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<title>Toric stack</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Toric stack</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <b>toric stack</b> is a <a href="Stack_(mathematics)" title="Stack (mathematics)">stacky</a> generalization of a <a href="Toric_variety" title="Toric variety">toric variety</a>. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking <a href="GIT_quotient" title="GIT quotient">GIT quotients</a> with that of taking <a href="Quotient_stack" title="Quotient stack">quotient stacks</a>. Consequently, a toric variety is a coarse approximation of a toric stack. A <b>toric orbifold</b> is an example of a toric stack.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Stanley%E2%80%93Reisner_ring" title="Stanley–Reisner ring">Stanley–Reisner ring</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite id="CITEREFIwanari2009" class="citation journal cs1">Iwanari, Isamu (2009). "The category of toric stacks". <i>Compositio Mathematica</i>. <b>145</b> (3): <span class="nowrap">718–</span>746. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0610548">math/0610548</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2FS0010437X09003911">10.1112/S0010437X09003911</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13941792">13941792</a>.</cite></li>
<li><cite id="CITEREFGeraschenkoSatriano2015" class="citation journal cs1">Geraschenko, Anton; Satriano, Matthew (2015). "Toric stacks I: The theory of stacky fans". <i>Transactions of the American Mathematical Society</i>. <b>367</b> (2): <span class="nowrap">1033–</span>1071. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1107.1906">1107.1906</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-2014-06063-7">10.1090/S0002-9947-2014-06063-7</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:5667546">5667546</a>.</cite></li>
<li><cite id="CITEREFIwanari2009" class="citation journal cs1">Iwanari, I. (2009). "Integral Chow Rings of Toric Stacks". <i>International Mathematics Research Notices</i>. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0705.3524">0705.3524</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimrn%2Frnp110">10.1093/imrn/rnp110</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12047977">12047977</a>.</cite></li></ul>
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