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Weighted projective space
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<h2 id="properties">Properties</h2> <ul><li>If <i>d</i> is a positive integer then P(<i>a</i>0,<i>a</i>1,...,<i>a</i><i>n</i>) is isomorphic to P(<i>da</i>0,<i>da</i>1,...,<i>da</i><i>n</i>). This is a property of the <a href="/facts/Proj_construction/gcvPmjpU">Proj construction</a>; geometrically it corresponds to the <i>d</i>-tuple <a href="/facts/Veronese_embedding/mXhQ3ouK">Veronese embedding</a>. So without loss of generality one may assume that the degrees <i>a</i><i>i</i> have no common factor.</li> <li>Suppose that <i>a</i><i>0</i>,<i>a</i><i>1</i>,...,<i>a</i><i>n</i> have no common factor, and that <i>d</i> is a common factor of all the <i>a</i>i with <i>i</i>≠<i>j</i>, then P(<i>a</i>0,<i>a</i>1,...,<i>a</i><i>n</i>) is isomorphic to P(<i>a</i>0/d,...,<i>a</i>j-1/d,<i>a</i>j,<i>a</i>j+1/d,...,<i>a</i><i>n</i>/d) (note that <i>d</i> is coprime to <i>a</i><i>j</i>; otherwise the isomorphism does not hold). So one may further assume that any set of <i>n</i> variables <i>a</i><i>i</i> have no common factor. In this case the weighted projective space is called well-formed.</li> <li>The only singularities of weighted projective space are cyclic quotient singularities.</li> <li>A weighted projective space is a Q-<a href="/facts/Fano_variety/or8RszIv">Fano variety</a><a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> and a <a href="/facts/Toric_variety/m1wEbNWw">toric variety</a>.</li> <li>The weighted projective space P(<i>a</i>0,<i>a</i>1,...,<i>a</i><i>n</i>) is isomorphic to the quotient of projective space by the group that is the product of the groups of roots of unity of orders <i>a</i>0,<i>a</i>1,...,<i>a</i><i>n</i> acting diagonally.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul> <ul><li>Dolgachev, Igor (1982), "Weighted projective varieties", <i>Group actions and vector fields (Vancouver, B.C., 1981)</i>, Lecture Notes in Math., vol. 956, Berlin: Springer, pp. 34–71, <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.169.5185">10.1.1.169.5185</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBFb0101508">10.1007/BFb0101508</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-11946-3, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0704986">0704986</a></li> <li>Hosgood, Timothy (2016), <i>An introduction to varieties in weighted projective space</i>, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1604.02441">1604.02441</a>, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2016arXiv160402441H">2016arXiv160402441H</a></li> <li>Reid, Miles (2002), <a href="https://homepages.warwick.ac.uk/~masda/surf/more/grad.pdf"><i>Graded rings and varieties in weighted projective space</i></a> (PDF)</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>M. Rossi and L. Terracini, Linear algebra and toric data of weighted projective spaces. Rend. Semin. Mat. Univ. Politec. Torino 70 (2012), no. 4, 469--495, proposition 8 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>This should be understood as a GIT quotient. In a more general setting, one can speak of a weighted projective stack. See https://mathoverflow.net/questions/136888/. <a href="/wiki/GIT_quotient" target="_blank">/wiki/GIT_quotient</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>