multiple points. Intersection index of two curves ina point.

5. APPLICATIONS: Bezout theorem and applications: Pascal and Pappus and the aditive group structure of a cubic irreducible curve. Resolution of singularities. Quadratic and cubic surfaces

The theoretical contents will be presented in master classes following the basic references in the bibliography. These lectures will be complemented with problem classes (classroom practice), in which students will apply the knowledge acquired in the theoretical lectures in order to solve the problems. In the seminar sessions, exercises and representative examples will be considered. Such exercises will be given to the students in advance so that they will be bale to work on them to work out the solutions by themselves. Students must participate actively in the seminar sessions and they will be encouraged to discuss about the solutions.

• Ebaluazio Jarraituaren Sistema
• Azken Ebaluazioaren Sistema
• Kalifikazioko tresnak eta ehunekoak:
  • Garatu beharreko proba idatzia (%): 60
  • Praktikak egitea (ariketak, kasuak edo buruketak) (%): 25
  • Lanen, irakurketen... aurkezpena (%): 15

To pass the subject it will be enough to follow and to carry out correctly the activities in class including seminars and projects. If the student decides to go to the final exam, the final mark will be the maximum between the mark of the final exam and the weighted average under there following formula:



Final Mark= max{0,2x(Mark of the Partial Exam)+0,4x(Mark Oral Exposition, Projects and Problems)

+0,8x(Mark of the Final Exam -5), Mark of the Final Exam}

The interest and willingness of the student during the course will also be taken into account.

The final mark will be that which is obtained in the written exam corresponding to this call.

Oinarrizko bibliografia

BIBLIOGRAFIA

M. ATIYAH, I.G. MACDONALD. Introducción al Algebra Conmutativa, Ed. Reverté, 1973.

D. COX, J. LITTLE, D. O'SHEA. Using Algebraic Geometry, Springer, 1998.

W. FULTON. Curvas Algebraicas, Reverté, 1971.

F. KIRWAN. Complex Algebraic Curves, Cambridge Univ. Press, 1992.

E. KUNZ. Introduction to Commutative Algebra and Algebraic Geometry, Birkhaüser, 1985.

C. MUSILI. Algebraic Geometry for Beginners, Hindustan Book Agency, 2001.

M. REID. Undergraduate Algebraic Geometry, Cambridge University Press, 1988.