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Motivic Euler Products and Motivic Height Zeta Functions
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes,
in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the
study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant
compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a
topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic
behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above
moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools
for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of
Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with
exponentials constructed using Denef and Loeser’s motivic vanishing cycles.
eBook
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields
The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $\\mathbb F_p(t)$, when $p$ is prime and $r\\ge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $\\mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $\\mathbb F_q(t^1/d)$.
eBook
Sir Cumference and the Isle of Immeter : a math adventure
Young Per must figure out how to unlock the secrets of the mysterious island of Immeter.
Book
Inflectionary Invariants for Isolated Complete Intersection Curve Singularities
We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring
singular members. Let
eBook
Twistors, Quartics, and del Pezzo Fibrations
It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of
non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The
anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the
pluri-half-anti-canonical map from the twistor spaces.
Specifically, each of the present twistor spaces is bimeromorphic to a
double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by
a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the
twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of
rational curves, and it is an anti-canonical curve on smooth members of the pencil.
These twistor spaces are naturally classified
into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the
pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic
hypersurface has to be of a specific form.
Together with our previous result in Honda (“A new series of compact minitwistor
spaces and Moishezon twistor spaces over them”, 2010), the present result completes a classification of Moishezon twistor spaces whose
half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a
long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is
smaller than a pencil.
Twistor spaces which have a similar structure were studied in Honda (“Double solid twistor spaces: the
case of arbitrary signature”, 2008 and “Double solid twistor spaces II: General case”, 2015) and they are very special examples among
the present twistor spaces.
eBook
In the search for beauty : unravelling non-Euclidean geometry
Chronicles the historical attempts to prove the fifth postulate of Euclid on parallel lines that led eventually to the creation of non-Euclidean geometry.
Book
Effective faithful tropicalizations associated to linear systems on curves
For a connected smooth projective curve
Let
As an application, when
eBook