[Paper Review] Genus of curves in generic hypersurfaces

Bin Wang|arXiv (Cornell University)|Oct 2, 2011

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper introduces a new obstruction formula for the first-order deformation of a smooth curve in a smooth hypersurface in projective space, leading to a genus formula and proving that smooth elliptic curves cannot exist in generic hypersurfaces of degree $ h /geq 2n-1 $ in $ \mathbf{P}^n $ for $ n \geq 3 $. The result improves Clemens' bound and reveals subtle geometric obstructions in deformation theory of pairs of varieties.

ABSTRACT

This is the continuation of our paper [10]. In this paper which is self contained, we would like to give a different obstruction formula to the FIRST order deformation of the pair of a smooth curve and a smooth hypersurface. This obstruction formula leads to a genus formula for a smooth curve in a smooth hypersurface. As an application we show that smooth elliptic curves in a smooth hypersurface of degree $$h\geq 2n-1$$ in the projective space $\mathbf P^n, n\geq 3,$ can't deform in the first order to all hypersurfaces of the same degree. In particular, there are no smooth elliptic curves in generic hypersurfaces of degree $$h\geq 2n-1.$$ This application in return leads to a study of the deformation of the pair mentioned above.

Motivation & Objective

• To develop a new obstruction formula for the first-order deformation of a smooth curve in a smooth hypersurface in $ \mathbf{P}^n $, distinct from prior work.
• To derive a genus formula for smooth curves embedded in smooth hypersurfaces via this new obstruction condition.
• To investigate the geometric distinction between first-order deformation of a pair and full deformation in the moduli space of hypersurfaces.
• To establish that smooth elliptic curves cannot deform to all hypersurfaces of degree $ h \geq 2n-1 $, implying their non-existence in generic such hypersurfaces.

Proposed method

• Formalizes the first-order deformation of the pair $ C_0 \subset f_0 $ using the map $ P_S^s: H^0(\bar{c}_0^*(TX_S)) \to T_{f_0}S $, representing infinitesimal deformations of the pair.
• Considers two parameter spaces: $ S = \mathbf{C}^{h+1} $ parametrizing hypersurfaces of the form $ f_0 + \sum a_i L_0 \cdots \hat{L}_i \cdots L_h $, and $ S = \mathbf{P}(H^0(\mathcal{O}_{\mathbf{P}^n}(h))) $, the full space of degree-$ h $ hypersurfaces.
• Constructs a subspace $ E_q \subset T_{f_0}|_q $ of codimension 2 at a point $ q \in C_0 $, such that certain monomials $ <L_0L\cdots L> $ lie in $ E_q $, but $ <L_0JL\cdots L> $ does not.
• Uses local analytic parametrization via $ U_i = \{ L + x_i J \} $ to define a subvariety $ S_{E_q,\epsilon} $, showing it is smooth and irreducible near $ (L_0, L, \dots, L) $ via non-vanishing Jacobian determinant.
• Applies $ GL(n+1) $-symmetry to construct sections in $ H^0(c_0^*(N_{C_0}f_0(1))) $ not lying in the image of the obstruction map, proving dimension inequality $ \dim H^0(c_0^*(N_{C_0}f_0(1))) > \dim B $.
• Establishes that if $ L_0 <L_1 \cdots L_h> - L_k <L_0L_1 \cdots \hat{L}_k \cdots L_h> \notin E_q $ at $ q $, then the corresponding section is not in the obstruction subspace $ B $, implying non-vanishing obstruction.

Experimental results

Research questions

• RQ1 What is the necessary condition (obstruction) for a smooth curve $ C_0 $ to deform in the first order within a family of smooth hypersurfaces of fixed degree in $ \mathbf{P}^n $?
• RQ2 How does the new obstruction formula differ from the one in [9], and what geometric insight does it provide about the deformation of pairs $ (C_0, f_0) $?
• RQ3 Under what conditions on the degree $ h $ and dimension $ n $ do smooth elliptic curves fail to exist in generic hypersurfaces of degree $ h $ in $ \mathbf{P}^n $?
• RQ4 What is the precise relationship between the vanishing of $ H^1(N_{C_0}f_0) $ and the existence of full deformations of the pair $ C_0 \subset f_0 $, especially when $ H^1(N_{C_0}f_0) \neq 0 $?
• RQ5 Can the dimension of the space of first-order deformations of the pair be strictly larger than the dimension of the obstruction subspace $ B $, and what does this imply for the existence of such curves?

Key findings

• A new obstruction formula for first-order deformation of a smooth curve in a smooth hypersurface is derived, differing from the one in [9], and applicable to the study of deformation obstructions in pairs of varieties.
• The genus of a smooth curve in a smooth hypersurface is shown to depend on the geometry of the embedding and the obstruction structure, leading to a genus formula via the new obstruction condition.
• It is proven that smooth elliptic curves cannot exist in generic hypersurfaces of degree $ h \geq 2n - 1 $ in $ \mathbf{P}^n $ for $ n \geq 3 $, improving Clemens' bound by one.
• The obstruction to first-order deformation is non-vanishing when $ h \geq 2n - 1 $, implying that such curves cannot deform to all nearby hypersurfaces of the same degree.
• The dimension of $ H^0(c_0^*(N_{C_0}f_0(1))) $ exceeds that of the obstruction subspace $ B $, showing that the space of first-order deformations is strictly larger than the image of the obstruction map.
• The result reveals a subtle geometric distinction in deformation theory: even when $ H^1(N_{C_0}f_0) \neq 0 $, the pair may still fail to deform due to higher-order obstructions, challenging the necessity of Kodaira’s vanishing condition.