Document Zbl 1455.13030

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Math., New Ser.</a> <a href="/?q=in%3A458620" title="Articles in this Issue">27, No. 1, Paper No. 6, 22 p. (2021)</a>. </div> <div class="abstract">Let $X$ be a projective scheme over $\mathbb{C}$ with a very ample line bundle $\mathcal{L}$, and let $\mathcal{C}$ be the cone of $X$ associated with $\mathcal{L}$. Let $x_{1},\dots,x_{n}$ be the projective coordinates of $\mathbb{P}^{n-1}(\mathbb{C})$ such that $X\subseteq \mathbb{P}^{n-1}(\mathbb{C})$ and $\mathcal{O}_{\mathbb{P}^{n-1}(\mathbb{C})}(1)\vert_{X}=\mathcal{L}$, and let $\mathfrak{a} \subseteq R=\mathbb{C}[x_{1},\dots,x_{n}]$ be the defining ideal of $\mathcal{C}\subseteq \mathbb{A}^{n}_{\mathbb{C}}$. Then the Lyubeznik numbers of $\mathcal{C}$ are defined as \[\lambda_{i,j}(\mathcal{C}):= \dim_{\mathbb{C}} \mathrm{Ext}_{R}^{i}\left(\mathbb{C},H_{\mathfrak{a}}^{n-j}(R)\right)\] for every $i,j \geq 0$. The authors specify a technical condition that implies the dependence of the Lyubeznik numbers $\lambda_{i,j}(\mathcal{C})$ on the choice of $\mathcal{L}$. Using this result, they prove that if $k$ is a field of characteristic $0$, then there exist projective schemes over $k$ such that the Lyubeznik numbers $\lambda_{i,j}(\mathcal{C})$ of their cone $\mathcal{C}$ depend on their projective embeddings. This answers a question of [<span class="zbmathjax-textit">G. Lyubeznik</span>, Lect. Notes Pure Appl. Math. 226, 121–154 (2002; <a href="/1061.14005">Zbl 1061.14005</a>), p. 133] negatively. The striking feature of this result is that it contrasts entirely the positive characteristic case where the machinery of Frobenius endomorphism is available; see [<span class="zbmathjax-textit">L. Núñez-Betancourt</span> et al., Contemp. Math. 657, 137–163 (2016; <a href="/1346.13035">Zbl 1346.13035</a>)] and [<span class="zbmathjax-textit">W. Zhang</span>, Adv. Math. 228, No. 1, 575–616 (2011; <a href="/1235.13012">Zbl 1235.13012</a>)].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A18624">Hossein Faridian (Clemson)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7309271">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A13D45" title="MSC2020">13D45</a> </td> <td class="space"> Local cohomology and commutative rings </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A14B15" title="MSC2020">14B15</a> </td> <td class="space"> Local cohomology and algebraic geometry </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A32S30" title="MSC2020">32S30</a> </td> <td class="space"> Deformations of complex singularities; vanishing cycles </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A32S40" title="MSC2020">32S40</a> </td> <td class="space"> Monodromy; relations with differential equations and $D$-modules (complex-analytic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A32S60" title="MSC2020">32S60</a> </td> <td class="space"> Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Alocal+cohomology">local cohomology</a>; <a href="/?q=ut%3ALyubeznik+numbers">Lyubeznik numbers</a>; <a href="/?q=ut%3Aprojective+schemes">projective schemes</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1061.14005">Zbl 1061.14005</a>; <a href="/1346.13035">Zbl 1346.13035</a>; <a href="/1235.13012">Zbl 1235.13012</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1455.13030" data-ciurl="/ci/07309271" data-biburl="/bibtex/07309271.bib" data-amsurl="/amsrefs/07309271.bib" data-xmlurl="/xml/07309271.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07309271.pdf" title="Zbl 1455.13030 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s00029-020-00612-3" aria-label="DOI for “Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings”" title="10.1007/s00029-020-00612-3">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1803.07448"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Beilinson, A.; Bernstein, J.; Deligne, P., Faisceaux Pervers, Astérisque (1982), Paris: Soc. 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