Collections of Hypersurfaces Containing a Curve

Abstract

We consider the closed locus parameterizing k-tuples of hypersurfaces that have positive dimensional intersection and fail to intersect properly, and show in a large range of degrees that its unique irreducible component of maximal dimension consists of tuples of hypersurfaces whose intersection contains a line. We then apply our methods in conjunction with a known reduction to positive characteristic argument to find the unique component of maximal dimension of the locus of hypersurfaces with positive dimensional singular loci. We will also find the components of maximal dimension of the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected. Problem 1.2. Does Z have a unique component of maximal dimension, consisting of tuples (F 1 , . . . , F k ) of hypersurfaces all containing the same r − k + 1 dimensional linear space?The answer to Problem 1.2 is negative as it stands. For example, if r = 3 and the degrees are d 1 = 2, d 2 = 2, and d 3 = 100, then the locus of 3-tuples of hypersurfaces all containing the same line is codimension 103, while the second quadric will be equal to the first quadric in codimension 9. Even if the degrees are all equal, we can let r = 4, k = 2, and the degrees be d 1 = 2, d 2 = 2, where the two quadrics will contain a plane in codimension 16, but are equal in codimension 14.

Disconnected moduli spaces of stable bundles on surfaces

“…When 𝑛 > 5, it suffices to show that the codimension of the locus of forms (𝐷 1 , … , 𝐷 𝑛−3 ) that contain a curve in ℙ 𝑛−3 is greater than the dimension of the Grassmannian 𝔾(𝑛 − 3, 𝑛). This follows from the dimension counts in [26,Lemmas 4.2 and 4.3]. □…”

“…When $$n>5$$, it suffices to show that the codimension of the locus of forms $$(D_1, \dots , D_{n-3})$$ that contain a curve in $$\mathbb {P}^{n-3}$$ is greater than the dimension of the Grassmannian $$\mathbb {G}(n-3,n)$$. This follows from the dimension counts in [26, Lemmas 4.2 and 4.3].$$\Box$$…”

The enumerative geometry of cubic hypersurfaces: point and line conditions

“…, dim V(h) sing ≥ 1}. This space has been studied in [Sla15;Tse20]. In the case of cubic surfaces, [Suk20] gives a classification of the cubic surfaces with positive-dimensional singular locus: it consists of the reducible cubics (which forms a variety of dimension 12) and of the orbit closure under the PGL 4 -action of an irreducible cubic surface corresponding to [Suk20, Table 1, 6C], giving a variety of dimension 13.…”

On degenerate sections of vector bundles

“…We show that after applying a high enough twist, any maximal component of this locus consists entirely of sections vanishing along a subscheme of minimal degree. In fact, we will give a more refined description of this locus, which will allow us to deduce its limit in the Grothendieck ring of varieties.Previously, the author has obtained a quantative version of Corollary 1.3, where the results are cleaner when k = r [14]. Even in this special case of a total split vector bundle on projective space, there are easy counterexamples to the conclusion of Theorem 1.2 if we don't apply a large twist.…”

“…Previously, the author has obtained a quantative version of Corollary 1.3, where the results are cleaner when k = r [14]. Even in this special case of a total split vector bundle on projective space, there are easy counterexamples to the conclusion of Theorem 1.2 if we don't apply a large twist.…”