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<title>Minimal model program</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Minimal model program</span></span>

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<p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the <b>minimal model program</b> is part of the birational classification of <a href="Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a>. Its goal is to construct a birational model of any complex <a href="Projective_variety" title="Projective variety">projective variety</a> which is as simple as possible. The subject has its origins in the classical <a href="Birational_geometry" title="Birational geometry">birational geometry</a> of surfaces studied by the <a href="Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school</a>, and is currently an active research area within algebraic geometry.

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<div class="mw-heading mw-heading2"><h2 id="Outline">Outline</h2></div>

<p>The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational <a href="Equivalence_class" title="Equivalence class">equivalence class</a>, a variety which is "as simple as possible". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> for which any birational <a href="Regular_map_(algebraic_geometry)" class="mw-redirect" title="Regular map (algebraic geometry)">morphism</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to X'}">

<msup>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to X'}</annotation>

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</math></span><img src="./3ffc344b65c8cbb94d8c4692848e680fb0671cbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.588ex; height:2.843ex;" alt="{\displaystyle f\colon X\to X'}" loading="lazy"></span> with a smooth surface <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

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</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> is an <a href="Isomorphism" title="Isomorphism">isomorphism</a>.

</p><p>In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, which for simplicity is assumed non-singular. There are two cases based on its <a href="Kodaira_dimension" title="Kodaira dimension">Kodaira dimension</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (X)}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (X)}</annotation>

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</math></span><img src="./029c79d249c3d9ec812c291ea1a197aef3ddba8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle \kappa (X)}" loading="lazy"></span>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle \kappa (X)=-\infty .}</annotation>

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</math></span><img src="./e8d4c5981c8d3b1001b15f68ea606d5c6113de27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.005ex; height:2.843ex;" alt="{\displaystyle \kappa (X)=-\infty .}" loading="lazy"></span> We want to find a variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

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</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> birational to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, and a morphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X'\to Y}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X'\to Y}</annotation>

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</math></span><img src="./e9da033dde2ede00387fd0f369a86ab557e69c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.381ex; height:2.843ex;" alt="{\displaystyle f\colon X'\to Y}" loading="lazy"></span> to a projective variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim Y<\dim X',}">

<msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim Y<\dim X',}</annotation>

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</math></span><img src="./a48477993c6b9cc775a881e176b4d96abb7cfa6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.725ex; height:2.843ex;" alt="{\displaystyle \dim Y<\dim X',}" loading="lazy"></span> with the <a href="Canonical_class" class="mw-redirect" title="Canonical class">anticanonical class</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -K_{F}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -K_{F}}</annotation>

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</math></span><img src="./ccf25ab372d71e0f7e3c7603c03d8633c56e13e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.244ex; height:2.509ex;" alt="{\displaystyle -K_{F}}" loading="lazy"></span> of a general fibre <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

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</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> being <a href="Ample_line_bundle" title="Ample line bundle">ample</a>. Such a morphism is called a <i><a href="Fano_fibration" title="Fano fibration">Fano fibre space</a></i>.</li>

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<annotation encoding="application/x-tex">{\displaystyle \kappa (X)\geqslant 0.}</annotation>

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</math></span><img src="./da1d63065492659511f075d3be541470ca633ca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.036ex; height:2.843ex;" alt="{\displaystyle \kappa (X)\geqslant 0.}" loading="lazy"></span> We want to find <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

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</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> birational to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, with the canonical class <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{X^{\prime }}}">

<msub>

<msup>

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</msup>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle K_{X^{\prime }}}</annotation>

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</math></span><img src="./7f5b25e9cc38e9b9f80c95cd07c6f36e686697d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.149ex; height:2.676ex;" alt="{\displaystyle K_{X^{\prime }}}" loading="lazy"></span> <a href="Numerically_effective" class="mw-redirect" title="Numerically effective">nef</a>. In this case, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

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</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> is a <i>minimal model</i> for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>.</li></ul>

<p>The question of whether the varieties <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

</semantics>

</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> appearing above are non-singular is an important one. It seems natural to hope that if we start with smooth <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, this is not true, and so it becomes necessary to consider singular varieties also. The singularities that appear are called <a href="Terminal_singularities" class="mw-redirect" title="Terminal singularities">terminal singularities</a>.

