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The Kodaira dimension of spaces of rational curves on low degree hypersurfaces

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arxiv math/0305432 v1 pith:4XYH3POM submitted 2003-05-29 math.AG

classification math.AG
keywords degreecurvesdimensionhypersurfacehypersurfaceskodairamodulirational
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abstract

For a hypersurface in complex projective space $X\subset \PP^n$, we investigate the singularities and Kodaira dimension of the Kontsevich moduli spaces $\Kbm{0,0}{X,e}$ parametrizing rational curves of degree $e$ on $X$. If $d+e \leq n$ and $X$ is a general hypersurface of degree $d$, we prove that $\Kbm{0,0}{X,e}$ has only canonical singularities and we conjecture the same is true for the coarse moduli space $\kbm{0,0}{X,e}$.This investigation is motivated by the question of which Fano hypersurfaces are unirational.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0

    math.AG 2025-06 conditional novelty 8.0 of 10

    The irreducible components of genus-0 stable map spaces to toric varieties are indexed by stable decorated trees that maximize a combinatorial invariant d_GGG.

  2. Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method

    math.AG 2024-12 conditional novelty 8.0 of 10

    For large ambient dimension n and curve degree e, the moduli space of genus g degree e maps into a smooth degree d hypersurface has at worst terminal singularities.

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