Families of hypersurfaces with many prescribed singularities [Elektronische Ressource] / Eric Westenberger

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• Mathematics

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• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2004
• Nombre de lectures: 17
• Langue: English

Extrait

Families of hypersurfaces with many
prescribed singularities
Eric Westenberger
Vom Fachbereich Mathematik der Universit at Kaiserslautern
zur Verleihung des akademischen Grades Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.) genehmigte Dissertation
1. Gutachter: Prof. Dr. G.-M. Greuel
2.hter: Prof. Dr. E. Shustin
Vollzug der Promotion: 18.06.2004
D 386The image on the title page shows the surface of degree 8 de ned by the equation
4 2 2 4 2 2 4 2 2(x x + 1) + (y y + 1) + (z z + 1) = 1:
It has 144 real ordinary double points. The equation is similar to Chmutov’s construction
[Ch92], and I would like to thank O. Labs for showing it to me.
The image was produced using the programme Surf written by S. Endrass.Families of hypersurfaces with
many prescribed singularities
Eric Westenberger
Vom Fachbereich Mathematik
der Universit at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation
1. Gutachter: Prof. Dr. G.-M. Greuel
2.hter: Prof. Dr. E. Shustin
Vollzug der Promotion: 18.06.2004
D 386Introduction
Studying varieties with prescribed properties is one of the fundamental tasks in
algebraic geometry. It was one of the main achievements in the last century to
realize that it is not only necessary to study these varieties for themselves, but also
to consider families of varieties. This means that a \good" space has to be found
parametrizing all varieties with the xed prescribed properties. Once such a moduli
space has been constructed, then understanding its geometry allows us to answer a
lot of interesting questions.
In this thesis, we consider reduced hypersurfaces W in a smooth, projective variety
V de ned over the complex numbers, and we assume that W has at most isolated
singularities. The singular points are considered up to analytical or topological
equivalence, and the equivalence classes are called singularity types. In the case of
topological types we restrict ourselves to singularities, which are either quasihomo-
geneous or of corank less than 2.
LetS ;:::;S be types of isolated singularities, and letH be an ample divisor ofV1 r
and d 0. The space
V (S + +S ) := fReduced hypersurfaces W2jdHj with r isolated singulard 1 r
points z ;:::;z of types S ;:::;S as its only singularitiesg1 r 1 r
is a locally closed subspace of the linear systemjdHj, and it is called the equisingular
stratum. The fundamental questions concerning this space are:
Is V (S + +S ) non-empty, i.e. does there exist a hypersurface with sin-d 1 r
gularities of the prescribed types and belonging to the given linear system?
Is V (S + +S ) smooth and what is its dimension? In particular, doesd 1 r
V (S + +S ) have the \expected dimension", i.e. do the singular pointsd 1 r
impose independent conditions on hypersurfaces in the given linear system?
IsV (S ++S ) irreducible and what is its degree in the linear spacejdHj?d 1 r
These questions have attracted the continuous attention of algebraic geometers since
the beginning of the 20th century, where the foundations were laid by Severi, Pluc ker,
iii
Segre, Zariski and others. It was realized quite early that all these questions ap-
peared to be rather hard, the last one being the most di cult. In fact the only case
for which a complete answer is known, is the classical case of plane, nodal curves
solved essentially by Severi [Sev21] and completed by Harris [Har85]. Severi showed
irrthat V (rA ) is non-empty if and only if1d
(d 1)(d 2)
r ;
2
irrwhereV (rA ) is the open subset ofV (rA ) corresponding to irreducible curves.1 d 1d
irrFurthermore, ifV (rA ) is non-empty, then it is smooth of the expected dimension1d
d(d+3)
k and also irreducible.
2
For other singularities and more general hypersurfaces, examples were found where
the spacesV (S +:::+S ) are reducible or non-reduced or singular or have dimensiond 1 r
bigger than the expected one. Note that such pathological behaviour of singular
hypersurfaces has already been observed for plane curves with only nodes and cusps.
Since there are no more complete answers, the problem is to nd necessary and
su cient conditions (in terms of numerical invariants of the singular points and the
linear system) for the \good" properties to hold.
1Our approach consists of reformulating the questions above in terms ofH -vanishing
conditions for ideal sheaves of zero-dimensional schemes associated to the singular
