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The Kodaira dimension of spaces of rational curves on low degree hypersurfaces
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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.
Forward citations
Cited by 2 Pith papers
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Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0
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.
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Terminal singularities of the moduli space of curves on low degree hypersurfaces and the circle method
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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