For concreteness, we focus on a particular example in this paper,
although our analysis is general.
The moduli space 
of all
complex structures on a quintic threefold X can be described in
terms of the defining equations of the quintics: the general
such equation takes the form of a homogeneous quintic polynomial
and involves 126 coefficients. Some 25 of these are redundant
due to the action of
on the
homogeneous coordinates 
, leaving 101 independent parameters
in the moduli space.
For general values of 
, the equation
1 defines a nonsingular Calabi--Yau manifold 
in 
, but
at certain special values (along a set 
of complex codimension
1 in the parameter space), the solution set of 1 becomes singular.
These singularities were analyzed many years ago by Lefschetz [15],
who showed that:
1) the ``generic'' singular space 
, 
has a single
node, i.e., a singular point with local equation 
,
(we distinguish between the singular points of the
Calabi--Yau---here called nodes---and the points in the moduli
space which label such Calabi--Yau's---called conifold points)
2) the singular point determines a ``vanishing cycle'' 
which shrinks to zero size when the singularity is approached (more
precisely, the period integrals over 
vanish in the limit), and
3) the homology of the Calabi--Yau manifolds undergoes a monodromy transformation
upon transport around a loop in 
encircling the singular locus
. (Here,
denotes the number of (oriented)
intersections of
with 
.) If all of the singular points on 
are nodes,
then 
is called
a ``Calabi--Yau conifold'' (so named [9] because of the conical
nature of the singularities).
If we introduce a holomorphic 3-form 
(depending on the moduli),
and use the ``periods'' 
of 
to describe the
complex structure, we find that some of the periods become multiple-valued
near the singular locus. In fact, if we let
be the period corresponding to the vanishing cycle, then we find
that the singular locus 
in 
is locally described by Z=0, and
that other periods must take the form
near Z=0 in order to have the correct monodromy property.
Let us now consider the set of quintic conifolds in 
which have k
singular points. Since asking for a single node places a single condition
on
the parameters, one's initial expectation is that asking for a conifold
with k singular
points will place k conditions on the parameters, leading
to a locus of complex codimension k in 
. When this is true, the
generic
point of that locus will locally be an intersection of k hypersurfaces,
meeting transversally. Near the intersection of all of these, there
will be k different monodromy transformations with vanishing cycles
, ..., 
, and the periods
must take the form
near 
, where 
are among
the local coordinates near the intersection.
However, in the example which we will study in detail there are k=16 singular points on the conifold which impose only 15 conditions on the parameters. In fact, the vanishing cycles in our example in the next section will satisfy the homology relation
which implies a corresponding relation among the periods:
The locus of conifolds with (at least) 15 singular points is described by
, but since the hypersurface 
passes through this locus (as a consequence of 7), the
point is also present, without imposing
any further conditions. This is clearly a special property of
the particular collection of 16 points which we are considering.
In this situation, we have 16 monodromy transformations near a locus
of codimension 15. The asymptotic form 5 still holds
near 
, but
the 
and 
are related by 6 and 7.
In particular, only a subset of the 
's can be included among
a list of local coordinates near the intersection locus.
