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{
    "pk": 64874,
    "title": "Lazy tournaments and multidegrees of a projective embedding of \\(\\overline{M}_{0,n}\\)",
    "subtitle": null,
    "abstract": "We consider the (iterated) Kapranov embedding \\(\\Omega_n:\\overline{M}_{0,n+3} \\hookrightarrow \\mathbb{P}^1 \\times \\cdots \\times \\mathbb{P}^n\\), where \\(\\overline{M}_{0,n+3}\\) is the moduli space of stable genus \\(0\\) curves with \\(n+3\\) marked points. In 2020, Gillespie, Cavalieri, and Monin gave a recursion satisfied by the multidegrees of \\(\\Omega_n\\) and showed, using two combinatorial insertion algorithms on certain parking functions, that the total degree of \\(\\Omega_n\\) is \\((2n-1)!!=(2n-1)\\cdot (2n-3) \\cdots 5 \\cdot 3 \\cdot 1\\). In this paper, we give a new proof of this fact by enumerating each multidegree by a set of boundary points of \\(\\overline{M}_{0,n+3}\\), via an algorithm on trivalent trees that we call a lazy tournament. The advantages of this new interpretation are twofold: first, these sets project to one another under the forgetting maps used to derive the multidegree recursion. Second, these sets naturally partition the complete set of boundary points on \\(\\overline{M}_{0,n+2}\\), of which there are \\((2n-1)!!\\), giving an immediate proof of the total degree formula.\n \nMathematics Subject Classifications: 05E14, 14N10, 05C05, 14H10, 05A19, 05C85\n \nKeywords: Moduli spaces of curves, projective embeddings, multidegrees, trivalent trees",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Moduli spaces of curves"
        },
        {
            "word": "projective embeddings"
        },
        {
            "word": "multidegrees"
        },
        {
            "word": "trivalent trees"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/24j1h4dk",
    "frozenauthors": [
        {
            "first_name": "Maria",
            "middle_name": "",
            "last_name": "Gillespie",
            "name_suffix": "",
            "institution": "Department of Mathematics, Colorado State University, Fort Collins, CO, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Sean",
            "middle_name": "T.",
            "last_name": "Griffin",
            "name_suffix": "",
            "institution": "Department of Mathematics, University of California Davis, Davis, CA, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Jake",
            "middle_name": "",
            "last_name": "Levinson",
            "name_suffix": "",
            "institution": "Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada",
            "department": ""
        }
    ],
    "date_submitted": "2023-03-14T14:36:20Z",
    "date_accepted": "2023-03-14T14:36:20Z",
    "date_published": "2023-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64874/galley/49684/download/"
        }
    ]
}