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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-12-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/24/3952 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |