On the projective normality of double coverings over a rational surface

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• 作者：Biswajit Rajaguru ^{biswajit.rajaguru@ku.edu" class="auth_mail" title="E-mail the corresponding author} ; Lei Song ; ^{lsong@ku.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Projective normality ; Double covering ; Adjoint divisor
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：251-266
• 全文大小：411 K

文摘

We study the projective normality of a minimal surface m>Xm> which is a ramified double covering over a rational surface m>S  m> with mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302782&_mathId=si1.gif&_user=111111111&_pii=S0021869316302782&_rdoc=1&_issn=00218693&md5=9bdecc0877c01ae37dc802fb36fd1a0f" title="Click to view the MathML source">dim⁡|&minus;K_S|≥1mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mi mathvariant="normal">dimmi><mo>⁡mo><mo stretchy="false">|mo><mo>&minus;mo><msub><mrow><mi>Kmi>mrow><mrow><mi>Smi>mrow>msub><mo stretchy="false">|mo><mo>≥mo><mn>1mn>math>. In particular Horikawa surfaces, the minimal surfaces of general type with mmlsi2" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302782&_mathId=si2.gif&_user=111111111&_pii=S0021869316302782&_rdoc=1&_issn=00218693&md5=42e9354f1ea32af208995ad5153c8e2d">mg class="imgLazyJSB inlineImage" height="19" width="128" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021869316302782-si2.gif">mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msubsup><mrow><mi>Kmi>mrow><mrow><mi>Xmi>mrow><mrow><mn>2mn>mrow>msubsup><mo>=mo><mn>2mn><msub><mrow><mi>pmi>mrow><mrow><mi>gmi>mrow>msub><mo stretchy="false">(mo><mi>Xmi><mo stretchy="false">)mo><mo>&minus;mo><mn>4mn>math>, are of this type, up to resolution of singularities. Let m>πm> be the covering map from m>Xm> to m>S  m>. We show that the mmlsi3" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302782&_mathId=si3.gif&_user=111111111&_pii=S0021869316302782&_rdoc=1&_issn=00218693&md5=61d59fbf2b063572e6a0514d55465ab3" title="Click to view the MathML source">Z₂mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi mathvariant="double-struck">Zmi>mrow><mrow><mn>2mn>mrow>msub>math>-invariant adjoint divisors mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302782&_mathId=si4.gif&_user=111111111&_pii=S0021869316302782&_rdoc=1&_issn=00218693&md5=ce40b0f2de47daec74c9da57ab1e5182" title="Click to view the MathML source">K_X+rπ^⁎AmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msub><mrow><mi>Kmi>mrow><mrow><mi>Xmi>mrow>msub><mo>+mo><mi>rmi><msup><mrow><mi>πmi>mrow><mrow><mo>⁎mo>mrow>msup><mi>Ami>math> are normally generated, where the integer mmlsi37" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021869316302782&_mathId=si37.gif&_user=111111111&_pii=S0021869316302782&_rdoc=1&_issn=00218693&md5=73acd8f2f75ad68157ace20182bf6710" title="Click to view the MathML source">r≥3mathContainer hidden">mathCode"><math altimg="si37.gif" overflow="scroll"><mi>rmi><mo>≥mo><mn>3mn>math> and m>Am> is an ample divisor on m>Sm>.