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<div class="mw-heading mw-heading2"><h2 id="Minimal_models_of_surfaces">Minimal models of surfaces</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Enriques%E2%80%93Kodaira_classification" title="Enriques–Kodaira classification">Enriques–Kodaira classification</a></div>

<p>Every irreducible complex <a href="Algebraic_curve" title="Algebraic curve">algebraic curve</a> is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the <a href="Castelnuovo's_contraction_theorem" title="Castelnuovo's contraction theorem">contraction theorem</a> of <a href="Guido_Castelnuovo" title="Guido Castelnuovo">Guido Castelnuovo</a> essentially describes the process of constructing a minimal model of any smooth projective surface. The theorem states that any nontrivial birational morphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation>

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</math></span><img src="./07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"></span> must contract a −1-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here a −1-curve is a smooth rational curve <i>C</i> with self-intersection <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\cdot C=-1.}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle C\cdot C=-1.}</annotation>

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</math></span><img src="./2d44683940a12f63a80af0ec50b36b5ff389773e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.928ex; height:2.343ex;" alt="{\displaystyle C\cdot C=-1.}" loading="lazy"></span> Any such curve must have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\cdot C=-1}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle K\cdot C=-1}</annotation>

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</math></span><img src="./03bc2c914e42277a941ea61e88f630b13c0b74f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.58ex; height:2.343ex;" alt="{\displaystyle K\cdot C=-1}" loading="lazy"></span> which shows that if the canonical class is nef then the surface has no −1-curves.

</p><p>Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply <a href="Contraction_morphism" title="Contraction morphism">contract</a> all the −1-curves on the surface, and the resulting variety <i>Y</i> is either a (unique) minimal model with <i>K</i> nef, or a <a href="Ruled_surface" title="Ruled surface">ruled surface</a> (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface over a curve). In the second case, the ruled surface birational to <i>X</i> is not unique, though there is a unique one isomorphic to the product of the projective line and a curve. A somewhat subtle point is that even though a surface might have infinitely many -1-curves, one need only contract finitely many of them to obtain a surface with no -1-curves.

</p>

<div class="mw-heading mw-heading2"><h2 id="Higher-dimensional_minimal_models">Higher-dimensional minimal models</h2></div>

<p>In dimensions greater than 2, the theory becomes far more involved. In particular, there exist <a href="Smooth_scheme" title="Smooth scheme">smooth varieties</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> which are not birational to any smooth variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

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</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> with <a href="Nef_line_bundle" title="Nef line bundle">nef canonical class</a>. The major conceptual advance of the 1970s and early 1980s was that the construction of minimal models is still feasible, provided one is careful about the types of singularities which occur. (For example, we want to decide if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{X'}}">

<msub>

<msup>

</msup>

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<annotation encoding="application/x-tex">{\displaystyle K_{X'}}</annotation>

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</math></span><img src="./44902d1da3863ff3a6af2dcbc52ca8d3ef421f36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.149ex; height:2.676ex;" alt="{\displaystyle K_{X'}}" loading="lazy"></span> is nef, so intersection numbers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{X'}\cdot C}">

<msub>

<msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{X'}\cdot C}</annotation>

</semantics>

</math></span><img src="./43e206b6ff0d66e47287826894fb291682a984c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.594ex; height:2.676ex;" alt="{\displaystyle K_{X'}\cdot C}" loading="lazy"></span> must be defined. Hence, at the very least, our varieties must have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nK_{X'}}">

<msub>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle nK_{X'}}</annotation>

</semantics>

</math></span><img src="./4a63ce1df11d4cba593ede6a72257c5bf4530dea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.544ex; height:2.676ex;" alt="{\displaystyle nK_{X'}}" loading="lazy"></span> to be a <a href="Cartier_divisor" class="mw-redirect" title="Cartier divisor">Cartier divisor</a> for some positive integer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>.)

</p><p>The first key result is the <a href="Cone_of_curves" title="Cone of curves">cone theorem</a> of <a href="Shigefumi_Mori" title="Shigefumi Mori">Shigefumi Mori</a>, describing the structure of the cone of curves of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>. Briefly, the theorem shows that starting with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span>, one can inductively construct a sequence of varieties <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span>, each of which is "closer" than the previous one to having <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{X_{i}}}">

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K_{X_{i}}}</annotation>

</semantics>

</math></span><img src="./782d19c46d6725ec9eaf5fb93eae33c9112d01ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.191ex; height:2.843ex;" alt="{\displaystyle K_{X_{i}}}" loading="lazy"></span> nef. However, the process may encounter difficulties: at some point the variety <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> may become "too singular". The conjectural solution to this problem is the <a href="Flip_(algebraic_geometry)" title="Flip (algebraic geometry)">flip</a>, a kind of codimension-2 surgery operation on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{i}}</annotation>

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span>. It is not clear that the required flips exist, nor that they always terminate (that is, that one reaches a minimal model <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X'}</annotation>

</semantics>

</math></span><img src="./865f8505e90120a535a4ee68ca253dbd8ce7eb6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.682ex; height:2.509ex;" alt="{\displaystyle X'}" loading="lazy"></span> in finitely many steps). <a href="#CITEREFMori1988">Mori (1988)</a> showed that flips exist in the 3-dimensional case.

