Program

on projections of surfaces to P^2 - preliminary program.

1. Curves in P^3, and projecting them to P^2. Generic projections give nodal curves in

P^2 (with no other singularities). Projecting further to P^1.

2. Surfaces in P^5, and projecting them to P^3. Standard singularities of surfaces in P^3:

double curves, triple points, Whitney pinch points.

Illustration: Whitney umbrella drawn in 'surf'.

3. Projecting smooth surfaces from P^3 to P^2: projective geometry.

3.1 "Visible contour" in P^3 and in P^2 (ramification curve and branch curve).

Ramification curve as the intersection of the surface and the polar surface.

References: Salmon.

3.2 Branch curve as the discriminant of a polinomial z^d + ... + a_d = 0

with coefficients in k[x,y,w].

3.3. Bisecant lines to the visible curve, and formation of nodes of branch curve.

3.4 (Projective geometry of) formation of cusps of branch curves. Example: (affine)

cubic surface z^3 + x z + y, and its projection to (x,y) - plane. Discriminant of cubic

polinomials. The visible curve as a twisted normal cubic, and its projections to P^2.

3.5 Segre-Severi variety of degree n plane curves with (given number of) nodes and cusps

V(n, k, delta).

3.6 Generic projection of a smooth degree d surface in P^3 gives a curve

of degree n = d(d-1) with k = d(d-1)(d-2) cusps and delta = d(d-1)(d-2)(d-3)/2 nodes.

The reculting correspondence

(smooth degree d surfaces in P^3) -> V( d(d-1), d(d-1)(d-2), d(d-1)(d-2)(d-3)/2 )

Example: Smooth cubic surfaces, and V(6,6,0).

References: Salmon.

3.7 Segre's theorem: if C is a branch curve of a smooth cubic surface,

then all 6 cusps of C belong to a conic (which is unusual). The notion of a special 0-cycle with

respect to a linear system on a surface; thus Segre gives an example of a special 0-cycle of

w.r.t. the linear system |2h|.

Geometry of the second polar, linear system on C given by cusps.

Equality (linear system given by cusps on C) = (plane embedding of C), and contemporary

interpretation of Segre's proof. Linear systems on cuspidal-nodal curves.

3.8 The second component of V(6,6,0): there are plane curves of degree 6 with 6 cusps,

which are not discriminants, (and their 6 cusps are not on a conic). Both

components of V(6,6,0) have the same dimension (!).

The expected dimension of V(n,k,delta), and the "speciality" (take a look at Hollcroft).

3.9 Zariski's interpretaion in terms of fundamental groups of compliment;

computation of the fundamental group for the both types of curves. Fundamental

group distingushes the components.

References: Castelnuovo, Segre, Zariski.

4. Projecting surfaces with double curves in P^3 to P^2: projective geometry.

4.1 Decomposition

(discriminant curve) = 2 (image of the double curve) + (geometric branch curve)

Possible degrees of the double curve, and geography in the (d, n) - plane,

where d is a degree of the surface, and n is the degree of geometric branch curve.

4.2 Example: classification of cubic surfaces in P^3 with standard singularities.

Cubic surface in P^3 with a double line, and the "deltoid" curve in V(4,3,0).

Computation of the braidmonodromy and the fundamental group of compliment of the deltoid curve, and

monodromy representation of the cover.

4.2b Deltoid curve as a cycloid curve, and its rational parametrisation. Cycloids and spyrograph.

Cycloid curves as points in V(n,d,k). Example: Astroid curve (cycloid of degree 6), and astroid surface.

4.3 Example: classification of degree 4 surfaces in P^3 with standard singularities. Five cases.

References: Griffiths and Harris.

5. Reconstructing a surface from monodromy representation: Grauert-Remmert theorem.

References: SGA1.

6. Chisini conjecture.

5.1 Chisini's counterexample. Ramified coverings of P^2 associated with

dual curves.

References: Catanese.

5.2 Statement of the Chisini's conjecture.