Degree of irrationality of very general abelian surfaces

Chen, Nathan

doi:10.2140/ant.2019.13.2191

Cited by 6 publications

(6 citation statements)

References 15 publications

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“…Proof of Corollary D. The quotient map from A to the Kummer surface A/ ± 1 has degree 2, so it suffices to prove irr(A/ ± 1) = 2. The main result of [4] can be rephrased as saying that a very general Kummer surface has degree of irrationality equal to 2. It follows that one can put A/ ± 1 in a family over a curve such that the very general member has degree of irrationality equal to 2.…”

Section: Maps To Ruled Varieties Specializementioning

confidence: 99%

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Fano Hypersurfaces With Arbitrarily Large Degrees of Irrationality

Chen

Stapleton

2020

Forum of Mathematics, Sigma

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We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index $e$ , then the degree of irrationality of a very general complex Fano hypersurface of index $e$ and dimension n is bounded from below by a constant times $\sqrt{n}$ . To our knowledge, this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic $p$ argument, which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way, we show that in a family of varieties, the invariant ‘the minimal degree of a dominant rational map to a ruled variety’ can only drop on special fibers. As a consequence, we show that for certain low-dimensional families of varieties, the degree of irrationality also behaves well under specialization.

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Section: Maps To Ruled Varieties Specializementioning

confidence: 99%

“…By work of the first author [4], it is known that a very general abelian surface A has irr(A) 4. From Proposition C, we are able to deduce: COROLLARY D. Every complex abelian surface A has irr(A) 4.…”

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confidence: 99%

Fano Hypersurfaces With Arbitrarily Large Degrees of Irrationality

Chen

Stapleton

2020

Forum of Mathematics, Sigma

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“…The case when X is a curve leads to the classical notion of gonality, whereas for dim(X) ≥ 2 very little is known in general and the problem has recently received considerable attention. The main results in this direction are for hypersurfaces [5,6,9,21,25] or abelian varieties [2,8,11,15,16,20,22,23,26] or hyper-Kähler varieties [24] or more specific examples [1,13]. Giving a sharp lower bound is in general a very difficult problem.…”

Section: Introductionmentioning

confidence: 99%

The polarized degree of irrationality of $K3$ surfaces

Moretti¹

2023

Preprint

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Given a polarized variety (X, L) we use an elementary observation to produce projections of low degree X ⊂ P(H 0 (L ∨ ))P r via vector bundles. In the case of surfaces this observation can be used to show that maps of the degree at most d move in families. In the case of polarized K3 surfaces we study the most special projections up to genus 6 and we give a new upper bound for higher genera. As a different application of our construction we exhibit new rational map of low degree for some hyper-Kähler varieties, abelian varieties and Gushel-Mukai threefolds.

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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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Degree of irrationality of very general abelian surfaces

Cited by 6 publications

References 15 publications

Fano Hypersurfaces With Arbitrarily Large Degrees of Irrationality

Fano Hypersurfaces With Arbitrarily Large Degrees of Irrationality

The polarized degree of irrationality of $K3$ surfaces

Cones of lines having high contact with general hypersurfaces and applications

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