Genus of non-complete intersections

Question

Suppose $X\subset \mathbb{P}_k^N$ a nonsingular curve is a complete intersection of hypersurfaces $F_1, \cdots, F_{N-1}$ (of degrees $d_1, \cdots, d_{N-1}$ resp). Then, we know that the canonical divisor on $X$ is $\mathcal{O}_X(d_1 + \cdots + d_{N-1} - n - 1)$. Hence, intersection theory on projective space gives a formula for the genus of $X$ entirely in terms of the various $d_i$. Specifically, $$2g - 2 = d_1\cdots d_{N-1} (d_1 + \cdots + d_{N-1} - N - 1)$$

$\textbf{Question:}$ Suppose $X$ is a nonsingular curve in $\mathbb{P}^N$, and $X = V(F_1, \cdots, F_m)$ is not a complete intersection. Can one get a similarly simple formula for the genus of $X$, perhaps entirely in terms of the degrees of the $F_i$? Is this too much to ask? I can't even simply describe the canonical divisor.

Answer 1

Perhaps Liaison Theory could be useful to you. Roughly speaking two subschemes $V,W\subset\mathbb{P}^n$ are linked if there is an arithmetically Cohen-Macaulay scheme $X$ such that $X = V\cup W$. In particular they may be linked by a complete intersection $X$. If $V$, and hence $W$, is locally Cohen-Macaulay then $$deg(V)+deg(W) = deg(X).$$ Furthermore one can express the arithmetic genus of $V$ in terms of the arithmetic genera of $W$ and $X$. This theory works for schemes of arbitrary codimension but it is particularly well behaved in codimension two.

Assume to have a curve $C\subset\mathbb{P}^n$ which is not a complete intersection but such that there exists an arithmetically Cohen-Macaulay curve $X\subset\mathbb{P}^n$ with minimal free resolution $$0\mapsto S(-d)\rightarrow E_r\rightarrow E_{r-1}\rightarrow ...\rightarrow E_1\rightarrow I_X\mapsto 0$$ such that $C\cup\Gamma = X$. That is $C$ and $\Gamma$ are linked via $X$. Let $g_1$ and $g_2$ be the arithmetic genera of $C$ and $\Gamma$. Then $$p_a(X)-1-deg(\Gamma)(d-n-1) = g_1-g_2.$$ On the other hand $$p_a(X) = \frac{1}{2}deg(X)(d-n-1)+1.$$ Combining these with $$deg(X)-2deg(\Gamma) = deg(C)-deg(\Gamma)$$ we get $$g_1-g_2 = \frac{1}{2}(d-n-1)(deg(C)-deg(\Gamma)).$$ In particular, if $\deg(C)=deg(\Gamma)$ then $g_1 = g_2$. Assume $X\subset\mathbb{P}^3$ is complete intersection of two surfaces of degree $d_1,d_2$. Then we get $$g_1-g_2 = \frac{1}{2}(d_1+d_2-4)(deg(C)-deg(\Gamma)).$$ An easy example: take a twisted cibic $C\subset\mathbb{P}^4$ and two quadrics $Q_1,Q_2$ containing $C$. Then $C\cup L = X =Q_1\cap Q_2$. The residual curve $L$ is a line. Furthermore we know that $p_a(X) = 1$. From the formula we get $g_1-0 = \frac{1}{2}(4-3-1)(3-1) = 0$.

Another example: Let $C\subset\mathbb{P}^3$ be a smooth curve of degree $5$. Then $C$ is contained in a unique quadric surface $Q$. Take a cubic surface $S$ containing $C$. Then $C\cup L = Q\cap S = X$. We have $deg(X) = 6$ and $p_a(X) = 4$. Therefore $g_1 = \frac{1}{2}(2+3-4)(5-1) = 2$ and $C$ is a curve of genus two. In particular it is hyperelliptic.

Some references:

• J. Migliore, Introduction to Liaison Theory and Deficiency Modules, Springer 1998.
• L. Gruson - C. Peskine, Genre des courbes de lespace projectif, Springer LNM, 687 (1977), pp. 3159.
• J. Migliore, Geometric invariants for liaison of space curves, Journal of Algebra Volume 99, Issue 2, 1 April 1986, Pages 548–572.
• C. Peskine - L. Szpiro, Liaison des variétés algébriques, I, Inv. Math. 26 (1974), 271-302