On irrationality of surfaces in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">P</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup></mml:math>

Bastianelli, Francesco

doi:10.1016/j.jalgebra.2017.06.020

Cited by 8 publications

(14 citation statements)

References 25 publications

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“…The purpose of this note is to study various measures of irrationality for hypersurfaces in projective spaces which were proposed recently by [4], [1]. In particular, we answer the question raised by Bastianelli that if X ⊂ P n+1 is a very general smooth hypersurface of dimension n and degree d ≥ 2n + 2, then stab.irr(X) = uni.irr(X) = d − 1.…”

Section: Introductionmentioning

confidence: 92%

“…, whose degeneracy locus is thus given by a linear form of degree n on P 1 . Since a general fiber doesn't lie in the ramification locus, we must have l · K W ′ /P = n. 1 Notice that even though we are working on an open variety, this intersection product still makes sense because we are intersecting a divisor with the fiber of a proper map. 2 Bastianelli pointed out to me that it is possible to avoid this assertion by passing to the Grassmannian and argue as in [4].…”

Section: One Finds Thatmentioning

confidence: 99%

“…where π 1 is the first projection map from Γ to P n and Γ is any subvariety of P n × X that both dominates P n and X. 1 Our first results concern corr(X):…”

Section: Introductionmentioning

confidence: 96%

“…There has been recent interest in studying measures of irrationality for algebraic varieties [4], [1]. For example, given an irreducible projective variety X of dimension n, the degree of irrationality of X is defined as irr(X) := min {δ > 0 | ∃ degree δ rational covering X P n }.…”

Section: Introductionmentioning

confidence: 99%

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On irrationality of hypersurfaces in $\mathbf {P}^{n+1}$

Yang

2018

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 92%

Section: One Finds Thatmentioning

confidence: 99%

“…where π 1 is the first projection map from Γ to P n and Γ is any subvariety of P n × X that both dominates P n and X. 1 Our first results concern corr(X):…”

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

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On irrationality of hypersurfaces in $\mathbf {P}^{n+1}$

Yang

2018

Proc. Amer. Math. Soc.

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“…The connecting gonality of very general surfaces X ⊂ P 3 of degree d ≥ 5 is computed by tangent hyperplane sections X ∩ T p X, so that conn. gon(X) = d − 2 (see e.g. [1]).…”

Section: K-irrationality Degree and Connecting Gonality Of General Hy...mentioning

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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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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On irrationality of surfaces in P3

Cited by 8 publications

References 25 publications

On irrationality of hypersurfaces in $\mathbf {P}^{n+1}$

On irrationality of hypersurfaces in $\mathbf {P}^{n+1}$

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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