Measures of irrationality of the Fano surface of a cubic threefold

Gounelas, Frank; Kouvidakis, Alexis

doi:10.1090/tran/7565

Cited by 8 publications

(7 citation statements)

References 22 publications

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“…Proof. The proof is analogue to the one of [GK,Lemma 2.1]. We consider the Hilbert scheme H of all 1-dimensional subschemes of X, which is not of finite type, but has countably many components.…”

Section: G Gonality and Covering Gonalitymentioning

confidence: 97%

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Quotients of higher-dimensional Cremona groups

2021

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Section: G Gonality and Covering Gonalitymentioning

confidence: 97%

“…gon(X) to get a dominant map to X. We then look at the set of gonality i, for each i, and obtain algebraic varieties parameterising these, as in [GK,Lemma 2.1]. Having finitely many constructible subsets in the image, at least one integer i⩽cov.…”

Section: G Gonality and Covering Gonalitymentioning

confidence: 99%

Quotients of higher-dimensional Cremona groups

2021

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“…In recent years, there has been a great deal of interest concerning measures of irrationality of projective varieties, that is birational invariants, which somehow measure the failure of a given variety to be rational (see, e.g., [3,5,8,10,16,18,19]), and several interesting results have been obtained in this direction for very general hypersurfaces of large degree (cf. [2][3][4]20]).…”

Section: Introductionmentioning

confidence: 99%

Cones of lines having high contact with general hypersurfaces and applications

Bastianelli

Ciliberto

Flamini

et al. 2022

Mathematische Nachrichten

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Given a smooth hypersurface X⊂Pn+1$X\subset \mathbb {P}^{n+1}$ of degree d⩾2$d\geqslant 2$, we study the cones Vph⊂Pn+1$V^h_p\subset \mathbb {P}^{n+1}$ swept out by lines having contact order h⩾2$h\geqslant 2$ at a point p∈X$p\in X$. In particular, we prove that if X is general, then for any p∈X$p\in X$ and 2⩽h⩽minfalse{n+1,dfalse}$2 \leqslant h\leqslant \min \lbrace n+1,d\rbrace$, the cone Vph$V^h_p$ has dimension exactly n+2−h$n+2-h$. Moreover, when X is a very general hypersurface of degree d⩾2n+2$d\geqslant 2n+2$, we describe the relation between the cones Vph$V^h_p$ and the degree of irrationality of k‐dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k‐dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies d−⌊⌋16n+25−32⩽prefixconn.gon(X)⩽d−⌊⌋8n+1+12$d-\left\lfloor \frac{\sqrt {16n+25}-3}{2}\right\rfloor \leqslant \operatorname{conn.gon}(X)\leqslant d-\left\lfloor \frac{\sqrt {8n+1}+1}{2}\right\rfloor$.

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“…In recent years there has been a great deal of interest concerning measures of irrationality of projective varieties, that is birational invariants which somehow measure the failure of a given variety to be rational (see e.g. [3,5,15,8,10,16,17]), and several interesting results have been obtained in this direction for very general hypersurfaces of large degree (cf. [2,3,4,18]).…”

Section: Introductionmentioning

confidence: 99%

Cones of lines having high contact with general hypersurfaces and applications

Bastianelli¹,

Ciliberto²,

Flamini³

et al. 2020

Preprint

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Given a smooth hypersurface X ⊂ P n+1 of degree d 2, we study the cones V h p ⊂ P n+1 swept out by lines having contact order h 2 at a point p ∈ X. In particular, we prove that if X is general, then for any p ∈ X and 2 h min{n + 1, d}, the cone V h p has dimension exactly n+2−h. Moreover, when X is a very general hypersurface of degree d 2n+2, we describe the relation between the cones V h p and the degree of irrationality of k-dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k-dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies d− √ 16n+25−3 2 conn. gon(X) d− √ 8n+1+1 2.

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Measures of irrationality of the Fano surface of a cubic threefold

Cited by 8 publications

References 22 publications

Quotients of higher-dimensional Cremona groups

Quotients of higher-dimensional Cremona groups

Cones of lines having high contact with general hypersurfaces and applications

Cones of lines having high contact with general hypersurfaces and applications

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