Let $X$ be a complex minimal surface of general type, id est $K_X$ is big and nef. It is well-known that $\displaystyle\int_X3c_2(X)-c_1(X)^2\geq0$, and the equality holds if and only if $X$ is uniformized by $\mathbb{B}\subset\mathbb{C}^2$ (the open ball). Ever in this case: $X$ does not contain neither rational smooth curves nor smooth curves of genus $1$.

In the opposite case, id est assuming that $\displaystyle\int_X3c_2(X)-c_1(X)^2>0$, there are examples of such $X$'s which do not contain any smooth rational curves.

Questions: let $X$ be a complex minimal surface of general type such that $\displaystyle\int_X3c_2(X)-c_1(X)^2>0$: does $X$ contain any smooth curves of genus $1$? Does the previous answer change if one assumes $\Omega^1_X$ nef but not ample and $K_X$ ample?

  • 1
    $\begingroup$ The Fano surface of lines $F(X)$ of a cubic threefold $X$ is a surface of general type. It contains an elliptic curve when $X$ has an Eckardt point. This happens exactly when $\Omega_X^1$ is not ample. $\endgroup$
    – AG learner
    Commented Dec 21, 2023 at 4:47

1 Answer 1


Take a product $X= C_1\times C_2$, where $C_i$ are smooth curves of genus greater than $1$. $X$ has general type, and is uniformized by a product of two disks. Also $X$ won't contain an elliptic curve because it maps trivially to each $C_i$.

  • $\begingroup$ Ok: $X$ is of general type with $K_X$ ample and $\Omega^1_X$ nef. However, why $\Omega^1_X$ is not ample? $\endgroup$ Commented Sep 1, 2023 at 12:50
  • 1
    $\begingroup$ Restrict $\Omega_X^1$ to $C_1\times pt$, then one factor is a trivial line bundle, so it can't be ample. $\endgroup$ Commented Sep 1, 2023 at 13:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.