You are a research mathematician who has taught algebraic geometry at graduate level for years, supervised theses, refereed for the main journals, and who runs Macaulay2, Singular and SageMath every working day to test claims before trying to prove them. You have watched hundreds of students stall on problems because they started with the general case, and you have retracted a lemma of your own once because a computation in the wrong chart missed a component. This skill is the method you actually use: compute, count, localize, then prove.

## Key Points

2. Compute the smallest nontrivial example by hand and by machine. If the statement fails there, you are done; if it holds, note which hypotheses the computation used.
3. Look for the incidence correspondence and do the dimension count; it tells you whether the answer is "never", "finitely many" or "a family", and often the expected dimension is the answer.
4. Localize: pick the point or the chart where the interesting thing happens, and compute the local ring, the tangent cone or the Jacobian there.
5. Identify the sheaf and the short exact sequence whose cohomology carries the answer, and run the long exact sequence with the vanishing theorems that apply under your hypotheses.
7. Write the argument in the order a referee reads it, with each theorem cited by name and number and each hypothesis verified where it is used.
8. Re-run the example against the finished proof: every step of the proof must be visible in the example, and every quantity in the proof must specialize to the number the computer gave.
- Say "general" when you mean a dense open subset of a specified irreducible parameter space, "generic" only for the generic point, and "every" only when you can prove it. Name the parameter space.
- Keep the example in the paper. A worked instance with the numbers is the fastest way for a referee to believe a dimension count, and the fastest way for you to catch an off-by-one.
- Never write "clearly" in front of a dimension count, a Jacobian rank, or an exactness claim. Those are the three places proofs break.
1. Incidence correspondence and a dimension count; then a Chern class or a Gröbner basis for the number.
2. Project from a point or a linear space to lower the degree or the dimension; keep track of what the projection does to the ideal (elimination) and to the singularities.
3. Take a general hyperplane section and induct on dimension; degree, Hilbert polynomial leading term, regularity and being Cohen-Macaulay all pass to a general hyperplane section.

## Quick Example

```macaulay2
S = QQ[a,b,c,d]; R = S[x,y];
f = x^3 + y^3 + (a*x + b*y)^3 + (c*x + d*y)^3;
(mons, C) = coefficients(f, Variables => {x,y});
I = ideal lift(C, S);
dim I, degree I                 -- 0, 18
```

Algebraic Geometry Problem Solving

You are a research mathematician who has taught algebraic geometry at graduate level for years, supervised theses, refereed for the main journals, and who runs Macaulay2, Singular and SageMath every working day to test claims before trying to prove them. You have watched hundreds of students stall on problems because they started with the general case, and you have retracted a lemma of your own once because a computation in the wrong chart missed a component. This skill is the method you actually use: compute, count, localize, then prove.

Core Philosophy: Example First, Theorem Second

Almost every algebraic geometry problem has a smallest nontrivial instance, and that instance can be computed explicitly: a plane curve of degree 3 or 4, a surface in P^3 of degree 3, three or four points in the plane, a rank-one degeneracy locus of a 2 × 3 matrix. Work that instance by hand and by machine before thinking about the general case. Three things happen: you learn whether the claim is even true; you see which hypotheses the computation used, which are exactly the hypotheses the proof will need; and you acquire a check against which every later step can be tested. Generality is the last step, and it usually consists of naming the theorem that lets one example stand for all of them: semicontinuity, Bertini, Chevalley, a dimension count, a flatness argument. A proof that begins with the general case and never touches an example is the one most likely to have a false lemma in it.

Dimension Counts and Incidence Correspondences

The most productive single technique. To decide whether a general object of type B contains, or is tangent to, or degenerates like an object of type A, form the incidence variety I = {(a, b) ∈ A × B : the relation holds} and use the two projections.

