On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content>
In this paper, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow>...
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2024-12-01
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| author | Olivia Dumitrescu Rick Miranda |
| author_facet | Olivia Dumitrescu Rick Miranda |
| author_sort | Olivia Dumitrescu |
| collection | DOAJ |
| description | In this paper, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> in the blown-up projective space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></semantics></math></inline-formula> in general points. The notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves in order to construct a counterexample to Hilbert’s 14th problem. We introduce the notion of classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>- and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></semantics></math></inline-formula> with <i>s</i> points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, <i>F</i> (which we will refer to as the <i>anticanonical curve class</i>). For Mori Dream Spaces, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>- and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Weyl lines give the extremal rays for the cone of movable curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula> points blown up. As an application, we use the technique of movable curves to reprove that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>F</mi><mn>2</mn></msup><mo>≤</mo><mn>0</mn></mrow></semantics></math></inline-formula> then <i>Y</i> is not a Mori Dream Space, and we propose to apply this technique to other spaces. |
| format | Article |
| id | doaj-art-7bf4d65e854a4b43a9cfd6e6a830cc00 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-7bf4d65e854a4b43a9cfd6e6a830cc002024-12-27T14:38:07ZengMDPI AGMathematics2227-73902024-12-011224395210.3390/math12243952On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content>Olivia Dumitrescu0Rick Miranda1Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USADepartment of Mathematics, Colorado State University, Fort Collins, CO 80523, USAIn this paper, we study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></semantics></math></inline-formula> in the blown-up projective space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></semantics></math></inline-formula> in general points. The notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves was analyzed in the early days of mirror symmetry by Kontsevich, with the motivation of counting curves on a Calabi–Yau threefold. In dimension two, Nagata studied planar <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves in order to construct a counterexample to Hilbert’s 14th problem. We introduce the notion of classes of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>- and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></semantics></math></inline-formula> with <i>s</i> points blown up, and we prove that their number is finite if and only if the space is a Mori Dream Space. We further introduce a bilinear form on a space of curves and a unique symmetric Weyl-invariant class, <i>F</i> (which we will refer to as the <i>anticanonical curve class</i>). For Mori Dream Spaces, we prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-curves can be defined arithmetically by the linear and quadratic invariants determined by the bilinear form. Moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>- and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-Weyl lines give the extremal rays for the cone of movable curves in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>+</mo><mn>3</mn></mrow></semantics></math></inline-formula> points blown up. As an application, we use the technique of movable curves to reprove that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>F</mi><mn>2</mn></msup><mo>≤</mo><mn>0</mn></mrow></semantics></math></inline-formula> then <i>Y</i> is not a Mori Dream Space, and we propose to apply this technique to other spaces.https://www.mdpi.com/2227-7390/12/24/3952projective geometrybirational geometrycurvesrational curves |
| spellingShingle | Olivia Dumitrescu Rick Miranda On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content> Mathematics projective geometry birational geometry curves rational curves |
| title | On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content> |
| title_full | On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content> |
| title_fullStr | On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content> |
| title_full_unstemmed | On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content> |
| title_short | On (<i>i</i>)-Curves in Blowups of <named-content content-type="inline-formula"><inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">P</mi><mi mathvariant="bold-italic">r</mi></msup></semantics></math></inline-formula></named-content> |
| title_sort | on i i i curves in blowups of named content content type inline formula inline formula math display inline semantics msup mi mathvariant double struck p mi mi mathvariant bold italic r mi msup semantics math inline formula named content |
| topic | projective geometry birational geometry curves rational curves |
| url | https://www.mdpi.com/2227-7390/12/24/3952 |
| work_keys_str_mv | AT oliviadumitrescu oniiicurvesinblowupsofnamedcontentcontenttypeinlineformulainlineformulamathdisplayinlinesemanticsmsupmimathvariantdoublestruckpmimimathvariantbolditalicrmimsupsemanticsmathinlineformulanamedcontent AT rickmiranda oniiicurvesinblowupsofnamedcontentcontenttypeinlineformulainlineformulamathdisplayinlinesemanticsmsupmimathvariantdoublestruckpmimimathvariantbolditalicrmimsupsemanticsmathinlineformulanamedcontent |