# Matematica | GEOMETRY IV

## Matematica 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 TYPE OF ACTIVITY MAT/03 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 FABRIZIO PUGLIESE T
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)