</p><p>The existence of the more general log flips was established by <a href="Vyacheslav_Shokurov" title="Vyacheslav Shokurov">Vyacheslav Shokurov</a> in dimensions three and four. This was subsequently generalized to higher dimensions by <a href="Caucher_Birkar" title="Caucher Birkar">Caucher Birkar</a>, Paolo Cascini, <a href="Christopher_Hacon" title="Christopher Hacon">Christopher Hacon</a>, and <a href="James_McKernan" title="James McKernan">James McKernan</a> relying on earlier work of Shokurov and Hacon, and McKernan. They also proved several other problems including finite generation of log canonical rings and existence of minimal models for varieties of log general type.

</p><p>The problem of termination of log flips in higher dimensions remains the subject of active research.

</p>

<ul><li><a href="Abundance_conjecture" title="Abundance conjecture">Abundance conjecture</a></li>

<li><a href="Minimal_rational_surface" class="mw-redirect" title="Minimal rational surface">Minimal rational surface</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Note that the Kodaira dimension of an <i>n</i>-dimensional variety is either <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

</semantics>

</math></span><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"></span> or an integer in the range 0 to <i>n</i>.</span>

</li>

</ol></div></div>

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</style><cite id="CITEREFBirkarCasciniHaconMcKernan2010" class="citation cs2"><a href="Caucher_Birkar" title="Caucher Birkar">Birkar, Caucher</a>; Cascini, Paolo; <a href="Christopher_Hacon" title="Christopher Hacon">Hacon, Christopher</a>; <a href="James_McKernan" title="James McKernan">McKernan, James</a> (2010), "Existence of minimal models for varieties of log general type", <i><a href="Journal_of_the_American_Mathematical_Society" title="Journal of the American Mathematical Society">Journal of the American Mathematical Society</a></i>, <b>23</b> (2): <span class="nowrap">405–</span>468, <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/0610203">math/0610203</a></span>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010JAMS...23..405B">2010JAMS...23..405B</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-09-00649-3">10.1090/S0894-0347-09-00649-3</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2601039">2601039</a></cite></li>

<li><cite id="CITEREFClemensKollárMori1988" class="citation cs2"><a href="Herbert_Clemens" title="Herbert Clemens">Clemens, Herbert</a>; <a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a>; <a href="Shigefumi_Mori" title="Shigefumi Mori">Mori, Shigefumi</a> (1988), "Higher-dimensional complex geometry", <i>Astérisque</i> (166): 144 pp. (1989), <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0303-1179">0303-1179</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1004926">1004926</a></cite></li>

<li><cite id="CITEREFFujino2009" class="citation cs2">Fujino, Osamu (2009), "New developments in the theory of minimal models", <i>Sugaku</i>, <b>61</b> (2), Mathematical Society of Japan: <span class="nowrap">162–</span>186, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0039-470X">0039-470X</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2560253">2560253</a></cite></li>

<li><cite id="CITEREFKollár1987" class="citation cs2"><a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a> (1987), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bams/1183554173">"The structure of algebraic threefolds: an introduction to Mori's program"</a>, <i><a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, New Series, <b>17</b> (2): <span class="nowrap">211–</span>273, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-1987-15548-0">10.1090/S0273-0979-1987-15548-0</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9904">0002-9904</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0903730">0903730</a></cite></li>

<li><cite id="CITEREFKollár1989" class="citation cs2"><a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a> (1989), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=SB_1988-1989__31__303_0">"Minimal models of algebraic threefolds: Mori's program"</a>, <i>Astérisque</i> (177): <span class="nowrap">303–</span>326, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0303-1179">0303-1179</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1040578">1040578</a></cite></li>

<li><cite id="CITEREFKollár1996" class="citation cs2"><a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a> (1996), <i>Rational curves on algebraic varieties</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Berlin: Springer-Verlag, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-03276-3">10.1007/978-3-662-03276-3</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-08219-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1440180">1440180</a></cite></li>

<li><cite id="CITEREFKollárMori1998" class="citation cs2"><a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a>; <a href="Shigefumi_Mori" title="Shigefumi Mori">Mori, Shigefumi</a> (1998), <i>Birational geometry of algebraic varieties</i>, Cambridge Tracts in Mathematics, vol. 134, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511662560">10.1017/CBO9780511662560</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-63277-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1658959">1658959</a></cite></li>

<li><cite id="CITEREFMatsuki2002" class="citation cs2">Matsuki, Kenji (2002), <i>Introduction to the Mori program</i>, Universitext, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-5602-9">10.1007/978-1-4757-5602-9</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-98465-0</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1875410">1875410</a></cite></li>

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<li><cite id="CITEREFKawamata2001" class="citation cs2"><a href="Yujiro_Kawamata" title="Yujiro Kawamata">Kawamata, Yujiro</a> (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Mori_theory_of_extremal_rays">"Mori theory of extremal rays"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li></ul></div><div><div class="zim-footer">

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