candidate_required_fields: rank dimension count

policy_visibility: public

prompt: Compute exactly ∫_Gr(2,4) c_4(Sym^3 S*) by torus localization. Return the rank, dimension, and integer count.

split: train


candidate_required_fields: rank dimension count

policy_visibility: public

prompt: Compute exactly ∫_Gr(2,5) c_6(Sym^5 S*) by torus localization. Return the rank, dimension, and integer count.

split: dev


candidate_required_fields: rank dimension count

policy_visibility: public

prompt: Compute exactly ∫_Gr(4,9) c_20(Sym^3 S*) by torus localization. Return the rank, dimension, and integer count.

split: eval


policy_visibility: public

prompt: For the supplied five quadratic jets, compute the exact 20×20 multiplication map, its kernel, and the quadratic obstruction from the supplied second-order term.

split: train


policy_visibility: public

prompt: Repeat the local Fano-section calculation for a scaled fifth jet and certify whether the obstruction is nonzero.

split: dev


policy_visibility: public

prompt: Compute the exact corank and second-order obstruction for the weighted fifth jet.

split: eval


policy_visibility: public

prompt: A line moves in a 4-dimensional family in P^3. For surfaces of degree 7, compute the line-incidence codimension.

split: train


policy_visibility: public

prompt: For a quadric threefold of index 3 with a 3-dimensional line family and d=8, compute the incidence codimension.

split: dev


policy_visibility: public

prompt: For a cubic threefold of index 2 with a 2-dimensional Fano surface of lines and d=9, compute the incidence codimension.

split: eval


policy_visibility: public

prompt: Use Koszul resolutions to compute the linear-strand source/target dimensions and the maximal-rank nonlinear Betti prediction.

split: train


policy_visibility: public

prompt: Compute the same Tor benchmark when both vertices and the edge are cut out only by linear equations.

split: dev


policy_visibility: public

prompt: Compute the maximal-rank cokernel prediction for two quadratic hypersurface coordinate rings glued along a linear-plus-quadratic complete intersection.

split: eval


policy_visibility: public

prompt: Normalize the raw free anticanonical degrees 2 and 3, compute the Apéry set and conductor, and test the stated bound 2.

split: train


policy_visibility: public

prompt: Compute the normalized numerical semigroup generated by 3 and 4, including its Apéry set, Frobenius number, conductor, and bound test.

split: dev


policy_visibility: public

prompt: For the product benchmark with raw degrees 6 and 7, compute the exact conductor and test the quadratic bound 36.

split: eval


policy_visibility: public

prompt: Verify the standard rank-two skew lattice, the span index of the two cycles, and the Picard-Lefschetz transvection in the first cycle.

split: train


policy_visibility: public

prompt: Compute the integral span index and verify the transvection for the supplied rank-four vanishing-cycle configuration.

split: dev


policy_visibility: public

prompt: Verify primitivity, pairing-one connectivity data, and an exact transvection for the rank-four configuration.

split: eval


policy_visibility: public

prompt: Verify the chain condition and integral homology of the one-cell CW chain complex for S^1.

split: train


policy_visibility: public

prompt: Compute the integral homology of the two-term complex Z --2--> Z.

split: dev


policy_visibility: public

prompt: Compute integral homology, including torsion, for the supplied three-term chain complex.

split: eval


policy_visibility: public

prompt: Verify that the supplied operator strictly raises a two-step filtration and compute the ideal nilpotence exponent.

split: train


policy_visibility: public

prompt: Verify strict filtration increase and compute the minimal exponent killing every product.

split: dev


policy_visibility: public

prompt: Compute the nilpotence exponent of the ideal generated by the two exact filtration-raising matrices.

split: eval


policy_visibility: public

prompt: Compute the torsion cokernel and regular-character count for independent contact orders 2 and 3.

split: train


policy_visibility: public

prompt: Compute the Smith invariants and character count for a cyclic order-four root-choice group.

split: dev


policy_visibility: public

prompt: Distinguish the order-four torsion group with invariant factors 2,2 from the cyclic order-four case.

split: eval


policy_visibility: public

prompt: Compute the Hodge-line exponents p^h-1 and their product for p=3 and h=1,2,3.

split: train


policy_visibility: public

prompt: Compute the first two higher-Hasse weights and cycle coefficient for p=5.

split: dev


policy_visibility: public

prompt: Compute all ten odd-characteristic Hasse weights at p=3, their exact product, and state the terminal supersingular multiplicity used by the task.

split: eval


policy_visibility: public

prompt: For p=5 and ramification index e=3, determine tameness and the p-completed relative logarithmic differential module.

split: train


policy_visibility: public

prompt: For p=5 and e=5, compute the p-primary annihilator and whether the dlog class is nonzero.

split: dev


policy_visibility: public

prompt: For p=3 and e=9, compute the exact p-adic valuation, tameness, and p-primary annihilator of the Kummer dlog class.

split: eval