15 research outputs found
Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds
We prove that the moduli spaces A_3(D) of polarized abelian threefolds with
polarizations of types D=(1,1,2), (1,2,2), (1,1,3) or (1,3,3) are unirational.
The result is based on the study of families of simple coverings of elliptic
curves of degree 2 or 3 and on the study of the corresponding period mappings
associated with holomorphic differentials with trace 0. In particular we prove
the unirationality of the Hurwitz space H_{3,A}(Y) which parameterizes simply
branched triple coverings of an elliptic curve Y with determinants of the
Tschirnhausen modules isomorphic to A^{-1}.Comment: 43 pages, latex2e, some typos corrected, references updated, to
appear in Annali di Matematica Pura ed Applicat
Irreducibility of Hurwitz spaces
Graber, Harris and Starr proved, when n >= 2d, the irreducibility of the
Hurwitz space H^0_{d,n}(Y) which parametrizes degree d coverings of a smooth,
projective curve Y of positive genus, simply branched in n points, with full
monodromy group S_d (math.AG/0205056). We sharpen this result and prove that
H^0_{d,n}(Y) is irreducible if n >= max{2,2d-4} and in the case of elliptic Y
if n >= max{2,2d-6}. We extend the result to coverings simply branched in all
but one point of the discriminant. Fixing the ramification multiplicities over
the special point we prove that the corresponding Hurwitz space is irreducible
if the number of simply branched points is >= 2d-2. We study also simply
branched coverings with monodromy group different from S_d and when n is large
enough determine the corresponding connected components of H_{d,n}(Y). Our
results are based on explicit calculation of the braid moves associated with
the standard generators of the n-strand braid group of Y.Comment: latex2e, 34 pages, 5 figure
Polarization types of isogenous Prym-Tyurin varieties
Let p:C-->Y be a covering of smooth, projective curves which is a composition
of \pi:C-->C' of degree 2 and g:C'-->Y of degree n. Let f:X-->Y be the covering
of degree 2^n, where the curve X parametrizes the liftings in C^{(n)} of the
fibers of g:C'-->Y. Let P(X,\delta) be the associated Prym-Tyurin variety,
known to be isogenous to the Prym variety P(C,C'). Most of the results in the
paper focus on calculating the polarization type of the restriction of the
canonical polarization of JX on P(X,\delta). We obtain the polarization type
when n=3. When Y=P^1 we conjecture that P(X,\delta) is isomorphic to the dual
of the Prym variety P(C,C'). This was known when n=2, we prove it when n=3, and
for arbitrary n if \pi:C-->C' is \'{e}tale. Similar results are obtained for
some other types of coverings.Comment: 28 page
Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve
Given a smooth, projective curve , a finite group and a positive
integer we study smooth, proper families of Galois
covers of with Galois group isomorphic to branched in points,
parameterized by algebraic varieties . When is with trivial center we
prove that the Hurwitz space is a fine moduli variety for this
category and construct explicitly the universal family. For arbitrary we
prove that is a coarse moduli variety. For families of pointed
Galois covers of we prove that the Hurwitz space is a
fine moduli variety, and construct explicitly the universal family, for
arbitrary group . We use classical tools of algebraic topology and of
complex algebraic geometry.Comment: v3: 42 pages, minor changes, introduction rewritte
Hurwitz spaces of Galois coverings of P^1 with Galois groups Weyl groups
We prove the irreducibility of the Hurwitz spaces which parametrize Galois
coverings of P^1 whose Galois group is an arbitrary Weyl group and the local
monodromies are reflections. This generalizes a classical theorem due to
Clebsch and Hurwitz.Comment: Latex, 15 page
Unirationality of Hurwitz spaces of coverings of degree <= 5
Let be a smooth, projective curve of genus over the complex
numbers. Let be the Hurwitz space which parametrizes coverings
of degree , simply branched in points, with monodromy
group equal to , and isomorphic to a fixed line bundle
of degree . We prove that, when or and is
sufficiently large (precise bounds are given), these Hurwitz spaces are
unirational. If in addition (when ), (when ) and
(when ), then these Hurwitz spaces are rational.Comment: Proposition 2.11 and Lemma 2.13 are corrected. The corrections do not
affect the other statements of the paper. Corrigendum submitted to IMR
A criterion for extending morphisms from open subsets of smooth fibrations of algebraic varieties
Given a smooth morphism Y→S and a proper morphism P→S of algebraic varieties we give a sufficient condition for extending an S-morphism U→P, where U is an open subset of Y, to an S-morphism Y→P, analogous to Zariski’s main theorem
Hurwitz spaces of quadruple coverings of elliptic curves and the moduli space of Abelian threefolds A_3(1,1,4)
We prove that the moduli space A_3(1,1,4) of polarized abelian threefolds with polarization of type (1,1,4) is unirational. By a result of Birkenhake and Lange this implies the unirationality of the isomorphic moduli space A_3(1,4,4). The result is based on the study the Hurwitz space H_{4,n}(Y) of quadruple coverings of an elliptic curve Y simply branched in n points. We prove the unirationality of its codimension one subvariety H^{0}_{4,A}(Y) which parametrizes quadruple coverings \pi:X --> Y with Tschirnhausen modules isomorphic to A^{-1}, where A\in Pic^{n/2}Y, and for which \pi^*:J(Y)--> J(X) is injective. This is an analog of the result of Arbarello and Cornalba that the Hurwitz space H_{4,n}(P^1) is unirational