THE  REPORTS of  Symposium on Algebraic Geometry at Niigata 2004
                                                February 4 - 6, 2004

                                              ***** February 4 *****

Hiroyuki Sakai : 13:00-14:00
"Infinitesimal deformations of Galois covering space and its applications to Galois closure curves"
Here is the report.

Atsushi Ikeda 14:30-15:30
"On the Hodge structure of degenerating hypersurface in projective toric"
Here is the report.

Hiro-o Tokunaga: 16:00-17:00
@"Zariski k-plats for rational curve arrangement and dihedral covers"
Examples of Zariski $k$-plets of rational curve arrangements are given for any $k$. We use dihedral covers to distinguish the embeddings of the curves in the plane.
Here is the report.

                                              ***** February 5 *****

Takashi Kishimoto : 10:00-11:00
"On the logarithmic Kodaira dimension of affine threefolds"
In the theory of affine algebraic varieties, it is important to investigate the structure associated to the log Kodaira dimension. In two-dimensional case, there are satisfactory structure theorems concerning
to log Kodaira dimension due to Miyanishi, Sugie, Kawamata and so on. Meanwhile, for higher dimensional case, there are no satisfactory result. In this talk, we shall restrict ourselves to the 3-dimensional case and
make the framework under a certain geometric condition. As a result, under this condition, we can describe the construction of affine 3-folds with log Kodaira dimension $-\infty$ fairly explicitly, and show that an affine
3-fold with log Kodaira dimension $2$ has the structure of a $\C^*$-fibration.
Here is the report.

Pietro Pirola: 11:20-12:20
"Alternating groups and rational functions of curves and surfaces"
We will discuss some classical and new results on the monodromy of algebraic curves and on the Galois theory of algebraic complex function fields. In particular we will analyze the geometry related with the alternating group .
We prove the following
Theorem
Let F be a finite extension of the complex field of transcendental degree one or two. Then there is an integer m(F) , such that for any n>m(F) there is a subfield L of F with extension degree [F:L ] =n, where the monodromy group ,i.e. the Galois group of the splitting field extension, is the alternating group.
Here is the report.


Tatsuhiro Minagawa: 14:00-15:00
"On weakened Fano 3-folds"
My talk is on clssification of weakened Fano 3-folds, That is, a smooth weak Fano 3-fold which will deform to a Fano 3-fold.
Here is the report.


Enrico Schlesinger  15:30-16:30
"Galois groups of projections"
Let $C$ be an irreducible algebraic curve in $\mathbb{P}^3 (\mathbb{C})$that is not contained in a plane. A line $L \subset \mathbb{P}^3$ is said to be uniform with respect to $C$ if the Galois group of the projection of $C$ from $L$ is the full symmetric group. The uniform position principle states the generic line is uniform. We refine this result showing that in the Grassmannian of lines in $\mathbb{P}^3$ the locus of non uniform  lines  has dimension at most two. This is sharp because, if there is a point $x \in \mathbb{P}^3$ such that projection from $x$ induces a non birational map from $C$ to its image in $\mathbb{P}^{2}$, then the surface $\sigma(x)$ of lines through $x$ is contained in the non-uniform locus. For a smooth curve $C$, we show any irreducible surface of non-uniform lines is a Schubert cycle $\sigma(x)$ as above, unless $C$ is a rational curve of degree three, four or six.
Joint work with professor Gian Pietro Pirola.
Here is the report.


                                               ***** February 6 *****

Yoshiaki Fukuma 10:00-11:00 
"On sectional invariants of polarized manifolds"
Here is the report.


Takayuki Hayakawa: 11:20-12:20
"Divisors with small discrepacies over 3-dimensional terminal singularities"
For each $3$-dimensional terminal singularity $P \in X$,we shall construct birational morphisms $\pi : \bar{X} \to X$ such that the exceptional divisors of $\pi$ are irreducible and have small discrepancies. We shall use classification of terminal singularities, and our construction is very concrete.
If $\bar{X}$ has only terminal singularities, then $\pi : \bar{X} \to X$ is a contraction of an extremal ray which contracts a divisor to a point. By our method, we can classify all such contractions with small discrepancies explicitly.
Here is the report.


Masaaki Homma 14:00-15:00
"Conics with a Hermitian curve"
Here is the report.


Hajime Kaji  15:30-16:30
"Projective geometry of Freudenthal's varieties of certain type"
We discuss projective geometry of a certain series of algebraic varieties which appear in the construction of exceptional Lie algebras by H. Freudenthal. (tentative):
Here is the report.