1points. Developing new techniques for deducing H -vanishing criteria is then one
of the fundamental tasks in this area. This approach was applied very successfully,
mainly by Greuel, Lossen and Shustin in the study of families of plane curves,
and generalized to some extent by Keilen and Tyomkin to curves on more general
nsurfaces. In this thesis we study the problem also for hypersurfaces in P .
We are particularly interested in \asymptotically proper" conditions, i.e. criteria
where the necessary and su cien t parts are asymptotically of the same order. Let
us explain this more closely by means of the existence problem of hypersurfaces in
nP with many singularities of a certain xed type S. A su cien t condition
m m 1r(S) d R(d); > 0; R(d)2O(d ); (0.0.1)1 1
for the existence of hypersurfaces in V (rS) will be called asymptotically proper ifd
2there exists an absolute constant > and in nitely many pairs (d;r)2 N for2 1
mwhich V (rS) =; but r(S) d .d 2
In fact, if (S) = (S) then (0.0.1) is asymptotically proper if and only if m = n,
as we shall see later.
Main results
The main results of this thesis deal with the existence and the smoothness prob-
nlem. We improve several conditions for the of hypersurfaces in P withiii
prescribed singularities and develop the rst asymptotically proper existence the-
orems for higher dimensional hypersurfaces with many singularities. On the other
hand, some of the results are new even in the case of plane curves. Our main tool
for constructing hypersurfaces with prescribed singularities is the so-called patch-
working method (also called Viro’s glueing method) combined with the theory of
zero-dimensional schemes.
For obtaining new conditions for T-smoothness we study the Castelnuovo function
3of zero-dimensional schemes on surfaces in P , which generalizes the theory in the
1plane case. This can be applied to derive H -vanishing theorems for these schemes.
nExistence of hypersurfaces in P with prescribed singularities
We introduce an invariant of sets of singularity types, which is essentially the leading
ncoe cien t in a su cien t condition for the existence of hypersurfaces in P with these
singularities.
sLetS be a set of singularity types and denote by (S) the equianalytic or equi-
singular Tjurina number of S2S. Consider the set of all 0 such that for all
fS ;:::;SgS, the condition1 r
rX
s n n 1k (S )d +O(d );i i
i=1
nimplies the existence of a non-emptyT-smooth component ofV (k S +:::+k S ).1 1 r rd
The T-smoothness requirement implies that
1
0 ;
n!
and the existence result is asymptotically proper if > 0. The supremum of all
regsatisfying the property above is denoted by (S).n
Let n > 2, and letS =S [S whereS is a set of analytic singularity types ofa t a
corank < n andS is a set of topological singularity types of corank 2. If thet
s(analytic, respectively topological) Tjurina number (S) is bounded as S varies in
S, then
reg (S)n 1reg (S) : (0.0.2)n n
Hence, asymptotically proper existence results automatically carry over to higher
dimensions, so this result can be seen as a kind of stabilization of the existence
problem. Generalizing a result of [Sh01] we obtain as an immediate consequence
that
2reg (S) (0.0.3)n 9n!iv
for a setS of (analytical or topological) singularity types of corank 2 with bounded
Tjurina number. Note that the lower bound (0.0.3) di ers from the natural upper
regbound (S) 1=n! only by a constant factor.n
1For plane curves with only tacnodes and cusps, the sharpest known lower bound of
6
was found by J. Roe in [Ro01]. Using our methods we are able to improve this result
substantially and even determine the precise asymptotic factor for hypersurfaces of
arbitrary dimension. IfS is a nite set of simple singularity types, then
1
reg (S) = : (0.0.4)n
n!
This is also an extension of the results of [Sh93], where it was shown for plane curves
with only ordinary nodes and cusps.
We also prove an asymptotically proper existence result for hypersurfaces with quasi-
n 1homogeneous singularities. For xed a = (a ;:::;a )2 N , denote byS the set2 n a
of all analytical types of singularities de ned by polynomials of the form
k a a2 nf(x ;:::;x ) =c x +c x +::: +c x ;1 n 1 2 n1 2 n
with k2 N and c ;:::;c 2 Cnf0g. Explicit constructions in combination with a1 n
local patchworking method allows us to deduce
creg (S ) > 0; (0.0.5)an n n2 nl
P Qn n
where l = 1 + a andc = (a 1). Note that inS the Tjurina number isi i ai=2 i=2
not bounded. We can use this result to deduce an asymptotically proper existence
result for hypersurfaces with singularities of modality 2.
3Zero-dimensional schemes on surfaces in P
We study the behaviour of the Castelnuovo function for zero-dimensional