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Complements of Geometry

Code: M3004     Acronym: M3004     Level: 300

Classification Keyword
OFICIAL Mathematics

Instance: 2016/2017 - 1S

Active? Yes
Responsible unit: Department of Mathematics
Course/CS Responsible: Bachelor in Mathematics

Cycles of Study/Courses

Acronym No. of Students Study Plan Curricular Years Credits UCN Credits ECTS Contact hours Total Time
L:B 0 Official Study Plan 3 - 6 56 162
L:M 5 Official Study Plan 2 - 6 56 162
L:Q 0 study plan from 2016/17 3 - 6 56 162

Teaching language



Study of affine and  projective geometry over a field and  an introduction to plane algebraic curves.  Essencial use will be made of methods from theory of group actions, linear algera, bilinear algebra and quadratic forms, and elementary results from ring theory,  polynomial algebras and field extensions.



Learning outcomes and competences

Good knoweldge of the concepts and results (including proofs) lectured in classes, and capacity to solve geometric problems of affine geometry, projective geometry and  algebraic plane curves, using methods from group actions, linear algebra, bilinear algebra and quadratic forms and polynomial algebras.

Working method


Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Required knowlege:  one year course in Linear Algebra ( (the units Linear Algebra I and II, Analysis in one and several  real variables (the unit real Analysis I or Infinitesimal Calculus I),  basic knowledge of group theory (the unit Algebra I or Group Theory)


A.Group Actions: equivalent definitions, invariant and fixed subspaces. induced actions: restriction to invariant subspaces, restricition to a subgroup, regular action in a space of functions with domain equal to the space of the base action.  Actions on a group by  a subgroup by translation and by conjugation. Orbits, quocient space of a group action. Isotropy group of a point, conjugacy property of isotropy groups.  faithful and k-transitive actions. Equation of the classes. Burnside theorem. Semi-direct product of  groups. G-spaces, morphisms of G-spaces (equivariant maps).  Homogeneous spaces  Semi-direct product of groups.

B.  Complements of Linear Algebra over a field: sums and direct sums of subspaces of a vector space, transversal subspaces,  direct sums of  vector spaces. Projections. Quotient of vector space by a subspace, dual space, transpose of a linear map, and bidual spaces. isomoprhims of biduality. Anihilator of a subspace, dual of a subspace and of a quotient space,   General Group, GL(E),  and Special Linear Group, SL(E), of a vector space E. Subgroup H(E) of homotheties. GL(E) as semi-direct product of SL(E) by H(E). Diltations. Generation of H(E) by dilatations. Transvections. Generation of  of SL(E) by transvections. Any two transvections are conjugate in GL(E): Center of GL(E) i(.e. H(E).)
Conjugation in GL(E) of any two transvections.  Center of  GL(E)  is  H(E).
  Natural action of GL(E)  on E,    E\{0}, on the space of ordered bases of E,  and on the lattice of subspaces of  E. Isomorphism  between  GL(E/F) and the subgroup  GL_F(E) of elelemnts of GL(E) that leav invariant subspace  F deixam invariante um subespaço vectorial  F. Bijection between the grassmannian  G(k, E) (conjunto dos subespaços and the  ehomogeneous space  GL(E)/GL_F(E).  Natural  bijection  between  G(k,E) e  G(dim(E)-k, E^*).

C. Linear Affine Geometry: Affine space X with direction space a vector space E as a regular homogeneous spcae (a torsor). The translation subgroup T(X). Affine morphisms. Affine isomorphisms. Affine frames, cartesian coordinates.  The Affine groups GA(X) and SA(X). . The affine group GA(X)   as a semi-direct product of GL(E) with the T(X).    Baricienters   and baricentric coordinates.  Affine subspaces. Intersection and join operations of  affine subspaces.  Parallelism and transversality of subspaces. Affine projectiions,  affine symmetries. Affine isomorphisms that  fix an affine hyperplane: affine dilataions, and affine transvections. Generation of GA(X) by the affine dilatations. Generation of SA(X) by the affine ttransvections. Classic theorems of affine geometry:  Thales, Desargues,  Pappus, Ceva e Menelaus. Fundamental theorem of affine geometry.

D. Linear Projective Geometry: Projective subspaces.  (Partial) projective morphisms. Homographies (projective isomorphisms). Projective frames. Homogeneous coordinates.  Affine atlas.  Projective subspaces.   Lattice of projective subspaces. Intersection and join operations. Tranversal (suplementary) subspaces.

Projective groups  PGL(E) and PSL(E). Homologies and elations.
Generation of PGL(E) by homologies and elations. 

Projection, section and perspectivity with center in a projective subespace. Projective Line; cross ratio, harmonic sets, projective group,  involutions, perspectives, global parametrizations.

Classic theorems of projective geometry: Desarges, Pappus. Fundamental theorem of projective geometry. 

 Projective duality: Projective space H(P(E)) of the hyperplanes of P(E) and  isomorphism with P(E^*)  where E^* is  dual space of E, colineations, correlations and polarities. Principle of projective duality.

Projective completion of affine spaces and morphisms. Affine structure in the complement of a projective  hyperplane . Restriction of a homology to the affine hyperplane associated to the homology. Restriction of an elation to the affine hyperplane assocaited to the homology. 

E. Complementary results on quadratic forms

Vector space of bilinear forms on a finite dimensional vector space. Gram matrix relative to a basis.  Characteristic of a bilinear form. Non degenerate bilinear form
 Refexive, symmetric and antisymetric bilinear forms. Quadratic form associated to a symmetric bilinear form. Polarization formula. Correspondence between symmetric bilinear forms and quadratic forms on a finite dimensional vector space over a field of characteristic not equal to 2. 
Orthogonal vectors, orthogonal of a subset. Existence of orthogonal basis-
Kernel (radical) of a quadratic form.  Congruent quadratic forms. Classification of quadratic forms over the real field, finite fields and algebraically closed fields.
Sylvester theorem of inertia index of a real quadratic form.

F. Projective space of quadrics (K is a field of characteristic different from 2)
 Vector space of space of quadratic forms. Projective spce of quadrics. Projective quadric- Proper quadric.  Reduction of a quadric to a proper quadri in the complementar subspace of the kernel. Equation of a quadric in a projective frame. Projective variety of points of a projective quadric. .Quadrics of a projective line.  de uma recta projectiva. Projective subspaces tangent to a quadric.
Polarities and conjugation of proper quadric.
Tangent hyperplanes.
Theorems of Pascal and Brianchon.



Mandatory literature

Carlos Menezes; Apontamentos de Complementos de Geometria

Complementary Bibliography

Samuel Pierre; Projective geometry. ISBN: 0-387-96752-4
Berger Marcel; Geometry. ISBN: 3-540-11658-3 (Vol. I)
Berger, Marcel; Geometrie vol 1 a 6, Nathan, 1977

Teaching methods and learning activities

Expository lectures of the theorical concepts with full proofs. Exercises for application and further development of concepts

Evaluation Type

Evaluation with final exam

Assessment Components

designation Weight (%)
Exame 100,00
Participação presencial 0,00
Total: 100,00

Calculation formula of final grade

The final classification is the mark obtained in the final exam.
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