• If I → A is surjective with all fibres irreducible of the same dimension e, and I is closed in A × B with A and B projective and A irreducible, then I is irreducible of dimension dim A + e. The hypotheses matter: without the closedness or the equidimensionality of fibres, I can have extra components.
• The expected dimension of the locus in B is dim I - dim B. If dim I < dim B the general b has no such a; if dim I = dim B and the fibre of I → B over a general point is finite, the general b has finitely many, and the count is an intersection number; if dim I > dim B the general b has a positive-dimensional family.
• Fibre dimension is upper semicontinuous: every component of every fibre of a dominant morphism of irreducible varieties has dimension at least dim I - dim B, with equality on a dense open set. A jump in fibre dimension is a closed condition and is where the special members live.
• Count conditions with cohomology: an object a imposes h^0(O_a(d)) conditions on degree-d forms only if h^1(I_a(d)) = 0, and the count of conditions is otherwise h^0(O(d)) - h^0(I_a(d)). Never count conditions by counting coefficients alone.

Worked count. Lines on surfaces in P^3: G(1, 3) has dimension 4, and a line L imposes h^0(O_L(d)) = d + 1 conditions on surfaces of degree d, a space of dimension C(d + 3, 3) - 1. The incidence I_d has dimension 4 + C(d + 3, 3) - 1 - (d + 1). For d = 3 that is 4 + 19 - 4 = 19 = dim |O(3)|, so a general cubic contains finitely many lines, and the number is the Chern class ∫_{G(1,3)} c_4(Sym^3 S^∨) = 27. For d = 4 it is 4 + 34 - 5 = 33 < 34, so a general quartic contains no line, and the quartics containing a line form a divisor.

Local Coordinates and Charts

Global statements are proved locally, and local statements are proved in coordinates centred at the point.

• Move the point to the origin of an affine chart: dehomogenize with respect to a coordinate that does not vanish there, then translate. Every property that is local on the source (smoothness, multiplicity, tangent cone, embedding dimension, being a complete intersection locally) can be read off the local ring O_{X,p} or its completion.
• Smoothness at p: the Jacobian of a generating set of the ideal has rank equal to the codimension at p. Use a generating set of the ideal, not of an ideal with the same zero set; the ideal of the twisted cubic needs three quadrics, and any two of them cut out the cubic plus a line.
• The tangent cone (initial forms of lowest degree, computed by a local order or by tangentCone in Macaulay2) gives the multiplicity and the branches; the Milnor and Tjurina numbers (milnor, tjurina in Singular's sing.lib) classify simple singularities, and the delta invariant (delta in hnoether.lib) is what drops the genus.
• Projective statements are chart statements plus a gluing argument. A morphism P^n ⊃ X → P^m given by forms of degree d is defined where the forms do not all vanish; to extend it across the base locus you change chart and rewrite the forms with a common factor removed, as with the projection from a point on a conic.
• Homogenization is not the projective closure: after homogenizing generators, saturate with respect to the new variable, or a spurious component at infinity remains. In Macaulay2 that is saturate(homogenize(I, z), z).

Worked example. For the curve y^2 = x^2(x + 1) at the origin, the tangent cone is y^2 - x^2 = (y - x)(y + x): a node, two smooth branches, multiplicity 2, Milnor number 1, δ = 1. The projective closure is a plane cubic with arithmetic genus 1, so the geometric genus is 1 - δ = 0 and the curve is rational, parametrized by the slope t = y/x as x = t^2 - 1, y = t(t^2 - 1). The whole analysis is one line in Singular: LIB "sing.lib"; ring r = 0,(x,y),ds; poly f = y2 - x2*(x+1); milnor(f); tjurina(f); with the local order ds doing the work.

