# Matematica | HIGHER GEOMETRY

## Matematica HIGHER GEOMETRY

 0512300022 DIPARTIMENTO DI MATEMATICA MATHEMATICS 2013/2014

 YEAR OF COURSE 3 YEAR OF DIDACTIC SYSTEM 2010 SECONDO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/03 6 48 LESSONS OPTIONAL SUBJECTS
 LUCA VITAGLIANO T
Objectives
ACQUIRING KNOWLEDGE AND UNDERSTANDING: THE COURSE AIMS AT 1) SUPPLYING STUDENTS WITH SOME BASIC KNOWLEDGE ON CLASSICAL ALGEBRAIC GEOMETRY AND SOME PRELIMINARY KNOWLEDGE OF MODERN ALGEBRAIC GEOMETRY, AND 2) TEACHING A USEFUL LANGUAGE TO ARIENT IN THE HUGE LITERATURE ABOUT THE COURSE'S SUBJECT.
Prerequisites
THE REQUIRED PREREQUISITE IS A BASIC KNOWLEDGE OF: VECTOR SPACES, AFFINE SPACES, FIELDS, FIELD EXTENSIONS, RINGS, IDEALS, MODULES OVER A RING. THE KNOLWLEDGE OF BASIC GENERAL TOPOLOGY, WHOSE NECESSARY NOTIONS WILL BE INTRODUCED IN DEDICATED LECTURES.
Contents
PROGRAM:
[1] AFFINE ALGEBRAIC SETS AND QUASI-AFFINE VARIETIES.
[2] PROJECTIVE SPACES.
[3] PROJECTIVE ALGEBRAIC SETS AND QUASI-PROJECTIVE VARIETIES.
[4] DIMENSION OF VARIETIES.
[6] MORPHISMS OF VARIETIES.
[7] RINGS OF FUNCTIONS ON VARIETIES.
[8] CATEGORIES AND FUNCTORS.
[9] PRODUCT OF VARIETIES.
[10] RATIONAL MAPS.
[11] SHEAVES.
[12] SCHEMES.
Teaching Methods
FRONT LECTURES
Verification of learning
THE FINAL TEST ON THE STUDENT'S LEARNING CONSISTS IN:
I) A TRADITIONAL ORAL EXAMINATION ON THE FIRST PART OF THE PROGRAM (ITEMS [1]-[9]), WITH GENERAL QUESTIONS AND EXERCISES AIMING AT TESTING
A) THE KNOWLEDGE OF BASIC NOTIONS OF CLASSICAL ALGEBRAIC GEOMETRY, ESPECIALLY REGARDING THE ALGEBRA/GEOMETRY DUALITY, ON ONE SIDE, AND
B) THE AUTONOMY IN HANDLING SIMPLE PROBLEMS ABOUT THE ALGEBRA AND GEOMETRY OF ALGEBRAIC VARIETIES ON THE OTHER SIDE.
CONDITIONS A) AND B) ARE NECESSARY FOR PASSING THE EXAM.
II) A SHORT PRESENTATION BY THE STUDENT ON A TOPIC TO STUDENT'S CHOICE, AMONG THE MODERN ALGEBRAIC GEOMETRY ONES PROPOSED IN THE SECOND PART OF THE PROGRAM (ITEMS [10]-[11]), AIMING AT TESTING THE EXPOSITION ABILITY AND THE LINGUISTIC RIGOUR ON THE SUBJECT OF THE COURSE.
Texts
ROBIN HARTSHORNE, ALGEBRAIC GEOMETRY (GRADUATE TEXTS IN MATHEMATICS), SPRINGER
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