Let $f:X to mathbb{P}^1$ be a non-isotrivial family of semi-stable curves of genus $ggeq 1$ defined over an algebraically closed field $k$ with $s_{nc}$ singular fibers whose Jacobians are non-compact. We prove that $s_{nc}geq 5$ if $k=mathbb C$ and $ggeq 5$; we also prove that $s_{nc}geq 4$ if ${rm char}~k>0$ and the relative Jacobian of $f$ is non-smooth.
This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct ima
ges of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of genus g>1 can not contain compact Shimura curves, and that it only contains a non-compact Shimura curve for g=3.
This paper considers the moduli spaces (stacks) of parabolic bundles (parabolic logarithmic flat bundles with given spectrum, parabolic regular Higgs bundles) with rank 2 and degree 1 over $mathbb{P}^1$ with five marked points. The stratification str
uctures on these moduli spaces (stacks) are investigated. We confirm Simpsons foliation conjecture of moduli space of parabolic logarithmic flat bundles for our case.
We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t in K$ and the Galois group of the polynomial $h(x)$ over $K$ is very bi
g and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)
We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equati
on for the dual line arrangement $mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.
We study generating series of Gromov-Witten invariants of $Etimesmathbb{P}^1$ and their tropical counterparts. Using tropical degeneration and floor diagram techniques, we can express the generating series as sums of Feynman integrals, where each sum
mand corresponds to a certain type of graph which we call a pearl chain. The individual summands are --- just as in the case of mirror symmetry of elliptic curves, where the generating series of Hurwitz numbers equals a sum of Feynman integrals --- complex analytic path integrals involving a product of propagators (equal to the Weierstrass-$wp$-function plus an Eisenstein series). We also use pearl chains to study generating functions of counts of tropical curves in $E_{mathbb{T}}timesmathbb{P}^1_mathbb{T}$ of so-called leaky degree.
Xin Lu
,Sheng-Li Tan
,Wan-Yuan Xu
.
(2014)
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"On the minimal number of singular fibers with non-compact Jacobians for families of curves over $mathbb P^1$"
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Xin Lu
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