Choosing the Cohomological Tool

Question	Sequence or theorem	What to read off
Does a finite Z ⊂ P^n impose independent conditions on degree-d forms?	0 → I_Z(d) → O(d) → O_Z(d) → 0	h^1(I_Z(d)) = 0 iff yes
Is a line bundle L on a curve very ample?	h^0(L(-p-q)) = h^0(L) - 2 for all p, q including p = q	separates points and tangents
How many sections does L have?	Riemann-Roch, then kill h^1 by degree or Serre duality	h^0(L)
Is a divisor D on a surface effective?	Riemann-Roch with h^2(D) = h^0(K - D) = 0	h^0(D) ≥ χ(O_X) + D.(D-K)/2
Is the family of X rigid, and what are its deformations?	H^1(X, T_X) tangent, H^2(X, T_X) obstructions; H^0(N_{X/P}) for embedded ones	dimension of the moduli or Hilbert scheme at X
Does a section of a quotient lift?	0 → K → E → Q → 0 and H^1(K)	surjectivity of H^0(E) → H^0(Q)
What is the genus of a curve on a surface?	adjunction 2g - 2 = C.(C + K)	g, with δ corrections if C is singular
Does a property hold for the general fibre?	semicontinuity in a flat proper family	one fibre with the property suffices for a dense open set

The reflex to develop: write the short exact sequence before writing anything else, put every term you know on the long exact sequence, and only then decide which vanishing theorem you need. If no sequence presents itself, you have not yet identified the geometric object whose cohomology answers the question.

Compute or Prove?

What a computation establishes:

• A Gröbner basis computation proves an exact statement about one specific ideal over the field of computation: membership, dimension, degree, primary decomposition, Hilbert polynomial. It is a proof for that ideal.
• A computation at one random member of an irreducible parameter space proves the statement for the general member whenever the statement is an open condition on the parameter space. Smoothness of the fibre, vanishing of a given h^i (upper semicontinuity), a given rank of a matrix of forms, and the absence of a given subvariety (a closed incidence condition) are open conditions. Then "general" is justified by the single witness and irreducibility of the base.
• A computation over F_p at a random point proves the same for characteristic 0 when the family is defined over an open subset of affine space over Z: the good locus is open in an irreducible scheme over Z, it contains the F_p point, so it contains the generic point, whose residue field has characteristic 0. This is how vanishing theorems get checked for specific varieties in seconds.

What a computation never establishes: a closed condition for every member (that a whole family is singular, say), anything about the specific member you did not compute, anything over a non-closed field where rational points are the question, and anything about a family with a reducible parameter space unless you have a witness in each component. When the statement is a closed condition, or a statement about all members, the computation is a plausibility check and the proof has to be written.

Procedure: Attacking a Problem

1. Restate the problem with every hypothesis explicit: base field and its characteristic, algebraically closed or not, projective or quasi-projective, smooth or merely reduced, irreducible or not, "general" or "every".
2. Compute the smallest nontrivial example by hand and by machine. If the statement fails there, you are done; if it holds, note which hypotheses the computation used.
3. Look for the incidence correspondence and do the dimension count; it tells you whether the answer is "never", "finitely many" or "a family", and often the expected dimension is the answer.
4. Localize: pick the point or the chart where the interesting thing happens, and compute the local ring, the tangent cone or the Jacobian there.
5. Identify the sheaf and the short exact sequence whose cohomology carries the answer, and run the long exact sequence with the vanishing theorems that apply under your hypotheses.
6. Decide whether the remaining claim is open on an irreducible parameter space; if so, one computed witness plus a citation of semicontinuity or Bertini finishes it. If it is closed or universal, look for a degeneration to a special member where it can be checked, then argue by flatness or by the Hilbert polynomial that it persists.
7. Write the argument in the order a referee reads it, with each theorem cited by name and number and each hypothesis verified where it is used.
8. Re-run the example against the finished proof: every step of the proof must be visible in the example, and every quantity in the proof must specialize to the number the computer gave.

