On the minimal number of singular fibers with non-compact Jacobians for families of curves over

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• 作者：Xin Lu^a ; ^d ; ^{xin-l@ecnu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{x.lu@uni-mainz.de" class="auth_mail" title="E-mail the corresponding author} ; Sheng-Li Tan^b ; ^{sltan@math.ecnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Wan-Yuan Xu^c ; ^{xwydy1988@126.com" class="auth_mail" title="E-mail the corresponding author} ; Kang Zuo^d ; ^{zuok@uni-mainz.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14D06 ; 14H10 ; 14J25
• 刊名：Journal de mathematiques pures et appliquees
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：105
• 期：5
• 页码：724-733
• 全文大小：340 K

文摘

Let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si165.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=ed03601f02145f8f1cbc198b3aaf0458" title="Click to view the MathML source">f:X→P¹class="mathContainer hidden">class="mathCode">$f:X\to {\mathbb{P}}^1$ be a non-isotrivial family of semi-stable curves of genus class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si10.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=bd6755f44950840e85e2f8f3f9f833eb" title="Click to view the MathML source">g≥1class="mathContainer hidden">class="mathCode">$g\ge 1$ defined over an algebraically closed field k  . Denote by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si4.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=1397761aa0d749377c61cf7ff0053d5e" title="Click to view the MathML source">s_ncclass="mathContainer hidden">class="mathCode">$s_{\mathit{nc}}$ the number of the singular fibers whose Jacobians are non-compact. We prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si22.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=9f39adc813fe7211d10edb464c4439bb" title="Click to view the MathML source">s_nc≥5class="mathContainer hidden">class="mathCode">$s_{\mathit{nc}}\ge 5$ if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si6.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=e97402407aa0465aebc0d13497d98cf3" title="Click to view the MathML source">k=Cclass="mathContainer hidden">class="mathCode">$k=\mathbb{C}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si25.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=ab3515f8046d8d5c38efad6cc0f72812" title="Click to view the MathML source">g≥5class="mathContainer hidden">class="mathCode">$g\ge 5$; we also prove that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si31.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=c6a259c092ed883d296cc526b8a2d00e" title="Click to view the MathML source">s_nc≥4class="mathContainer hidden">class="mathCode">$s_{\mathit{nc}}\ge 4$ if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021782415001610&_mathId=si9.gif&_user=111111111&_pii=S0021782415001610&_rdoc=1&_issn=00217824&md5=db7a3d137d95b52f33772364335e694b" title="Click to view the MathML source">char(k)>0class="mathContainer hidden">class="mathCode">$\mathrm{char}(k)>0$ and the relative Jacobian of f is non-smooth.