Matematica | GEOMETRY IV
Matematica GEOMETRY IV
cod. 0512300013
GEOMETRY IV
| 0512300013 | |
| DIPARTIMENTO DI MATEMATICA | |
| EQF6 | |
| MATHEMATICS | |
| 2016/2017 |
| OBBLIGATORIO | |
| YEAR OF COURSE 3 | |
| YEAR OF DIDACTIC SYSTEM 2010 | |
| PRIMO SEMESTRE |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/03 | 6 | 48 | LESSONS |
| Objectives | |
|---|---|
| THE COURSE'S MAIN AIM IS TO GIVE BASIC NOTIONS OF PROJECTIVE GEOMETRY AND ALGEBRAIC CURVES. - KNOWLEDGE AND UNDERSTANDING: THIS IS AN UNDERGRADUATE COURSE ON PROJECTIVE GEOMETRY AND PLANE ALGEBRAIC CURVES; THIS LATTER PART OF THE COURSE MUST BE VIEWED AS A GENTLE INTRODUCTION TO ALGEBRAIC GEOMETRY. - APPLYING KNOWLEDGE AND UNDERSTANDING: THE AIM IS TO ENABLE STUDENTS TO APPLY THE THEORETICAL NOTIONS AND COMPUTATIONAL TOOLS THEY WILL LEARN. TO THIS AIM, MANY LECTURES WILL BE DEVOTED TO PROBLEM SESSIONS. |
| Prerequisites | |
|---|---|
| THE ONLY NECESSARY PRELIMINARY KNOWLEDGES ARE THE BASIC ONES IN LINEAR ALGEBRA AND ABSTRACT ALGEBRA USUALLY GIVEN IN THE STANDARD UNDERGRADUATE COURSES. |
| Contents | |
|---|---|
| PART 1: COMPLEMENTS OF LINEAR ALGEBRA AND AFFINE GEOMETRY • JORDAN CANONICAL FORM • RUDIMENTS OF EXTERIOR ALGEBRA • AFFINE GEOMETRY OF REAL AND COMPLEX QUADRICS (SYMMETRY PROPERTIES, SINGULARITIES, CANONICAL FORMS) PART II: PROJECTIVE GEOMETRY • PROJECTIVE SPACES AND THEIR SUBSPACES; PROJECTIVE FRAMES AND HOMOGENEOUS COORDINATES; PROJECTIVE EXTENSION OF AN AFFINE SPACE; AFFINE CHARTS AND NONHOMOGENEOUS COORDINATES • PROJECTIVE MAPS; FUNDAMENTAL THEOREM ON PROJECTIVITIES; CROSS RATIO OF FOUR POINTS; CLASSIFICATION OF PROJECTIVITIES ON THE REAL LINE AND PLANE • REMARKABLE CONFIGURATIONS (DESARGUES AND PAPPUS THEOREMS); HARMONIC GROUPS AND DARBOUX THEOREM • DUALITY PRINCIPLE: DUAL PROJECTIVE SPACE, DUAL PROJECTIVITY, CROSS RATIO OF HYPERPLANES; RECIPROCITIES BETWEEN DUAL PROJECTIVE SPACES • PROJECTIVE GEOMETRY OF CONICS: CLASSIFICATION AND NORMAL FORMS; PROJECTIVE GENERATION OF A CONIC, STEINER'S THEOREM, POLARITY; PASCAL'S AND BRIANCHON'S THEOREMS AND THEIR DUALS. • PROJECTIVE CLASSIFICATION OF REAL AND COMPLEX QUADRICS; POLARITY ASSOCIATED WITH A QUADRIC; LINE COMPLEXES IN A QUADRIC • GRASSMANNIAN VARIETIES: PLUCKER COORDINATES AND DECOMPOSABILITY CONDITIONS; LINES IN P^3 AND KLEIN QUADRIC PART 3. ALGEBRAIC CURVES • COMPLEMENTS OF POLYNOMIAL ALGEBRA: RESULTANT OF TWO POLYNOMIALS AND ELIMINATION THEORY; DISCRIMINANT OF POLYNOMIAL EQUATION • PLANE ALGEBRAIC CURVES: AFFINE AND PROJECTIVE CURVES, IRREDUCUBLE CURVES, IRREDUCIBLE COMPONENTS OF A CURVE • SINGULARITIES OF PLANE ALGEBRAIC CURVES: LOCAL ANALYSIS AND CLASSIFICATION; BIRATIONSL MAPS; BLOWING-UP OF SINGULARITIES BY CREMONA TRANSFORMATIONS • INTERSECTIONS OF ALGEBRAIC CURVES: INTERSECTION MULTIPLICITY AND BEZOUT THEOREM • RATIONS CURVES: RATIONAL PARAMETRIZATION OF A CONIC; LUROTH'S THEOREM • PROJECTIVE GEOMETRY OF PLANE CUBICS: CLASSIFICATION AND NORMAL FORMS, FLEXES AND SALMON'S THEOREM, J INVARIANT OF A CUBIC; ABELIAN GROUP STRUCTURE OF A NON SINGULAR CUBIC • LINEAR SYSTEMS OF ALGEBRAIC CURVES: PENCILS AND NETS OF CONICS; BASE POINTS OF A LINEAR SYSTEMS, BERTINI'S THEOREM |
| Teaching Methods | |
|---|---|
| FRONTAL LECTURES |
| Verification of learning | |
|---|---|
| THE FINAL TEST CONSISTS IN: 1) AN ORAL TEST ON THE CONTENTS OF THE COURSE, WITH THE AIM OF CHECKING THE LEVEL OF UNDERSTANDING OF THE THEORETICAL PART OF THE COURSE; 2) GIVING THE CANDIDATE SOME SIMPLE EXERCISE TO SOLVE, IN ORDER TO TEST HIS/HER ABILITY TO PRACTICALLY APPLY THE THEORETICAL NOTIONS AND THE COMPUTATIONAL TOOLS. FURTHERMORE, DURING THE WHOLE COURSE THE STUDENTS WILL BE ASSIGNED AUTO-TESTING EXERCICES AS HOMEWORK, AND THEY WILL ALSO BE INVITED TO SOLVE, WITH THE HELP OF THE TEACHER, SOME SIMPLE PROBLEMS IN CLASS. |
| Texts | |
|---|---|
| TEXTBOOKS: • E. SERNESI, GEOMETRIA 1, BOLLATI BORINGHIERI 1989 • BELTRAMETTI ET AL., LEZIONI DI GEOMETRIA ANALITICA E PROIETTIVA, BOLLATI BORINGHIERI 2002 • BELTRAMETTI ET AL., LETTURE SU CURVE, SUPERFICIE E VARIETA' PROIETTIVE SPECIALI, BOLLATI BORINGHIERI 2002 (ALSO AVAILABLE IN ENGLISH: "LECTURES ON CURVES, SURFACES AND PROJECTIVE VARIETIES: A CLASSICAL VIEW OF ALGEBRAIC GEOMETRY", EMS 2009) FURTHER READING: • N. HITCHIN, PROJECTIVE GEOMETRY, LECTURE NOTES 2003 (ONLINE) • N. HITCHIN, ALGEBRAIC CURVES, LECTURE NOTES 2009 (ONLINE) • G. FISCHER, PLANE ALGEBRAIC CURVES, AMS 2001 • A. KUROSH, HIGHER ALGEBRA, MIR 1972 |
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