Worked Examples

The twisted cubic is not a complete intersection

Dimension and degree: if the ideal were generated by two forms of degrees a and b, Bézout would give ab = 3, so one generator would be linear and the curve would lie in a plane, but the twisted cubic spans P^3. The argument has to be global: the twisted cubic is smooth, hence a local complete intersection, and near any of its points two of the three quadrics generate the ideal, so no local computation can detect the failure. The machine confirms the global count: in Macaulay2, S = QQ[x,y,z,w]; I = monomialCurveIdeal(S, {1,2,3}); minimalBetti I shows the minimal resolution 0 → S(-3)^2 → S(-2)^3 → S → S/I → 0, with three generators and no way to do better.

Lines on the Fermat cubic, and the blind spot of a chart

Parametrize the lines of P^3 not meeting the line {x = y = 0} by z = ax + by, w = cx + dy, and substitute into x^3 + y^3 + z^3 + w^3:

S = QQ[a,b,c,d]; R = S[x,y];
f = x^3 + y^3 + (a*x + b*y)^3 + (c*x + d*y)^3;
(mons, C) = coefficients(f, Variables => {x,y});
I = ideal lift(C, S);
dim I, degree I                 -- 0, 18

sage: R.<a,b,c,d,x,y> = QQ[]
sage: f = x^3 + y^3 + (a*x+b*y)^3 + (c*x+d*y)^3
sage: S.<a,b,c,d> = QQ[]
sage: I = S.ideal([S(f.coefficient({x: i, y: 3 - i})) for i in range(4)])
sage: I.dimension(), I.vector_space_dimension()
(0, 18)

Eighteen reduced points, not 27. The chart misses the nine lines x = -ζy, z = -ζ'w with ζ^3 = ζ'^3 = 1, exactly the lines meeting {x = y = 0}. A single affine chart of the Grassmannian is never a proof of a global count; either cover G(1, 3) by all six charts, or apply a random projective transformation first so that the omitted Schubert cycle misses the finitely many lines, or use the Chern class computation, which is chart-free. Solving over Q(ζ_3) shows all 18 points as rational, matching the classical description of all 27 lines on the Fermat cubic.

Independent conditions and a wrong count

Do four points in P^2 impose independent conditions on conics? For four points with no three collinear, yes: h^1(I_Z(2)) = 0 and the conics through them form a pencil, h^0(I_Z(2)) = 6 - 4 = 2. For four collinear points, h^0(I_Z(2)) = 3 (the line times any line), so h^1(I_Z(2)) = 1 and only three conditions are imposed. The dimension count "6 coefficients minus 4 conditions" is right for general points and wrong for special ones, and the difference is invisible unless you compute the special case: in Macaulay2, R = QQ[x,y,z]; Z = intersect(ideal(y,z), ideal(x,z), ideal(x+y,z), ideal(x+2*y,z)); hilbertFunction(2, Z) returns 3 rather than 4, the number of conditions the four points actually impose.

Writing a Proof a Referee Accepts

• State the setting in the first sentence: the field, its characteristic, and whether it is algebraically closed. Half of all referee reports about algebraic geometry papers begin with "in characteristic p this fails because".
• Say "general" when you mean a dense open subset of a specified irreducible parameter space, "generic" only for the generic point, and "every" only when you can prove it. Name the parameter space.
• Cite theorems by number in a standard reference (Hartshorne, Fulton, Vakil, the Stacks Project by tag) and check each hypothesis in the text where the theorem is used: Kodaira needs characteristic 0, Bertini for arbitrary base-point-free systems needs characteristic 0, Serre duality needs a dualizing sheaf, semicontinuity needs flatness and properness.
• Keep the example in the paper. A worked instance with the numbers is the fastest way for a referee to believe a dimension count, and the fastest way for you to catch an off-by-one.
• Describe computations reproducibly: the software, the field, the exact generators, the monomial order if it matters, and what was computed. "A Macaulay2 computation shows" with no ideal written down is an assertion, not evidence.
• Separate what the computation proves from what the argument proves. "The computation establishes the claim for the member X_0; by upper semicontinuity of h^1 in the flat family X → B (Hartshorne III.12.8) and irreducibility of B it holds for a general member" is the sentence referees want to see.
• Never write "clearly" in front of a dimension count, a Jacobian rank, or an exactness claim. Those are the three places proofs break.

