Terracini Loci for Maps
Let <i>X</i> be a smooth projective variety and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | AppliedMath |
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| Online Access: | https://www.mdpi.com/2673-9909/3/3/36 |
| _version_ | 1797581523000492032 |
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| author | Edoardo Ballico |
| author_facet | Edoardo Ballico |
| author_sort | Edoardo Ballico |
| collection | DOAJ |
| description | Let <i>X</i> be a smooth projective variety and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></mrow></semantics></math></inline-formula> a morphism birational onto its image. We define the Terracini loci of the map <i>f</i>. Most results are only for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">dim</mo><mi>X</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>⊊</mo><mi>X</mi></mrow></semantics></math></inline-formula> or solutions containing a prescribed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools. |
| first_indexed | 2024-03-10T23:05:45Z |
| format | Article |
| id | doaj.art-b4dc0dc3374a4fd0ac4d94ca0b17588d |
| institution | Directory Open Access Journal |
| issn | 2673-9909 |
| language | English |
| last_indexed | 2024-03-10T23:05:45Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj.art-b4dc0dc3374a4fd0ac4d94ca0b17588d2023-11-19T09:20:50ZengMDPI AGAppliedMath2673-99092023-09-013369070110.3390/appliedmath3030036Terracini Loci for MapsEdoardo Ballico0Department of Mathematics, University of Trento, 38123 Povo, TN, ItalyLet <i>X</i> be a smooth projective variety and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi mathvariant="double-struck">P</mi><mi>r</mi></msup></mrow></semantics></math></inline-formula> a morphism birational onto its image. We define the Terracini loci of the map <i>f</i>. Most results are only for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">dim</mo><mi>X</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. With this new and more flexible definition, it is possible to prove strong nonemptiness results with the full classification of all exceptional cases. We also consider Terracini loci with restricted support (solutions not intersecting a closed set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo>⊊</mo><mi>X</mi></mrow></semantics></math></inline-formula> or solutions containing a prescribed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>). Our definitions work both for the Zariski and the euclidean topology and we suggest extensions to the case of real varieties. We also define Terracini loci for joins of two or more subvarieties of the same projective space. The proofs use algebro-geometric tools.https://www.mdpi.com/2673-9909/3/3/36Terracini locicurvecurves in projective spacesjoins of varietiessecant varieties |
| spellingShingle | Edoardo Ballico Terracini Loci for Maps AppliedMath Terracini loci curve curves in projective spaces joins of varieties secant varieties |
| title | Terracini Loci for Maps |
| title_full | Terracini Loci for Maps |
| title_fullStr | Terracini Loci for Maps |
| title_full_unstemmed | Terracini Loci for Maps |
| title_short | Terracini Loci for Maps |
| title_sort | terracini loci for maps |
| topic | Terracini loci curve curves in projective spaces joins of varieties secant varieties |
| url | https://www.mdpi.com/2673-9909/3/3/36 |
| work_keys_str_mv | AT edoardoballico terracinilociformaps |