Bigness of Very Moving Algebraic Cycles

Definitions and mathematical statement

Let $Y$ be smooth projective of dimension $n$ and let $W\subset Y$ have codimension $k$. Spell out the source's very-moving condition as follows: a deformation family $\{W_t\}$ of $W$ can realize a generic pair \[ (y,E),\qquad y\in Y,\quad E\subset T_yY,\quad\dim E=n-k, \] with $y$ a smooth point of some $W_t$ and $T_yW_t=E$. In an algebraic-family formulation, the incidence Gauss map from smooth points of the family to $\operatorname{Gr}_{n-k}(T_Y)$ is dominant. Merely having deformations cover $Y$ omits the prescribed-tangent condition and is not the stated hypothesis.

Let $V^k(Y)$ be the finite-dimensional real cycle-class space intended in the source, and let \[ \operatorname{Eff}^k(Y)= \left\{\sum_{j=1}^{s}a_j[Z_j]:a_j\geq0,\ Z_j\text{ effective codimension-}k\text{ cycles}\right\}. \] The conjecture asks whether \[ W\text{ very moving}\quad\Longrightarrow\quad [W]\in\operatorname{int}_{V^k(Y)}\operatorname{Eff}^k(Y). \] This is the supplied meaning of bigness: interior positivity, not just membership in the effective cone.

The extract does not specify the ground field or the cycle-class quotient and ambient topology. Common settings include the numerical class space $N^k(Y)_{{\mathbb{R}}}$ and, over ${\mathbb{C}}$, the real space of algebraic cohomology classes. These must not be silently identified: taking interior in all cohomology, rather than the intended algebraic-class span, can change the condition. The exact deformation-family and genericity convention also needs confirmation. No smoothness of $W$ everywhere or restriction to complete intersections is added.

Short English statement

Does a cycle whose deformations realize a generic point and a generic tangent space have class in the interior of the effective cone? The tangent-space requirement is essential, and the ambient numerical or cohomological class space must be specified.

Sources

• Supplied report: Classical Algebraic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/26, pp. 1425--1474; proposed year 2008. https://doi.org/10.4171/owr/2008/26.
• Supplied reference: Voisin, Coniveau 2 complete intersections, https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/gafa.pdf.
• Local UnsolvedMath registry, permanent ID 30000969, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note describes editorial cleanup. The supplied 21 August 2026 literature triage labels the very-moving bigness assertion open, reporting special cases and restricted numerical ranges rather than a general proof. No PDF access or independent updated assessment is asserted here.