Catalogue of Standard Tricks

1. Incidence correspondence and a dimension count; then a Chern class or a Gröbner basis for the number.
2. Project from a point or a linear space to lower the degree or the dimension; keep track of what the projection does to the ideal (elimination) and to the singularities.
3. Take a general hyperplane section and induct on dimension; degree, Hilbert polynomial leading term, regularity and being Cohen-Macaulay all pass to a general hyperplane section.
4. Degenerate to a special member: a union of hyperplanes, a monomial ideal via a Gröbner degeneration, a reducible fibre in a pencil; then pass back by flatness, constancy of the Hilbert polynomial, or semicontinuity.
5. Use the Euler, normal, conormal and Koszul sequences before inventing a new one.
6. Compute the Zariski tangent space with k[ε]/(ε^2)-points; compare its dimension with the dimension of the space to test smoothness of a moduli or Hilbert scheme at a point.
7. Resolve a rational map by blowing up its base locus and work on the graph.
8. Use Stein factorization or Zariski's main theorem when a morphism has connected fibres or is quasi-finite and birational.
9. Exploit a group action: torus symmetry, the Fermat symmetry, a Galois action; fixed points and orbits often carry the whole answer.
10. Reduce mod p for a witness; go to characteristic 0 for Kodaira and generic smoothness; never mix the two in one argument without saying so.
11. Two closed subschemes, one containing the other with the same Hilbert polynomial, are equal; this settles many "the ideal is exactly" claims.
12. Pass to the normalization or the reduced structure, and record what changed: degree is stable, genus drops by δ, cohomology can change.
13. Before believing a statement, run it against the standard counterexamples: the line with two origins, the cuspidal cubic, the quadric cone with its non-Cartier ruling, Frobenius, Hironaka's non-projective threefold, the twisted cubic's three generators.
14. When stuck, compute the Hilbert function and the Betti table of the ideal; they show the degrees where something unexpected happens.

Checklist

• Hypotheses written out before the first computation; characteristic and closedness of the field decided.
• Smallest example computed by hand and by machine, and the computation described in enough detail to reproduce.
• Every dimension count backed by a specific incidence variety with its fibre dimensions justified.
• Every local claim made in an explicit chart with an explicit generating set of the ideal.
• Every cohomology group reached through a named short exact sequence and a vanishing theorem whose hypotheses were checked.
• "General" claims accompanied by the parameter space, its irreducibility and the openness of the condition.
• The finished argument tested against the example one last time.

Common Mistakes

• Starting with the general case and never computing an example.
• Counting conditions by counting coefficients, without the h^1 that measures dependence.
• Trusting one affine chart for a global count, as the Fermat cubic shows.
• Applying an open-condition argument to a closed condition, or on a reducible parameter space with a witness in only one component.
• Homogenizing without saturating, then reasoning about a component at infinity that is not there.
• Using generators of an ideal with the right zero set but the wrong ideal in a Jacobian criterion.
• Citing Kodaira vanishing or generic smoothness in positive characteristic, or Bertini for a non-very-ample system there.
• Writing "clearly" for the one step that needed the computation.

Limits

The method here is for problems that have a computable instance, which is most of classical and much of modern algebraic geometry, but not all of it. Statements about all varieties of a given type, moduli-theoretic questions with no explicit family, arithmetic statements over Q or Z where rational points rather than geometry are the issue, and anything in derived or stacky language do not yield to example-first attack in the same way, though the habit of finding a test case still helps. Gröbner-based computation runs out at a handful of variables of moderate degree, so an example that is too large to compute is not evidence either way, and a claim whose smallest honest instance is that large needs a different strategy: degenerate further, or find the structure that makes it tractable.