[ { "id": "authors:rxncd-17d86", "collection": "authors", "collection_id": "rxncd-17d86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190510-105529171", "type": "article", "title": "Local Gromov-Witten invariants are log invariants", "author": [ { "family_name": "van Garrel", "given_name": "Michel", "clpid": "van-Garrel-M" }, { "family_name": "Graber", "given_name": "Tom", "clpid": "Graber-T-B" }, { "family_name": "Ruddat", "given_name": "Helge", "clpid": "Ruddat-H" } ], "abstract": "We prove a simple equivalence between the virtual count of rational curves in the total space of an anti-nef line bundle and the virtual count of rational curves maximally tangent to a smooth section of the dual line bundle. We conjecture a generalization to direct sums of line bundles.", "doi": "10.1016/j.aim.2019.04.063", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "2019-07-09", "volume": "350", "pages": "860-876" }, { "id": "authors:z00cp-dx986", "collection": "authors", "collection_id": "z00cp-dx986", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140130-082331813", "type": "book_section", "title": "Restriction of Sections for Families of Abelian Varieties", "book_title": "A Celebration of Algebraic Geometry", "author": [ { "family_name": "Graber", "given_name": "Tom", "clpid": "Graber-T-B" }, { "family_name": "Starr", "given_name": "Jason Michael", "clpid": "Starr-J-M" } ], "contributor": [ { "family_name": "Hassett", "given_name": "Brendan", "clpid": "Hassett-B" }, { "family_name": "McKernan", "given_name": "James", "clpid": "McKernan-J" }, { "family_name": "Starr", "given_name": "Jason", "clpid": "Starr-J" }, { "family_name": "Vakil", "given_name": "Ravi", "clpid": "Vakil-R" } ], "abstract": "Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base.", "isbn": "978-0-8218-8983-1", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2013", "pages": "311-327" }, { "id": "authors:yk2a8-1th60", "collection": "authors", "collection_id": "yk2a8-1th60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:ABRajm08", "type": "article", "title": "Gromov-Witten theory of Deligne-Mumford stacks", "author": [ { "family_name": "Abramovich", "given_name": "Dan", "clpid": "Abramovich-D" }, { "family_name": "Graber", "given_name": "Tom", "clpid": "Graber-T-B" }, { "family_name": "Vistoli", "given_name": "Angelo", "clpid": "Vistoli-A" } ], "abstract": "Given a smooth complex Deligne-Mumford stack X with a projective coarse moduli space. we introduce Gromov-Witten invariants of X and prove some of their basic properties, including the WDVV equation.", "issn": "0002-9327", "publisher": "Johns Hopkins University Press", "publication": "American Journal of Mathematics", "publication_date": "2008-10", "series_number": "5", "volume": "130", "issue": "5", "pages": "1337-1398" }, { "id": "authors:egwpq-w6310", "collection": "authors", "collection_id": "egwpq-w6310", "cite_using_url": "https://authors.library.caltech.edu/records/egwpq-w6310", "type": "article", "title": "The orbifold quantum cohomology of \u2102\u00b2/\u2124\u2083 and Hurwitz-Hodge integrals", "author": [ { "family_name": "Bryan", "given_name": "Jim" }, { "family_name": "Graber", "given_name": "Tom", "clpid": "Graber-T-B" }, { "family_name": "Pandharipande", "given_name": "Rahul", "orcid": "0009-0003-7258-5570" } ], "abstract": "

Let \u2124\u2083 act on \u2102² by non-trivial opposite characters. Let X = [\u2102²/\u2124\u2083] be the orbifold quotient, and let Y be the unique crepant resolution. We show that the equivariant genus 0 Gromov-Witten potentials F^x and F^y are equal after a change of variables—verifying the Crepant Resolution Conjecture for the pair (X,Y). Our computations involve Hodge integrals on trigonal Hurwitz spaces, which are of independent interest. In a self-contained Appendix, we derive closed formulas for these Hurwitz-Hodge integrals.

", "doi": "10.1090/s1056-3911-07-00467-5", "issn": "1056-3911", "publisher": "American Mathematical Society (AMS)", "publication": "Journal of Algebraic Geometry", "publication_date": "2008-01-01", "series_number": "1", "volume": "17", "issue": "1", "pages": "1-28" }, { "id": "authors:w8vaa-7jk64", "collection": "authors", "collection_id": "w8vaa-7jk64", "cite_using_url": "https://authors.library.caltech.edu/records/w8vaa-7jk64", "type": "article", "title": "Descendant invariants and characteristic numbers", "author": [ { "family_name": "Graber", "given_name": "Tom", "clpid": "Graber-T-B" }, { "family_name": "Kock", "given_name": "Joachim" }, { "family_name": "Pandharipande", "given_name": "Rahul", "clpid": "Pandharipande-Rahul" } ], "abstract": "

On a stack of stable maps, the cotangent line classes are modified by subtracting certain boundary divisors. These modified cotangent line classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow simple expressions in terms of modified cotangent line classes. Topological recursion relations are established among their top products in genus 0, yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety. In higher genus, the obtained numbers are only virtual, due to contributions from spurious components of the space of maps. For the projective plane, the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases.

", "doi": "10.1353/ajm.2002.0014", "issn": "1080-6377", "publisher": "Project MUSE", "publication": "American Journal of Mathematics", "publication_date": "2002-06", "series_number": "3", "volume": "124", "issue": "3", "pages": "611-647" } ]